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Sandip Chakraborty, Bivas Mitra +Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, INDIA 721302 +Email: {debasreedas1994, sandipchkraborty, bivasmitra}@gmail.com +Abstract—Driving is a complex task carried out under the +influence of diverse spatial objects and their temporal inter- +actions. Therefore, a sudden fluctuation in driving behavior +can be due to either a lack of driving skill or the effect of +various on-road spatial factors such as pedestrian movements, +peer vehicles’ actions, etc. Therefore, understanding the context +behind a degraded driving behavior just-in-time is necessary +to ensure on-road safety. In this paper, we develop a system +called DriCon that exploits the information acquired from a +dashboard-mounted edge-device to understand the context in +terms of micro-events from a diverse set of on-road spatial +factors and in-vehicle driving maneuvers taken. DriCon uses the +live in-house testbed and the largest publicly available driving +dataset to generate human interpretable explanations against the +unexpected driving events. Also, it provides a better insight with +an improved similarity of 80% over 50 hours of driving data +than the existing driving behavior characterization techniques. +Index Terms—Driving behavior, spatial events, context analysis +I. INTRODUCTION +With an increase in the traffic population, we witnessed +a phenomenal rise in road accidents in the past few years. +According to the World Health Organization (WHO) [1], the +loss is not only limited to humans but affects the GDP of +the country as well. The officially reported road crashes are +inspected mostly based on the macro circumstances, such as +the vehicle’s speed, the road’s situation, etc. Close inspection +of those macro circumstances reveals a series of micro-events, +which are responsible for such fatalities. For example, suppose +a driver hit the road divider and faced an injury while driving +on a non-congested road. From the macro perspective, we +might presume it is due to the driver’s amateurish driving +skill or the vehicle’s high speed. But, it is also possible that +some unexpected obstacles (say, crossing pedestrians/animals) +arrived at that moment out of sight. The driver deviated from +his lane while decelerating to avoid colliding with them. +Therefore, recording these micro-events are crucial in iden- +tifying the reasoning behind such accidents. Such contextual +information, or micro-events, thus, can help various stakehold- +ers like car insurance or app-cab companies to analyze the on- +road driving behavior of their drivers. Interestingly, an app-cab +company can penalize or incentivize their drivers based on how +they handle such context and take counter-measures to avoid +accidents. +A naive solution to extract the context information is +to analyze the traffic videos. Notably, CCTV cameras [2] +capture only static snapshots of the events concerning the +Live Deployment Setup +(a) +Output of DriCon +(b) +The Preceding +Vehicle Suddenly +Braked and +The Ego Vehicle +Abruptly Stopped +and +Faced Severe +Jerkiness +Fig. 1: DriCon: Hardware components and a running instance +when a vehicle faced severe jerks +moving vehicles. Existing works [2], [3] use dash-cam videos +along with IMU sensor data for manual or partly automated +investigation of the accident. Note that, human intervention is +error-prone and labor-intensive with higher costs. The situation +gets further complicated when multiple events are responsible +for the accident. For instance, suppose the preceding vehicle +suddenly brakes to avoid collision with a pedestrian or at +a run-yellow traffic signal. Consequently, the ego vehicle +has to decelerate abruptly, resulting in a two-step chain of +responsible events for the unexpected stop. Thus, identifying +spatiotemporal interactions among traffic objects are crucial in +characterizing the root cause behind such incidents. +Importantly, understanding the contexts behind the degraded +driving behavior on the fly is not trivial and poses multi- +ple challenges. First, this involves continuous monitoring of +the driving behavior of the driver as well as an exhaustive +knowledge of various on-road spatial micro-events. Expensive +vehicles use LiDAR, Radar etc., to sense the driver and +the environment [4], [5]; however, app-cab companies are +resistant to invest in such high-end vehicles due to low-profit +margin. Second, depending on the driving maneuvers taken, +temporally interlinking the micro-events based on the vehicle’s +interaction with on-road spatial objects is a significant research +challenge. For example, if adverse snowy weather is observed +on one day, its effect on traffic movements may last till the +next day. In contrast, reckless driving would impact only a few +other vehicles around and will not be temporally significant +after a few minutes. Such temporal impacts of an event +would vary depending on the type and space of the event. +Third, spatial positions of the surrounding objects impact the +driving maneuver. Precisely, along with temporal dependency, +the distance between the ego vehicle and the surrounding + +objects plays a vital role. For example, a far-sighted pedestrian +might cross the road at high speed, keeping a safe distance, +but it is fatal if the distance to the vehicle is low. Existing +literature [6], [7] have attempted to identify risky driving, +e.g., vehicle-pedestrian interaction, through IMU and video +analysis; however, they fail to capture such temporal scaling +or the spatial dependency among surrounding objects. Fourth, +identifying the context in real-time over an edge-device (such +as a dashcam) is essential for providing a just-in-time feed- +back. But, deploying such a system for context characterization +and analysis from multi-modal data over resource-constrained +edge-device is not straightforward. +To address these challenges, we propose DriCon that +develops +a +smart +dash-cam +mounted +on +the +vehicle’s +dashboard to characterize the micro-events to provide just-in- +time contextual feedback to the driver and other stakeholders +(like the cab companies). It senses the maneuvers taken by +the ego vehicle through IMU and GPS sensors. In addition, a +front camera mounted on the device itself, is used to analyze +the relationship between various on-road micro-events and the +driving maneuvers taken. This facilitates the system to run in +each vehicle in a silo and makes it low-cost and lightweight. +Fig. 1 shows a snapshot of the hardware components of +our system mounted on a vehicle, and an example scenario +where DriCon generates a live contextual explanation behind +a sudden jerk observed in the vehicle. In summary, our +contributions to this paper are as follows. +(1) Pilot Study to Motivate Micro-Event Characterization: +We perform a set of pilot studies over the Berkeley Deep +Drive (BDD) dataset [8], the largest public driving dataset +available on the Internet (as of January 16, 2023), to +investigate the variations in driving behavior depending on +various road types, time of the day, day of the week, etc., +and highlight the spatiotemporal micro-events causing abrupt +changes in driving maneuvers. +(2) Designing a Human Explainable Lightweight Causal +Model: The development of DriCon relies on the (i) IMU +& GPS data to infer the driving maneuvers, and (ii) object +detection model & perspective transformation [9] to detect the +surrounding objects and their actions to capture various spatial +micro-events. Subsequently, we identify the spatiotemporal +contexts whenever the driving behavior deteriorates during a +trip. Finally, we implement Self Organizing Maps (SOMs), +a lightweight but effective causal model to capture the +spatiotemporal dependency among features to learn the +context and generate human-interpretable explanations. +(3) +Deployment +on +the +Edge: We deploy the whole +architecture of DriCon on a Raspberry Pi 3 model, embedded +with a front camera, IMU and GPS sensors (Fig. 1). For this +purpose, we make both the IMU and visual processing of +the data lightweight and delay-intolerant. Following this, the +pre-trained model generates recommendations based on the +ongoing driving trip and makes it efficient to run live for +just-in-time causal inferences. +(4) Evaluating DriCon on a Live System Deployment and +with BDD Dataset: We evaluate DriCon on our live in-house +deployment, as well as on the BDD dataset [8] (over the +annotated data [10]), comprising 33 hours and 17 hours of +driving, respectively. We obtain on average 70% and 80% +similarity between the derived and the ground-truth causal +features, respectively, with top-3 and top-5 features returned +by the model, in correctly identifying the micro-events causing +a change in the driving behavior. Notably, in most cases, +we observe a good causal relationship (in terms of average +treatment effect) between the derived features and the observed +driving behavior. In addition, we perform different studies of +the resource consumption benchmarks on the edge-device to +get better insights into the proposed model. +II. RELATED WORK +Several works have been proposed in the literature on +understanding road traffic and its implications for road fa- +talities. Early research focused on traffic surveillance-based +techniques to prevent road accidents. For instance, National +Highway Traffic Safety Administration (NHTSA) [3] had +recorded statistics about fatal accident cases; TUAT [2] has +been collecting video records from taxis and drivers’ facial +images since 2005 to derive injury instances into several +classes along with driving behavior estimation. In India, the +source of information behind the causes of traffic injuries is the +local traffic police [11]. In contrast, works like [12], [13] learn +the crime type and aviator mobility pattern just-in-time from +street view images and raw trajectory streams, respectively. +Apart from harnessing videos and crowd-sourced information, +several works [14], [15] are done on abnormal driving behavior +detection by exploiting IMU and GPS data. To prevent fatal +accidents, authors [16]–[18] try to alert the drivers whenever +risky driving signature is observed, such as lane departure or +sudden slow-down indicating congestion. However, they have +not looked into the effect of neighboring vehicles or other +surrounding factors on various driving maneuvers. +Interaction among the ego vehicle and other obstacles, +such as pedestrians, adverse weather in complex city traffic, +often affects the vehicle’s motion, consequently affecting +the driving behavior. Existing studies [19] reveal that road +category, unsignalized crosswalks, and vehicle speed often +lead to a disagreement among pedestrians to cross the road, +leading to road fatalities. A more detailed study [20], [21] +focuses on causality analysis for autonomous driving, faces +infeasibility in real-time deployment. Moreover, they only use +a limited set of driving maneuvers, e.g., speed change only. +Particularly, causal inferencing is challenging due to high +variance in driving data and spurious correlation [22] between +traffic objects and maneuvers. The existing works limit their +study by considering only static road attributes or relying +on single or multi-modalities from a connected road network +system. Such methodologies will not be applicable for a +single vehicle in real-time deployment unless connected to the + +system. In contrast, leveraging multi-modalities from onboard +vehicle sensors can efficiently characterize the continuous +and dynamic contexts behind unexpected driving behavior +fluctuations. DriCon develops a system in this direction. +III. MOTIVATION +In an ideal scenario, two vehicles are likely to follow similar +maneuvers under the same driving environment; but this is +not the case in reality. Driving behavior varies according +to the driver’s unique skill set and is influenced by the +impact of various on-road events, such as the movement +of other heavy and light vehicles, movement of pedestrians, +road congestion, maneuvers taken by the preceding vehicle, +etc., which we call spatial micro-events or micro-events, in +short. In this section, we perform a set of pilot studies to +answer the following questions. (a) Does a driver’s driving +behavior exhibit spatiotemporal variations? (b) Do all +micro-events occurrences during a trip similarly impact the +driving behavior? (c) Does a sequence of inter-dependent +micro-events collectively influence the driving behavior? +Following this, we analyze the publicly-available open-source +driving dataset named Berkeley Deep Drive dataset (BDD) [8] +to answer these questions stating the impact of different micro- +events on the driving behavior. The dataset contains 100k +trips crowd-sourced by 10k voluntary drivers over 18 cities +across two nations – the USA and Israel. The dataset has been +annotated with a driving score on the Likert scale of 1 (worst +driving) to 5 (best driving) for each 5-second of driving trips. +A. Variation in Driving Behavior over Space and Time +We first check whether the on-road driving behavior exhibits +a spatiotemporal variation. For this purpose, we vary two +parameters – road type as the spatial parameter (say, “High- +way”, “City Street”, “Residential”), and time of the day as +the temporal one (say, “Daytime”, “Nighttime”, “Dawn/Dusk”) +in the BDD dataset. In this pilot study, we form 9 groups +with 30 trips each, in a total of 270, where the trips under +a group are randomly picked from the BDD dataset. We plot +the distribution of the driving scores over all the trips for each +group. From Fig. 2(a), it is evident that the score distribution +varies both (a) for a single type of road at different times of +the day, and (b) for different types of road at any given time of +the day (with p < 0.05 reflecting its statistical significance). +In the following, we investigate the role played by various +micro-events behind the variations in driving behavior. +B. Role of Spatial Micro-events +Next, we inspect whether various on-road micro-events, +which are characterized by the movements of other spatial +objects such as “cars”, “pedestrians”, “trucks”, “buses”, “mo- +torcycles”, “bicycles”, etc., impact a driver’s driving behavior +in the same way across different times of the day. We +perform this study by handpicking 30 trips along with their +annotated driving scores for both day and night time from +the BDD dataset. We compute the volume (say, count) of +spatial objects extracted using the existing object detection +algorithm [23] from the video captured during the trip and +take the average count of each object for a 5-second time +window. Thus, for both daytime and nighttime, we get two +time-series distributions, (a) the count of each on-road spatial +object captured over the trip video during each time window, +and (b) the annotated driving scores at those time windows. +Next, we compute the Spearman’s Correlation Coefficient +(SCC) among these two distributions for day time and night +time, respectively. From Fig. 2(b), we infer that mostly all the +on-road spatial objects adversely affect the driving behavior +(depicting a negative correlation). Cars and pedestrians affect +the driving score majorly during the daytime. Whereas, at +night time, trucks and buses, along with the cars, impact the +driving behavior because heavy vehicles such as trucks move +primarily during the nighttime. However, the effect of light +vehicles such as motorcycles and bicycles is insignificant due +to the dedicated lanes for their movements. This observation +is further extended to Fig. 2(c), where the same study is done +for weekdays vs. weekends. We extracted the day of the week +using already provided timestamps in the BDD dataset and +clubbed 30 trips from Monday to Friday for weekdays and 30 +trips from Saturday to Sunday for the weekend. From Fig. 2(c), +we observe that during the early days of the week, cars, +pedestrians, and trucks adversely affect the driving behavior, +whereas the impact is less during the weekend. Hence, we +conclude that different on-road objects exert diverse temporal +effects on the driving behavior. +C. Micro-events Contributing to Sudden Driving Maneuver: +Abrupt Stop as a Use-case +Finally, we explore whether multiple inter-dependent micro- +events can be responsible for a particular driving maneuver +that might degrade the driving behavior. For this purpose, +we choose abrupt stop as the maneuver, which we extract +from the GPS and the IMU data (the situations when a +stop creates a severe jerkiness [24]). We take 30 trips for +each scenario, including daytime, nighttime, weekdays, and +weekends. For each scenario, we extract the instances when +an abrupt stop is taken and record the corresponding micro- +events observed at those instances. Precisely, we extract the +presence/absence of the following micro-events: red traffic +signal, pedestrian movements, presence of heavy vehicles as +truck & bus, light vehicles as motorcycle & bicycle, and the +preceding vehicles’ braking action (as peer vehicle maneuver), +using well-established methodologies [10], [23]. We compute +the cumulative count of the presence of each micro-events and +the number of abrupt stops taken over all the trips for the +four scenarios mentioned above. From Fig. 2(d) and (e), we +observe that the red traffic signal, the peer vehicle maneuvers, +and heavy vehicles mostly cause an abrupt stop during the +nighttime and on weekdays. Therefore, we argue that multiple +on-road micro-events, such as the reckless movement of heavy +vehicles at night, force even an excellent driver to slam on the +brake and take an unsafe maneuver. + +Highway +City Street +Residential +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +Driving Score +(a) +Cars +Pedestrians +Truck +Bus +Motorcycle +Bicycle +−0.8 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +Spearman's Correlation +Day Time +Night Time +(b) +Cars +Pedestrians +Truck +Bus +Motorcycle +Bicycle +−0.6 +−0.4 +−0.2 +0.0 +0.2 +Spearman's Correlation +Week Day +Weekend +(c) +Red +Signal +Pedestrians +Heavy +Vehicles +Light +Vehicles +Peer +Vehicle +Action +0 +10 +20 +30 +40 +50 +60 +70 +%age of Occurences +Day Time +Night Time +(d) +Red +Signal +Pedestrians +Heavy +Vehicles +Light +Vehicles +Peer +Vehicle +Action +0 +10 +20 +30 +40 +50 +%age of Occurences +Week Day +Weekend +(e) +Fig. 2: (a) Variation of Driving Behavior with respect to Road Type and Time of the Day, (b)-(c) Impact of Spatial Micro-events +on the Driving Score at Different (i) Time of the Day, (ii) Day of the Week, (d)-(e) Contributing Factors Observed behind +Abrupt Stop at Different (i) Time of the Day, (ii) Day of the Week +IV. PROBLEM STATEMENT AND SYSTEM OVERVIEW +A. Problem Statement +Consider that FM denotes the set of driving maneuvers +and FS be the set of spatial micro-events. Fi be the set +of temporally-represented feature variables corresponding to +the driving maneuvers taken and on-road spatial micro-events +encountered during a trip i. Let Ri +T be the driving score +at time T during the trip i. We are interested in inspecting +the events occurred, representing the feature values Fi, when +|Ri +T − ˆRi +T −1| > ǫ (ǫ is a hyper-parameter, we set ǫ = 1), +reflecting the fluctuations in driving behavior. Here, ˆRi +T −1 = +⌈mean([Ri +1, Ri +T −1])⌉ represents the mean driving behavior till +T −1. The output of the system is a characterization of {Fi +M, +Fi +S}, as to whether a fluctuation in the driving behavior is due +to the driving maneuvers only (Fi +M) or forced by the spatially +causal micro-events (Fi +S). Finally, we target to generate the +explanations based on {Fi +M, Fi +S} to give feedback to the +stakeholders for further analysis of the driving profile. +B. Feature Selection +Leveraging the existing literature [24], we identified a +set of feature variables at timestamp T representing various +driving maneuvers FM of the ego vehicle. These features +are – Weaving (AW +T ), Swerving (AS +T ), Side-slipping (AL +T ), +Abrupt Stop (AQ +T ), Sharp Turns (AU +T ), and Severe Jerkiness +(AJ +T ). Similarly, we consider the following feature variables +corresponding to the spatial micro-events FS – Relative Speed +(ST ) and Distance (DT ) between the ego and the preceding +vehicle, preceding vehicle’s Braking Action (BT), volume +of the peer vehicles in front of the ego vehicle indicating +Congestion in the road (CT ), Pedestrian (PT ), and it’s speed +(QT ), Traffic light (LT ), Heavy vehicles: {Bus & Truck} +(HT ), Type of the Road (GT ), and Weather condition (WT ). +Note that, we empirically select these features based on the +existing literature and observations from the dataset; additional +features can also be incorporated in DriCon without losing its +generality. +We next broadly introduce our system architecture. DriCon +captures IMU, GPS, and video data from a dashcam (say, +an edge-device) and characterizes the context behind the +improved/degraded driving behavior on the fly. The system +comprises three components: (a) Data Preprocessing and +Feature Extraction, (b) Detection of Improved/Degraded +Driving Behavior, and (c) Identification of Possible Context +(see Fig. 3). +4 +Pedestrian Crossed and +The Ego Vehicle Abruptly +Stopped and Faced Severe +Jerkiness. +Inferred Features: Crossing +Pedestrians, Severe +Jerkiness, Abrupt Stops +INPUT: IMU, GPS & Video ++ +Driving Maneuvers +Spatial Micro-events +2 +4 +4 +Detect Change +in Driving +Behavior +Capture Time- +Series Dependency +Among Features +Output: Generated Explanations +(a) +(b) +(c) +(d) +(e) +Data Preprocessing and Feature Extraction +Identification of Possible Context +Model Output +Model +Construction +Fig. 3: DriCon System Flow and Modeling Pipeline +C. Data Preprocessing and Feature Extraction +The collected IMU and GPS sensor data are prone to +noise due to the earth’s gravitational force, signal attenuation, +and atmospheric interference. Hence, we implement a low- +pass filter to eliminate such noises from IMU and GPS to +compute inertial features for the extraction of the driving +maneuvers (FM). Next, we preprocess the video data before +extracting on-road spatial micro-events and their actions (FS). +We up/downsample the acquired videos to a resolution of +960 × 540p, preserving the signal-to-noise ratio above 20 dB. +1) Driving Maneuvers - FM: In order to generate the +features corresponding to different driving maneuvers (FM), +we extract the instances of Weaving (AW +T ), Swerving (AS +T ), +Side-slipping (AL +T ), Abrupt Stop (AQ +T ), Sharp Turns (AU +T ), +and Severe Jerkiness (AJ +T ) from the IMU data using standard +accelerometry analysis [10], [24]. +2) Spatial Micro-events - FS: Next, we implement the +state-of-the-art video data-based object detection algorithms +and further fine-tune them based on our requirements, as +developing vision-based algorithms is beyond the scope of our +work. We leverage the YOLO-V3 [23] algorithm trained on +the COCO dataset [25] to detect a subset of traffic objects such +as Pedestrians, Cars, Buses, Trucks, and Traffic Lights (de- +picted as FS). Next, we estimate the influence of pedestrians’ +interactions, the presence of heavy vehicles (buses & trucks), +traffic light signal transitions (red, yellow & green), and the +cars on the driving behavior of the ego vehicle. Next, we +discard the detected objects which depict a confidence score + +troficintao.50o6 +treffieliehtzo5 +cor. +0.90r0.680 +trotmcliehtr0.27Daytime +Nightime +Dawn/Dusk< 50% and bounding boxes of area < 10k, capturing the fact +that the far-sighted traffic objects around the ego vehicle exert +marginal impact compared to the near-sighted ones. Addition- +ally, the traffic objects in the mid-way of the road, broadly +visible from the driver’s dashboard, will be of more influence +than the left or right lanes, as the ego vehicle will follow +them immediately. Thus, we divide each of the frames into +0.2:0.6:0.2 ratio along the horizontal axis, as left:middle:right +lanes. Therefore, we keep the Pedestrians PT , Cars, Heavy +Vehicles as {Buses & Trucks} HT , which have bounding +box co-ordinates within the middle lane boundary, and Traffic +Light Signal Transitions LT (Red, Yellow & Green) without +the lane information as traffic lights are often positioned on +the left and right lanes. Since our pilot study demonstrated +that the pedestrians and peer vehicles’ action significantly +impact the driving maneuvers of the ego vehicle, (a) we extract +the Pedestrian Speed (QT ), as well as identify the crossing +pedestrians in the mid-way, and (b) we compute the preceding +vehicle’s Braking Action (BT ), and Congestion (CT), as +well as detect the Relative Speed (ST ) and Distance (DT ) +variation among the ego and the preceding vehicle. We apply +perspective transformation and deep learning methods [9], [26] +to infer the above. Finally, the above pipeline runs on each +frame where the video is re-sampled to 15 frames-per-second. +D. Detection of Driving Behavior Fluctuations +The crux of DriCon is to capture the temporal dependency +of various driving maneuvers and spatial micro-events when +a change in the driving behavior is observed during the trip. +For a run-time annotation of the driving behavior, we use an +existing study [10] that provides a driving behavior score on +the Likert scale [1 − 5] by analyzing driving maneuvers and +other surrounding factors. We divide the trip into continuous +non-overlapping time windows of size δ and compute the +driving score at the end of every window U (denoted as RP +U ), +using the feature values captured during that window [10]. +To quantitatively monitor whether there is a change in the +driving behavior during a window U, we compare RP +U and +ˆRP +U = +1 +U−1 +U−1 +� +i=1 +RP +i (mean driving score during previous U−1 +windows). Suppose this difference is significant (greater than +some predefined threshold ǫ). In that case, DriCon proceeds +towards analyzing the temporal dependency among the feature +vectors at different time windows to understand the reason +behind this difference. +E. Identification of Possible Context +In the final module, we use the feature vectors at different +windows to build the model that identifies which features +(FGEN) are responsible for the change in driving behavior +during the window U. The model reactively seeks explanations +behind such fluctuations by analyzing the effect of the micro- +events that occurred over the past windows [1, · · · , (U − 1)] +and the present window U. Finally, natural language-based +human interpretable explanations are generated and fed back +to the stakeholders for further analysis. +V. MODEL DEVELOPMENT +To develop the core model for DriCon, we leverage the +already extracted features F ∈ {FM +� FS} (details in §IV-C) +to capture the temporal dependency of the past as well as the +present events. In addition, DriCon derives the explanation be- +hind the detected events through explanatory features FGEN. +For this purpose, we need a self-explanatory model that +can capture the spatiotemporal dependency among different +driving maneuvers and micro-events associated with the on- +road driving behavior. We choose a Self Organizing Map +(SOM) [27] for constructing the model that can exploit such +spatiotemporal dependencies with minimum data availability. +The major limitation of the classical deep learning models +(such as CNN or RNN) stems from the fact that, (i) deep +networks consume heavy resources (say, memory), as well as +suffer from huge data dependency, and (ii) they act as a black +box, hence fail to generate human interpretable explanations +behind certain predictions [28]. On the other hand, SOM is +able to characterize the micro-events in runtime using feature +variability and unlabelled data. +Neighboring Radius +F1 +F2 +F3 +FU +Input +Layer +Learning Phase +Feature +Input +Converge +Final Map +Code Book +BMU +Weight +(a) +(b) +(c) +No Change +Change +Fig. 4: Working Principle of SOM +A. Inferring Explanatory Features using SOM +The key idea behind obtaining the explanatory features is +first to discover the spatiotemporal feature dependency. In +DriCon, we derive so using Kohonen’s Self Organizing Map +(see Fig. 4), as it is an unsupervised ANN-based technique +leveraging competitive learning methods. Since DriCon runs +on an edge-device, we employ a minimal number of model +parameters to expedite the processing without compromising +the performance. Precisely, we implement the codebook with +147 neurons, spread out over a two-dimensional array of +size 7 × 21 (where 7 is a hyperparameter depending on the +maximum influence of the past windows during a trip, 21 cor- +responds to the number of features in the feature space). These +neurons are initialized with a random weight (see Fig. 4(a)), +where the weight vector has the same length (of 21) as the +feature vector. Next, we represent each trip with a 2D grid of +size 8 × 21 (considering 8 consecutive windows in a trip) to +capture the influence of the past windows [1, · · · , (U −1)] and +the present window U. In principle, the inherent topological +ordering of SOM groups the similar feature space (in windows +[1, · · · , (U −1)]) into a single group, when there is no change +in the driving behavior. On the contrary, the dissimilar ones + +(say, during the window U), when there exists a change in +the driving behavior, are mapped into a different group, as +depicted in Fig. 4(b,c). +For instance, suppose on a trip, the ego vehicle abruptly +stops due to the preceding vehicle’s braking action following +a sudden change in the traffic signal. Hence the feature space +in window [1, · · · , (U − 1)] exhibits a similar signature (until +the abrupt stop occurs), and subsequently gets mapped to a +single neuron. However, during the abrupt stop, there will be +changes in the feature space (say, maneuvers and other spatial +events). These changes in the feature space will get it assigned +to a different neuron and settle the other neurons’ weight +automatically depending on the changes in the feature space +between the windows [1, · · · , (U −1)] and the window U. This +procedure allows SOM to harness the temporal dependency +among spatial events in an unsupervised mode, without using +the driving score explicitly. +1) Model Training: The input trip data is represented in +the 2D grid format for learning the best-matched neuron, +optimizing the Euclidean distance between the feature space +and weight vector of the corresponding neuron. To ensure the +best-fitting, the best-matched neuron tries to learn the weight +vector of the feature space at most. Also, the neurons in the +neighborhood try to tune their weights as nearest as possible +compared to the best-matched neuron. We train this model +for 500 epochs, where each neuron gets mapped with the +best matching trip instances and converges to their coordinate +position in the codebook. We implement the Bubble neigh- +borhood function [29] to update the neighborhood neurons’ +weights until the neighborhood radius converges to ≈ 0. We +ensure that both the distance and neighborhood functions are +computationally faster for accurate learning accelerating the +convergence. Upon completing the total number of epochs, we +obtain the converged codebook called the Map, where each trip +instance gets assigned to the best matching neuron called the +Best Matching Unit (BMU). The weight vector corresponding +to the BMU’s coordinate reveals the explanatory features +FGEN. +2) Model Execution: We leverage the constructed Map for +the runtime inference. First, we conduct the feature processing +of the current ongoing trip (following §IV-C), and in parallel, +the extracted feature space is fed as input to the constructed +Map. Eventually, we obtain the BMU’s coordinate and extract +its corresponding weight vector and the feature encoding for +the given trip instance. From the weight vector, we extract +the top-k weights and their corresponding feature names +(say, weather type) and their encoded values (say, weather +type: rainy). Finally, we populate them in FGEN (called +the Generative micro-events) for further generation of human +interpretable explanation. +B. Generating Textual Explanation +DriCon aims to generate the explanations in textual format +utilizing the output features FGEN for better readability and +human interpretation. As the features f ∈ FGEN are already +associated with some keywords (say, severe jerkiness), we +need to generate them in a sentential form, keeping the features +as “action” or “subject” depending on whether f ∈ FM +or f ∈ FS, respectively. For instance, if the feature is an +action, we assign the ego vehicle as the subject, replace the +corresponding output feature f with its describing keyword, +and finally concatenate them to obtain the sentential form. +For example, in case of severe jerkiness, the constructed +sentence becomes, “the ego vehicle severe jerks”. However, if +the output feature f represents a subject, then many possible +sentences can be generated out of one subject. Thus, we +mine several traffic guidelines [30] and compute the cosine +similarity among the features and existing guidelines using TF- +IDF vectorizer. Upon extracting the most relevant guidelines, +we fetch the object associated with the sentence and construct +a single sentence for each output feature (e.g., “pedestrian +crossing” → “pedestrian crossing the intersection”). Next, for +all the generated sentences, the describing keywords corre- +sponding to each feature are converted to an adjective or +adverb using Glove [31] for better structuring of the sentences +(say, “the ego vehicle severe jerks” → “the ego vehicle severely +jerks”). Finally, each sentence is concatenated using the “and” +conjunction, and repetitive subjects are replaced using their +pronoun form using string manipulation to generate the whole +explanation, as depicted in Fig. 3(e). +VI. PERFORMANCE EVALUATION +This section gives the details of DriCon implemented over +a live setup as well as over the BDD dataset. We report the +performance of the SOM model and compare it against a +well-established baseline. Additionally, we show how well our +system has generated the textual explanations along with a +sensitivity analysis to distinguish how error-prone DriCon is. +We start with the experimental setup details as follows. +A. Experimental Setup +DriCon is implemented over a Raspberry Pi 3 Model B +microprocessor kit operating Raspbian OS with Linux kernel +version 5.15.65 − v7+ along with 1 GB primary memory +and ARMv7 processor. We primarily utilize the IMU, the +GPS, and the video data captured through the front camera +(facing towards the front windscreen) as different modalities. +For this purpose, we embed one MPU−9250 IMU sensor, +one u-blox NEO−6M GPS module, and one Logitech USB +camera over the Raspberry Pi board, as depicted in Fig. 1(a). +We deployed DriCon over three different types of vehicles +(e.g., SUV, Sedan, & Hatchback). We hired 6 different drivers +in the age group of [20 − 45] who regularly drive in practice. +Therefore, our whole experimentation ran for more than two +months over three cities, resulting in approximately 33 hours +of driving over 1000 km distance. The drivers drove freely +without any specific instructions given, with each trip varying +from approximately 20 minutes to 2 hours. In addition, each +driver drove over five different types of roads (city street, +highway, residential, parking & campus road) at three different +times of the day (day, dusk & night). We evaluate DriCon by +analyzing how well our proposed model extracts the generative + +micro-events FGEN (see §V-B). For implementing DriCon, +we consider δ = 5 seconds, ǫ = 1. The impact of other hy- +perparameters and resource consumption have been discussed +later during the analysis. We next discuss the ground-truth +annotation procedure used for the evaluation of DriCon. +B. Annotating Micro-events +We launched an annotation drive by floating a Google +form among a set of recruited annotators, where they had +to watch a video of at most 10 seconds and choose the +top-3 most influential factors impacting the driving behavior. +We do this annotation over the in-house data (video data +collected during the live experiments) and the videos over the +BDD dataset. For each video from both the datasets given +in the form, we showed only the clipped portion where the +score fluctuations had occurred. Next, out of the total 15 +factors (including driving maneuvers and spatial micro-events) +given in a list, they were instructed to choose the top-3 most +influential factors responsible for the poor driving behavior +based on their visual perception. Besides, we also provided +the model-generated sentences (§V-B) and asked how relevant +and well-structured the sentences are (on a scale of [1−5]) for +explaining the change in the driving behavior. The annotators +also had the option to write their own explanation if they +perceived a better reason behind the driving behavior change. +As the number of trips is quite large, we need to design a set +of Google forms (sample form1), each containing at most 20 +videos to ensure the least cognitive load on the annotators. We +also collected annotators’ demographic information such as +age, gender, city, etc. We find that most participants (> 67%) +had prior driving skills. At least three independent annotators +had annotated each instance. Upon receiving the annotated +factors, we need to find the agreement among the annotators +to ensure the received ground truth is unbiased and non- +random. As standard inter-annotator agreement policies (say, +Cohen’s kappa index) work on quantitative analysis or one-to- +one mapping, we cannot apply such metrics. Thus, we use the +majority voting technique where each listed factor is assigned +a percentage, signifying how many times the annotators choose +that factor. Each factor having a vote of at least 60% is kept in +FGT . We observe the minimum and the maximum cardinality +of FGT are 3 and 5, respectively. This also indicates that +the annotators agreed on selecting the factors that influenced +the driving behavior. FGT contains the annotated micro-events +against which FGEN is evaluated. +C. Performance Metric +We use the Dice Similarity Coefficient score [32] (N) +which computes the similarity between FGT and FGEN as fol- +lows: N = 2×|FGT ∩FGEN| +|FGT |+|FGEN| . We report the mean N across all +the trips to measure the accuracy of DriCon. Next, we also use +Average Treatment Effect [33] (ATE) to report comparatively +higher causal features out of the model identified features. +Finally, we define Percentage of Error as follows. First, we +1https://forms.gle/97N6uk4ujRaZSWbj8 (Accessed: January 16, 2023) +Top-3 +Top-5 +30 +50 +70 +90 +Dice Coefficient (in %) +(a) +Top-3 +Top-5 +10 +30 +50 +70 +90 +Dice Coefficient (in %) +(b) +Fig. 5: (a) Dice Coefficient Similarity (in %) between Human +Annotated and Model Generated Features (b) Ablation Study +compute the set-difference as {FGT \FGEN}, and extract the +corresponding feature category (say, FM, FS). Once we get +the count of each feature category, we compute its percentage +out of the total trips as the Percentage of Error. +D. Baseline Implementation +As a baseline for extracting FGEN, we implement a super- +vised rule-based Random Forest (RF) algorithm with 20 deci- +sion trees where each tree is expanded to an unlimited depth +over the training data. We optimize the labels RP +U with the +intuition that features will contribute differently to each of the +predicted scores. Although the RF-based model has a feature +importance score signifying the contribution of each feature +in constructing the model, we need to have an explanation of +how each feature contributes to predicting the driving scores +on a trip instance basis. Therefore, we use LIME [34] in the +background of the RF model for generating the explanatory +features. As LIME is a model-agnostic method, it tries to map +the relationship between the input features and output scores +by tweaking the feature values. Thus, it explains the range +of values and probability for each feature that contributes +to predicting the score. From the generated explanation, we +extract the contributing features FGEN along with their values +for further generation of textual explanation. This pipeline is +executed in a similar manner as described in §VI-A. +E. Accuracy of Characterized Context +We present the accuracy of DriCon using the SOM and +RF+LIME model over the in-house dataset using Dice Coeffi- +cient Similarity N. We extract the top-k features from FGEN +where k ∈ {3, 5} and compute N between the two sets of fea- +tures – FGEN and FGT with top-k. Fig. 5(a) shows the result. +For top-3, we get 69% & 40% similarity on average with SOM +and RF+LIME, respectively. Whereas for top-5, we observe +79% & 48% similarity on average with SOM and RF+LIME, +respectively. As the in-house dataset has more complex micro- +events, the slight performance drop over the in-house dataset +using the top-3 features is tolerable. Intuitively, the model can +capture more diversity as perceived by the human annotators; +therefore, the similarity improves as we move from k = 3 +to k = 5. However, as the RF+LIME considers each time +instance of a trip independently, its performance degrades. +It captures the dominant features responsible for the driving +behavior change within the current time window, contrary to +inspecting past time windows’ impact. + +DriCon +DriCon-man +DriCon-spat.SOM +RF WI LIMETABLE I: Similarity Measure among Human Annotated vs. Model Generated Output +Instance# +Human Annotated FGT +Model Generated FGEN +Similarity N(%) +ATE +1 +Poor Weather Conditions (Heavy Rainfall, Fog, etc.), Swerving, +Congestion, Overtaking, Taking Abrupt Stop +Congestion, Preceding Vehicle Braking, +Weaving, Abrupt Stop, Severe Jerkiness +40% +1.96 +2 +Sideslip, Taking Abrupt Stop, Traffic Lights: Red +Traffic Lights: Red, Congestion, Abrupt Stop +66.67% +2.5 +3 +Crossing Pedestrian, High Speed Variation among Cars, Weaving +Severe Jerkiness, Crossing Pedestrian, Weaving +66.67% +1.35 +To have a glimpse, we present the explanatory features +(FGEN) vs. human-annotated ones (FGT) in Table I for a +sample of three test instances where the similarity (Dice coef- +ficient) is comparatively lower. Interestingly, when there is a +mismatch, we observe that the corresponding features from the +model-generated and human-annotated ones are conceptually +related for most of the time. Additionally, a positive high +mean ATE value for the model-generated mismatched features +signifies that the model perceived those features as more causal +than normal human perception. It can be noted that an ATE +value ≥ 1 indicates high causal relationships between the +features and the corresponding effect (changes in the driving +behavior). For example, in test instance #2, the mismatched +features are Sideslip (for human generated) and Congestion +(for model generated), where Congestion was relatively more +causal, affecting the change in the driving behavior. By manu- +ally analyzing this instance and interviewing the corresponding +driver, we found that he indeed made a minor sideslip on a +congested road. Indeed, the driver was not very comfortable +in driving a manually-geared car on a congested road. +2 +3 +4 +5 +Fig. 6: Generated Map from SOM for a 7×7 Network (Scaled +Down) +F. Ablation Study +Next, we understand the importance of different feature +categories corresponding to the driving maneuvers and on-road +spatial events, as described in §IV-A, on the overall perfor- +mance of DriCon. To study the impact of driving maneuvers +and spatial features, we implement SOM, excluding each of +the above feature classes one at a time, and evaluate N to +inspect the importance of each. The two variants other than +DriCon are constructed in the following way. (a) DriCon- +man.: Here, we exclude the driving maneuvers FM and keep +FS only. (b) DriCon-spat.: Next, we exclude the spatial +features FS and keep FM only. We evaluate these two variants +over both top-3 and top-5 generated features, along with +DriCon containing all the features, as depicted in Fig. 5(b). +On excluding the driving maneuvers and spatial features, +performance drops to 45% and 31%, respectively, for top-5 +features. This drastic drop signifies the crucial importance of +spatial features, as these are the frequently changing features +responsible for fluctuating driving behavior. +G. Model Insight +To understand how the spatiotemporal dependency among +different features corresponding to the driving maneuvers and +various on-road spatial micro-events are derived, we use 49 +neurons spread over a 7 × 7 two-dimensional array (a smaller +variant of the SOM network originally used to develop the +model, as the original model having 147 neurons is difficult to +visualize), fitted over 200 trips. This instance produces a Map +as depicted in Fig. 6, where all the given trips are assigned +to each of the neurons. The scores RP +U are used only for +visual depiction purpose of how the trips are located on the +Map. Each trip captures the change in the driving behavior +using the feature variation. The neurons with multi-color are +of more importance than the mono-color, as in those, the +score fluctuations are most observed. During a stand-alone +trip, the features corresponding to each instance of the trip +will have a similar value until there is a change in the driving +behavior, thus getting assigned to the same neuron (mono- +color). However, the difference in the driving behavior induces +distinct feature values than the previous instances; thus, it gets +assigned to a different neuron in the Map. The neurons having +multi-color, as depicted in Fig. 6, map the trip instances where +a sudden change of driving behavior has occurred. +H. Dissecting DriCon +We next benchmark the resource consumption behavior of +DriCon, followed by an analysis of the model’s significance +and sensitivity. +1) Edge-device Resource Consumption: We benchmark +the CPU & memory usage, processing time, temperature rise, +and energy consumption over two cases: when (a) the device +is idle, & (b) DriCon is running. From Fig. 7(a), we observe +that in idle mode, on average, 2% of CPU (using “top” +command) is used. In contrary, running DriCon acquires at +most 10% of the processor, which is acceptable. However, +the memory usage is a bit high (≈ 500MB) mainly due to +video processing overhead as depicted in Fig. 7(b). Next, +we show the required processing time starting from data +acquisition to output generation on a number of trip basis. + +Idle +Live +2 +4 +6 +8 +10 +CPU Consumption (in %) +(a) +Idle +Live +100 +200 +300 +400 +500 +600 +Memory Consumption (in MB) +(b) +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +Processing Time (in mins) +0 +5 +10 +15 +20 +25 +#Trips +(c) +Idle +Live +43 +46 +49 +52 +55 +58 +61 +Temperature Rise (in ∘ C) +(d) +0 +12 +24 +36 +48 +60 +Time (in mins) +0 +5 +10 +15 +20 +25 +Energy Consumption (in W-Hr) +Nexar +Live +Idle +(e) +Fig. 7: Resource Consumption over the Edge-device (a) CPU Usage (b) Memory Usage (c) Histogram of Processing Time +w.r.t., #Trips (d) Temperature Rise, (e) Energy Consumed +Relevance +Well-Structured +3 +4 +5 +Annotation Score +(a) +Spatial +Maneuver +0 +5 +10 +15 +%age of Error +(b) +Top-3 +Top-5 +30 +50 +70 +90 +Dice Coefficient (in %) +(c) +Fig. 8: (a) Significance of DriCon (b) Sensitivity Analysis of +DriCon (c) Performance on BDD Dataset +DriCon generates the output within ≈ 3 minutes only for +majority of the trips, further validating shorter response time +(see Fig. 7(c)). To further delve deeper, we also log the +temperature hike (from “vcgencmd measure temp” command) +and total energy consumption using Monsoon High Voltage +Power Monitor [35] while running DriCon. From Fig. 7(d) & +(e), we observe that the temperature hiked at most to 59°C, +while on average, 13 Watt-hour energy is consumed, which is +nominal for any live system. To benchmark DriCon, we have +also measured the energy consumption of the Nexar dashcam, +which consumes 22 Watt-hour on an average, while capturing +very few driving maneuvers (say, hard brake) without any +context. This further justifies that DriCon never exhausts the +resources on the edge-device and is can accurately detect the +micro-events precisely. +2) Significance of Generated Explanation: +Next, we +check how significant our generated explanations are. As +reported in §VI-B, we plot the distribution of annotated +scores (given by the recruited annotators) for the two fields – +“Relevance” and “Well-Structured”. “Relevance” signifies the +generated explanation’s applicability in explaining unexpected +events. In contrast, “Well-structured” indicates how well inter- +pretative the generated sentences are as per human cognition. +Fig. 8(a) depicts a median value of 5 and 4 for “Relevance” +and “Well-Structured”, respectively, which further justifies +the credibility of DriCon. We also compute the similarity +between the human-annotated and model-generated sentences +and obtain a minimum, maximum, and mean similarity value +as 51.33%, 85.5% & 70.57%, respectively, using the TF-IDF +vectorizer. Thus DriCon resembles human cognition level up +to an indistinguishable level (between a human and model) of +auto-generating a contextual explanation, which further shows +its applicability to give feedback to the stakeholders for their +decision-making procedure. +3) Sensitivity of DriCon: Finally, we inspect the micro- +events that DriCon fails to capture. Because, apart from a +model’s efficiency, we must also look into its deficiency to +analyze how much that might affect the overall performance. +Especially, this is important in the case where stakeholders +are boosting/penalizing the driver’s profile. As depicted in +Fig. 8(b), incompetence to capture both the spatial and ma- +neuvers is low. Although this might lead to degraded model +performance, as studied in §VI-F; driving maneuvers (FM) +do not contribute superiorly to model performance due to the +inter-dependency on spatial features (FS). But for FS, the +Percentage of Error is still ≤ 13%, making the system less +sensitive into generating error-prone contextual explanations. +I. Offline Performance +Finally, we report the accuracy of our system over the BDD +dataset comprising 17 hours of driving data over 1.5k trips +using N. As depicted in Fig. 8(c), DriCon performs quite +well on pre-recorded data, with N = {71%, 84%}, for top- +3 and top-5 features. We observe that SOM can identify the +micro-events in a better way for offline analysis with a public +dataset. However, as running the system live is essential for a +realistic driving environment other than offline analysis, this +much of slight accuracy drop can be endured. +VII. CONCLUSION +This paper developed an intelligent system on the edge- +device called DriCon leveraging multi-modalities to detect the +micro-events responsible for unexpected fluctuations in driving +behavior. The human-interpretable explanations generated by +DriCon show their relevance and credibility in identifying +such context. Further, the spatiotemporal dependency among +various features is inspected in an unsupervised manner to +capture a diverse set of driving scenarios. Additionally, the +resource-friendly deployment over a live testbed further vali- +dates DriCon. Although our study captures the context where +each feature’s contribution is taken independently, inter-feature +dependency is not captured explicitly. For instance, say, a +driver suddenly weaves while taking a turn to avoid colliding +with a crossing pedestrian, making the following vehicle’s +driver slam the brake. Here, the first driver’s action is due +to the crossing pedestrian, which in turn impacts the second +driver’s action. The analysis of such complex and collective +interactions among the vehicles needs a more sophisticated + +SOM +RF W/ LIMEsystem, possibly a different modality that can connect the +inter-vehicle interactions. However, DriCon provides a simple, +in-the-silo solution that can be independently deployed over +vehicles with a dashboard-mounted edge-device or dashcam. +REFERENCES +[1] “Road +traffic +injuries, +by +world +health +organization +(who),” +https://www.who.int/news-room/fact-sheets/detail/road-traffic-injuries, +2022, (Online Accessed: January 16, 2023). +[2] “Institute +of +engineering +tokyo +university +of +agriculture +and +technology +(tuat). +smart +mobility +research +center +- +research.” +https://web.tuat.ac.jp/∼smrc/research.html, +2017, +(Online +Accessed: +January 16, 2023). +[3] N. H. T. S. A. 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Platel et al., “Evaluating +white matter lesion segmentations with refined sørensen-dice analysis,” +Scientific reports, vol. 10, no. 1, pp. 1–19, 2020. +[33] D. B. Rubin, “Estimating causal effects of treatments in randomized and +nonrandomized studies.” Journal of educational Psychology, vol. 66, +no. 5, p. 688, 1974. +[34] M. T. Ribeiro, S. Singh, and C. Guestrin, ““why should i trust you?” +explaining the predictions of any classifier,” in Proceedings of the 22nd +ACM SIGKDD, 2016, pp. 1135–1144. +[35] “Monsoon +high +voltage +power +monitor,” +https://www.msoon.com/online-store/High-Voltage-Power-Monitor-p90002590, +(Online Accessed: January 16, 2023). + diff --git a/2dE4T4oBgHgl3EQfzw3P/content/tmp_files/load_file.txt b/2dE4T4oBgHgl3EQfzw3P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bfedbf88925746f53a2ea29d2c8891a75fc8d5a6 --- /dev/null +++ b/2dE4T4oBgHgl3EQfzw3P/content/tmp_files/load_file.txt @@ -0,0 +1,688 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf,len=687 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='05277v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='HC] 12 Jan 2023 DriCon: On-device Just-in-Time Context Characterization for Unexpected Driving Events Debasree Das, Sandip Chakraborty, Bivas Mitra Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, INDIA 721302 Email: {debasreedas1994, sandipchkraborty, bivasmitra}@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='com Abstract—Driving is a complex task carried out under the influence of diverse spatial objects and their temporal inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Therefore, a sudden fluctuation in driving behavior can be due to either a lack of driving skill or the effect of various on-road spatial factors such as pedestrian movements, peer vehicles’ actions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Therefore, understanding the context behind a degraded driving behavior just-in-time is necessary to ensure on-road safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In this paper, we develop a system called DriCon that exploits the information acquired from a dashboard-mounted edge-device to understand the context in terms of micro-events from a diverse set of on-road spatial factors and in-vehicle driving maneuvers taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' DriCon uses the live in-house testbed and the largest publicly available driving dataset to generate human interpretable explanations against the unexpected driving events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Also, it provides a better insight with an improved similarity of 80% over 50 hours of driving data than the existing driving behavior characterization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Index Terms—Driving behavior, spatial events, context analysis I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' INTRODUCTION With an increase in the traffic population, we witnessed a phenomenal rise in road accidents in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' According to the World Health Organization (WHO) [1], the loss is not only limited to humans but affects the GDP of the country as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The officially reported road crashes are inspected mostly based on the macro circumstances, such as the vehicle’s speed, the road’s situation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Close inspection of those macro circumstances reveals a series of micro-events, which are responsible for such fatalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For example, suppose a driver hit the road divider and faced an injury while driving on a non-congested road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From the macro perspective, we might presume it is due to the driver’s amateurish driving skill or the vehicle’s high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' But, it is also possible that some unexpected obstacles (say, crossing pedestrians/animals) arrived at that moment out of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The driver deviated from his lane while decelerating to avoid colliding with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Therefore, recording these micro-events are crucial in iden- tifying the reasoning behind such accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Such contextual information, or micro-events, thus, can help various stakehold- ers like car insurance or app-cab companies to analyze the on- road driving behavior of their drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Interestingly, an app-cab company can penalize or incentivize their drivers based on how they handle such context and take counter-measures to avoid accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' A naive solution to extract the context information is to analyze the traffic videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Notably, CCTV cameras [2] capture only static snapshots of the events concerning the Live Deployment Setup (a) Output of DriCon (b) The Preceding Vehicle Suddenly Braked and The Ego Vehicle Abruptly Stopped and Faced Severe Jerkiness Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 1: DriCon: Hardware components and a running instance when a vehicle faced severe jerks moving vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Existing works [2], [3] use dash-cam videos along with IMU sensor data for manual or partly automated investigation of the accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Note that, human intervention is error-prone and labor-intensive with higher costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The situation gets further complicated when multiple events are responsible for the accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For instance, suppose the preceding vehicle suddenly brakes to avoid collision with a pedestrian or at a run-yellow traffic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Consequently, the ego vehicle has to decelerate abruptly, resulting in a two-step chain of responsible events for the unexpected stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Thus, identifying spatiotemporal interactions among traffic objects are crucial in characterizing the root cause behind such incidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Importantly, understanding the contexts behind the degraded driving behavior on the fly is not trivial and poses multi- ple challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' First, this involves continuous monitoring of the driving behavior of the driver as well as an exhaustive knowledge of various on-road spatial micro-events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Expensive vehicles use LiDAR, Radar etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', to sense the driver and the environment [4], [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' however, app-cab companies are resistant to invest in such high-end vehicles due to low-profit margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Second, depending on the driving maneuvers taken, temporally interlinking the micro-events based on the vehicle’s interaction with on-road spatial objects is a significant research challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For example, if adverse snowy weather is observed on one day, its effect on traffic movements may last till the next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In contrast, reckless driving would impact only a few other vehicles around and will not be temporally significant after a few minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Such temporal impacts of an event would vary depending on the type and space of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Third, spatial positions of the surrounding objects impact the driving maneuver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Precisely, along with temporal dependency, the distance between the ego vehicle and the surrounding objects plays a vital role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For example, a far-sighted pedestrian might cross the road at high speed, keeping a safe distance, but it is fatal if the distance to the vehicle is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Existing literature [6], [7] have attempted to identify risky driving, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', vehicle-pedestrian interaction, through IMU and video analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' however, they fail to capture such temporal scaling or the spatial dependency among surrounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Fourth, identifying the context in real-time over an edge-device (such as a dashcam) is essential for providing a just-in-time feed- back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' But, deploying such a system for context characterization and analysis from multi-modal data over resource-constrained edge-device is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' To address these challenges, we propose DriCon that develops a smart dash-cam mounted on the vehicle’s dashboard to characterize the micro-events to provide just-in- time contextual feedback to the driver and other stakeholders (like the cab companies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' It senses the maneuvers taken by the ego vehicle through IMU and GPS sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In addition, a front camera mounted on the device itself, is used to analyze the relationship between various on-road micro-events and the driving maneuvers taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' This facilitates the system to run in each vehicle in a silo and makes it low-cost and lightweight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 1 shows a snapshot of the hardware components of our system mounted on a vehicle, and an example scenario where DriCon generates a live contextual explanation behind a sudden jerk observed in the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In summary, our contributions to this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (1) Pilot Study to Motivate Micro-Event Characterization: We perform a set of pilot studies over the Berkeley Deep Drive (BDD) dataset [8], the largest public driving dataset available on the Internet (as of January 16, 2023), to investigate the variations in driving behavior depending on various road types, time of the day, day of the week, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', and highlight the spatiotemporal micro-events causing abrupt changes in driving maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (2) Designing a Human Explainable Lightweight Causal Model: The development of DriCon relies on the (i) IMU & GPS data to infer the driving maneuvers, and (ii) object detection model & perspective transformation [9] to detect the surrounding objects and their actions to capture various spatial micro-events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Subsequently, we identify the spatiotemporal contexts whenever the driving behavior deteriorates during a trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Finally, we implement Self Organizing Maps (SOMs), a lightweight but effective causal model to capture the spatiotemporal dependency among features to learn the context and generate human-interpretable explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (3) Deployment on the Edge: We deploy the whole architecture of DriCon on a Raspberry Pi 3 model, embedded with a front camera, IMU and GPS sensors (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For this purpose, we make both the IMU and visual processing of the data lightweight and delay-intolerant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Following this, the pre-trained model generates recommendations based on the ongoing driving trip and makes it efficient to run live for just-in-time causal inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (4) Evaluating DriCon on a Live System Deployment and with BDD Dataset: We evaluate DriCon on our live in-house deployment, as well as on the BDD dataset [8] (over the annotated data [10]), comprising 33 hours and 17 hours of driving, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We obtain on average 70% and 80% similarity between the derived and the ground-truth causal features, respectively, with top-3 and top-5 features returned by the model, in correctly identifying the micro-events causing a change in the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Notably, in most cases, we observe a good causal relationship (in terms of average treatment effect) between the derived features and the observed driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In addition, we perform different studies of the resource consumption benchmarks on the edge-device to get better insights into the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' RELATED WORK Several works have been proposed in the literature on understanding road traffic and its implications for road fa- talities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Early research focused on traffic surveillance-based techniques to prevent road accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For instance, National Highway Traffic Safety Administration (NHTSA) [3] had recorded statistics about fatal accident cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' TUAT [2] has been collecting video records from taxis and drivers’ facial images since 2005 to derive injury instances into several classes along with driving behavior estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In India, the source of information behind the causes of traffic injuries is the local traffic police [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In contrast, works like [12], [13] learn the crime type and aviator mobility pattern just-in-time from street view images and raw trajectory streams, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Apart from harnessing videos and crowd-sourced information, several works [14], [15] are done on abnormal driving behavior detection by exploiting IMU and GPS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' To prevent fatal accidents, authors [16]–[18] try to alert the drivers whenever risky driving signature is observed, such as lane departure or sudden slow-down indicating congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, they have not looked into the effect of neighboring vehicles or other surrounding factors on various driving maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Interaction among the ego vehicle and other obstacles, such as pedestrians, adverse weather in complex city traffic, often affects the vehicle’s motion, consequently affecting the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Existing studies [19] reveal that road category, unsignalized crosswalks, and vehicle speed often lead to a disagreement among pedestrians to cross the road, leading to road fatalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' A more detailed study [20], [21] focuses on causality analysis for autonomous driving, faces infeasibility in real-time deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Moreover, they only use a limited set of driving maneuvers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', speed change only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Particularly, causal inferencing is challenging due to high variance in driving data and spurious correlation [22] between traffic objects and maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The existing works limit their study by considering only static road attributes or relying on single or multi-modalities from a connected road network system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Such methodologies will not be applicable for a single vehicle in real-time deployment unless connected to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In contrast, leveraging multi-modalities from onboard vehicle sensors can efficiently characterize the continuous and dynamic contexts behind unexpected driving behavior fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' DriCon develops a system in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' MOTIVATION In an ideal scenario, two vehicles are likely to follow similar maneuvers under the same driving environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' but this is not the case in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Driving behavior varies according to the driver’s unique skill set and is influenced by the impact of various on-road events, such as the movement of other heavy and light vehicles, movement of pedestrians, road congestion, maneuvers taken by the preceding vehicle, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', which we call spatial micro-events or micro-events, in short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In this section, we perform a set of pilot studies to answer the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (a) Does a driver’s driving behavior exhibit spatiotemporal variations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (b) Do all micro-events occurrences during a trip similarly impact the driving behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (c) Does a sequence of inter-dependent micro-events collectively influence the driving behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Following this, we analyze the publicly-available open-source driving dataset named Berkeley Deep Drive dataset (BDD) [8] to answer these questions stating the impact of different micro- events on the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The dataset contains 100k trips crowd-sourced by 10k voluntary drivers over 18 cities across two nations – the USA and Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The dataset has been annotated with a driving score on the Likert scale of 1 (worst driving) to 5 (best driving) for each 5-second of driving trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Variation in Driving Behavior over Space and Time We first check whether the on-road driving behavior exhibits a spatiotemporal variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For this purpose, we vary two parameters – road type as the spatial parameter (say, “High- way”, “City Street”, “Residential”), and time of the day as the temporal one (say, “Daytime”, “Nighttime”, “Dawn/Dusk”) in the BDD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In this pilot study, we form 9 groups with 30 trips each, in a total of 270, where the trips under a group are randomly picked from the BDD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We plot the distribution of the driving scores over all the trips for each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2(a), it is evident that the score distribution varies both (a) for a single type of road at different times of the day, and (b) for different types of road at any given time of the day (with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='05 reflecting its statistical significance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In the following, we investigate the role played by various micro-events behind the variations in driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Role of Spatial Micro-events Next, we inspect whether various on-road micro-events, which are characterized by the movements of other spatial objects such as “cars”, “pedestrians”, “trucks”, “buses”, “mo- torcycles”, “bicycles”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', impact a driver’s driving behavior in the same way across different times of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We perform this study by handpicking 30 trips along with their annotated driving scores for both day and night time from the BDD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We compute the volume (say, count) of spatial objects extracted using the existing object detection algorithm [23] from the video captured during the trip and take the average count of each object for a 5-second time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Thus, for both daytime and nighttime, we get two time-series distributions, (a) the count of each on-road spatial object captured over the trip video during each time window, and (b) the annotated driving scores at those time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, we compute the Spearman’s Correlation Coefficient (SCC) among these two distributions for day time and night time, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2(b), we infer that mostly all the on-road spatial objects adversely affect the driving behavior (depicting a negative correlation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Cars and pedestrians affect the driving score majorly during the daytime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Whereas, at night time, trucks and buses, along with the cars, impact the driving behavior because heavy vehicles such as trucks move primarily during the nighttime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, the effect of light vehicles such as motorcycles and bicycles is insignificant due to the dedicated lanes for their movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' This observation is further extended to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2(c), where the same study is done for weekdays vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' weekends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We extracted the day of the week using already provided timestamps in the BDD dataset and clubbed 30 trips from Monday to Friday for weekdays and 30 trips from Saturday to Sunday for the weekend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2(c), we observe that during the early days of the week, cars, pedestrians, and trucks adversely affect the driving behavior, whereas the impact is less during the weekend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Hence, we conclude that different on-road objects exert diverse temporal effects on the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Micro-events Contributing to Sudden Driving Maneuver: Abrupt Stop as a Use-case Finally, we explore whether multiple inter-dependent micro- events can be responsible for a particular driving maneuver that might degrade the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For this purpose, we choose abrupt stop as the maneuver, which we extract from the GPS and the IMU data (the situations when a stop creates a severe jerkiness [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We take 30 trips for each scenario, including daytime, nighttime, weekdays, and weekends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For each scenario, we extract the instances when an abrupt stop is taken and record the corresponding micro- events observed at those instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Precisely, we extract the presence/absence of the following micro-events: red traffic signal, pedestrian movements, presence of heavy vehicles as truck & bus, light vehicles as motorcycle & bicycle, and the preceding vehicles’ braking action (as peer vehicle maneuver), using well-established methodologies [10], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We compute the cumulative count of the presence of each micro-events and the number of abrupt stops taken over all the trips for the four scenarios mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2(d) and (e), we observe that the red traffic signal, the peer vehicle maneuvers, and heavy vehicles mostly cause an abrupt stop during the nighttime and on weekdays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Therefore, we argue that multiple on-road micro-events, such as the reckless movement of heavy vehicles at night, force even an excellent driver to slam on the brake and take an unsafe maneuver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Highway City Street Residential 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 Driving Score (a) Cars Pedestrians Truck Bus Motorcycle Bicycle −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content="6 Spearman's Correlation Day Time Night Time (b) Cars Pedestrians Truck Bus Motorcycle Bicycle −0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content="2 Spearman's Correlation Week Day Weekend (c) Red Signal Pedestrians Heavy Vehicles Light Vehicles Peer Vehicle Action 0 10 20 30 40 50 60 70 %age of Occurences Day Time Night Time (d) Red Signal Pedestrians Heavy Vehicles Light Vehicles Peer Vehicle Action 0 10 20 30 40 50 %age of Occurences Week Day Weekend (e) Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2: (a) Variation of Driving Behavior with respect to Road Type and Time of the Day, (b)-(c) Impact of Spatial Micro-events on the Driving Score at Different (i) Time of the Day, (ii) Day of the Week, (d)-(e) Contributing Factors Observed behind Abrupt Stop at Different (i) Time of the Day, (ii) Day of the Week IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' PROBLEM STATEMENT AND SYSTEM OVERVIEW A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Problem Statement Consider that FM denotes the set of driving maneuvers and FS be the set of spatial micro-events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Fi be the set of temporally-represented feature variables corresponding to the driving maneuvers taken and on-road spatial micro-events encountered during a trip i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Let Ri T be the driving score at time T during the trip i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We are interested in inspecting the events occurred, representing the feature values Fi, when |Ri T − ˆRi T −1| > ǫ (ǫ is a hyper-parameter, we set ǫ = 1), reflecting the fluctuations in driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Here, ˆRi T −1 = ⌈mean([Ri 1, Ri T −1])⌉ represents the mean driving behavior till T −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The output of the system is a characterization of {Fi M, Fi S}, as to whether a fluctuation in the driving behavior is due to the driving maneuvers only (Fi M) or forced by the spatially causal micro-events (Fi S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Finally, we target to generate the explanations based on {Fi M, Fi S} to give feedback to the stakeholders for further analysis of the driving profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Feature Selection Leveraging the existing literature [24], we identified a set of feature variables at timestamp T representing various driving maneuvers FM of the ego vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' These features are – Weaving (AW T ), Swerving (AS T ), Side-slipping (AL T ), Abrupt Stop (AQ T ), Sharp Turns (AU T ), and Severe Jerkiness (AJ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Similarly, we consider the following feature variables corresponding to the spatial micro-events FS – Relative Speed (ST ) and Distance (DT ) between the ego and the preceding vehicle, preceding vehicle’s Braking Action (BT), volume of the peer vehicles in front of the ego vehicle indicating Congestion in the road (CT ), Pedestrian (PT ), and it’s speed (QT ), Traffic light (LT ), Heavy vehicles: {Bus & Truck} (HT ), Type of the Road (GT ), and Weather condition (WT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Note that, we empirically select these features based on the existing literature and observations from the dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' additional features can also be incorporated in DriCon without losing its generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We next broadly introduce our system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' DriCon captures IMU, GPS, and video data from a dashcam (say, an edge-device) and characterizes the context behind the improved/degraded driving behavior on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The system comprises three components: (a) Data Preprocessing and Feature Extraction, (b) Detection of Improved/Degraded Driving Behavior, and (c) Identification of Possible Context (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 4 Pedestrian Crossed and The Ego Vehicle Abruptly Stopped and Faced Severe Jerkiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Inferred Features: Crossing Pedestrians, Severe Jerkiness, Abrupt Stops INPUT: IMU, GPS & Video + Driving Maneuvers Spatial Micro-events 2 4 4 Detect Change in Driving Behavior Capture Time- Series Dependency Among Features Output: Generated Explanations (a) (b) (c) (d) (e) Data Preprocessing and Feature Extraction Identification of Possible Context Model Output Model Construction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 3: DriCon System Flow and Modeling Pipeline C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Data Preprocessing and Feature Extraction The collected IMU and GPS sensor data are prone to noise due to the earth’s gravitational force, signal attenuation, and atmospheric interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Hence, we implement a low- pass filter to eliminate such noises from IMU and GPS to compute inertial features for the extraction of the driving maneuvers (FM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, we preprocess the video data before extracting on-road spatial micro-events and their actions (FS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We up/downsample the acquired videos to a resolution of 960 × 540p, preserving the signal-to-noise ratio above 20 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 1) Driving Maneuvers - FM: In order to generate the features corresponding to different driving maneuvers (FM), we extract the instances of Weaving (AW T ), Swerving (AS T ), Side-slipping (AL T ), Abrupt Stop (AQ T ), Sharp Turns (AU T ), and Severe Jerkiness (AJ T ) from the IMU data using standard accelerometry analysis [10], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2) Spatial Micro-events - FS: Next, we implement the state-of-the-art video data-based object detection algorithms and further fine-tune them based on our requirements, as developing vision-based algorithms is beyond the scope of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We leverage the YOLO-V3 [23] algorithm trained on the COCO dataset [25] to detect a subset of traffic objects such as Pedestrians, Cars, Buses, Trucks, and Traffic Lights (de- picted as FS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, we estimate the influence of pedestrians’ interactions, the presence of heavy vehicles (buses & trucks), traffic light signal transitions (red, yellow & green), and the cars on the driving behavior of the ego vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, we discard the detected objects which depict a confidence score troficintao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='50o6 treffieliehtzo5 cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='90r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='680 trotmcliehtr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='27Daytime Nightime Dawn/Dusk< 50% and bounding boxes of area < 10k, capturing the fact that the far-sighted traffic objects around the ego vehicle exert marginal impact compared to the near-sighted ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Addition- ally, the traffic objects in the mid-way of the road, broadly visible from the driver’s dashboard, will be of more influence than the left or right lanes, as the ego vehicle will follow them immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Thus, we divide each of the frames into 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='2:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='6:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='2 ratio along the horizontal axis, as left:middle:right lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Therefore, we keep the Pedestrians PT , Cars, Heavy Vehicles as {Buses & Trucks} HT , which have bounding box co-ordinates within the middle lane boundary, and Traffic Light Signal Transitions LT (Red, Yellow & Green) without the lane information as traffic lights are often positioned on the left and right lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Since our pilot study demonstrated that the pedestrians and peer vehicles’ action significantly impact the driving maneuvers of the ego vehicle, (a) we extract the Pedestrian Speed (QT ), as well as identify the crossing pedestrians in the mid-way, and (b) we compute the preceding vehicle’s Braking Action (BT ), and Congestion (CT), as well as detect the Relative Speed (ST ) and Distance (DT ) variation among the ego and the preceding vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We apply perspective transformation and deep learning methods [9], [26] to infer the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Finally, the above pipeline runs on each frame where the video is re-sampled to 15 frames-per-second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Detection of Driving Behavior Fluctuations The crux of DriCon is to capture the temporal dependency of various driving maneuvers and spatial micro-events when a change in the driving behavior is observed during the trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For a run-time annotation of the driving behavior, we use an existing study [10] that provides a driving behavior score on the Likert scale [1 − 5] by analyzing driving maneuvers and other surrounding factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We divide the trip into continuous non-overlapping time windows of size δ and compute the driving score at the end of every window U (denoted as RP U ), using the feature values captured during that window [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' To quantitatively monitor whether there is a change in the driving behavior during a window U, we compare RP U and ˆRP U = 1 U−1 U−1 � i=1 RP i (mean driving score during previous U−1 windows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Suppose this difference is significant (greater than some predefined threshold ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In that case, DriCon proceeds towards analyzing the temporal dependency among the feature vectors at different time windows to understand the reason behind this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Identification of Possible Context In the final module, we use the feature vectors at different windows to build the model that identifies which features (FGEN) are responsible for the change in driving behavior during the window U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The model reactively seeks explanations behind such fluctuations by analyzing the effect of the micro- events that occurred over the past windows [1, · · · , (U − 1)] and the present window U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Finally, natural language-based human interpretable explanations are generated and fed back to the stakeholders for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' MODEL DEVELOPMENT To develop the core model for DriCon, we leverage the already extracted features F ∈ {FM � FS} (details in §IV-C) to capture the temporal dependency of the past as well as the present events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In addition, DriCon derives the explanation be- hind the detected events through explanatory features FGEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For this purpose, we need a self-explanatory model that can capture the spatiotemporal dependency among different driving maneuvers and micro-events associated with the on- road driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We choose a Self Organizing Map (SOM) [27] for constructing the model that can exploit such spatiotemporal dependencies with minimum data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The major limitation of the classical deep learning models (such as CNN or RNN) stems from the fact that, (i) deep networks consume heavy resources (say, memory), as well as suffer from huge data dependency, and (ii) they act as a black box, hence fail to generate human interpretable explanations behind certain predictions [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' On the other hand, SOM is able to characterize the micro-events in runtime using feature variability and unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Neighboring Radius F1 F2 F3 FU Input Layer Learning Phase Feature Input Converge Final Map Code Book BMU Weight (a) (b) (c) No Change Change Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 4: Working Principle of SOM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Inferring Explanatory Features using SOM The key idea behind obtaining the explanatory features is first to discover the spatiotemporal feature dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In DriCon, we derive so using Kohonen’s Self Organizing Map (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 4), as it is an unsupervised ANN-based technique leveraging competitive learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Since DriCon runs on an edge-device, we employ a minimal number of model parameters to expedite the processing without compromising the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Precisely, we implement the codebook with 147 neurons, spread out over a two-dimensional array of size 7 × 21 (where 7 is a hyperparameter depending on the maximum influence of the past windows during a trip, 21 cor- responds to the number of features in the feature space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' These neurons are initialized with a random weight (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 4(a)), where the weight vector has the same length (of 21) as the feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, we represent each trip with a 2D grid of size 8 × 21 (considering 8 consecutive windows in a trip) to capture the influence of the past windows [1, · · · , (U −1)] and the present window U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In principle, the inherent topological ordering of SOM groups the similar feature space (in windows [1, · · · , (U −1)]) into a single group, when there is no change in the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' On the contrary, the dissimilar ones (say, during the window U), when there exists a change in the driving behavior, are mapped into a different group, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 4(b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For instance, suppose on a trip, the ego vehicle abruptly stops due to the preceding vehicle’s braking action following a sudden change in the traffic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Hence the feature space in window [1, · · · , (U − 1)] exhibits a similar signature (until the abrupt stop occurs), and subsequently gets mapped to a single neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, during the abrupt stop, there will be changes in the feature space (say, maneuvers and other spatial events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' These changes in the feature space will get it assigned to a different neuron and settle the other neurons’ weight automatically depending on the changes in the feature space between the windows [1, · · · , (U −1)] and the window U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' This procedure allows SOM to harness the temporal dependency among spatial events in an unsupervised mode, without using the driving score explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 1) Model Training: The input trip data is represented in the 2D grid format for learning the best-matched neuron, optimizing the Euclidean distance between the feature space and weight vector of the corresponding neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' To ensure the best-fitting, the best-matched neuron tries to learn the weight vector of the feature space at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Also, the neurons in the neighborhood try to tune their weights as nearest as possible compared to the best-matched neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We train this model for 500 epochs, where each neuron gets mapped with the best matching trip instances and converges to their coordinate position in the codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We implement the Bubble neigh- borhood function [29] to update the neighborhood neurons’ weights until the neighborhood radius converges to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We ensure that both the distance and neighborhood functions are computationally faster for accurate learning accelerating the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Upon completing the total number of epochs, we obtain the converged codebook called the Map, where each trip instance gets assigned to the best matching neuron called the Best Matching Unit (BMU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The weight vector corresponding to the BMU’s coordinate reveals the explanatory features FGEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2) Model Execution: We leverage the constructed Map for the runtime inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' First, we conduct the feature processing of the current ongoing trip (following §IV-C), and in parallel, the extracted feature space is fed as input to the constructed Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Eventually, we obtain the BMU’s coordinate and extract its corresponding weight vector and the feature encoding for the given trip instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From the weight vector, we extract the top-k weights and their corresponding feature names (say, weather type) and their encoded values (say, weather type: rainy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Finally, we populate them in FGEN (called the Generative micro-events) for further generation of human interpretable explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Generating Textual Explanation DriCon aims to generate the explanations in textual format utilizing the output features FGEN for better readability and human interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' As the features f ∈ FGEN are already associated with some keywords (say, severe jerkiness), we need to generate them in a sentential form, keeping the features as “action” or “subject” depending on whether f ∈ FM or f ∈ FS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For instance, if the feature is an action, we assign the ego vehicle as the subject, replace the corresponding output feature f with its describing keyword, and finally concatenate them to obtain the sentential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For example, in case of severe jerkiness, the constructed sentence becomes, “the ego vehicle severe jerks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, if the output feature f represents a subject, then many possible sentences can be generated out of one subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Thus, we mine several traffic guidelines [30] and compute the cosine similarity among the features and existing guidelines using TF- IDF vectorizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Upon extracting the most relevant guidelines, we fetch the object associated with the sentence and construct a single sentence for each output feature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', “pedestrian crossing” → “pedestrian crossing the intersection”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, for all the generated sentences, the describing keywords corre- sponding to each feature are converted to an adjective or adverb using Glove [31] for better structuring of the sentences (say, “the ego vehicle severe jerks” → “the ego vehicle severely jerks”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Finally, each sentence is concatenated using the “and” conjunction, and repetitive subjects are replaced using their pronoun form using string manipulation to generate the whole explanation, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' PERFORMANCE EVALUATION This section gives the details of DriCon implemented over a live setup as well as over the BDD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We report the performance of the SOM model and compare it against a well-established baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Additionally, we show how well our system has generated the textual explanations along with a sensitivity analysis to distinguish how error-prone DriCon is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We start with the experimental setup details as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Experimental Setup DriCon is implemented over a Raspberry Pi 3 Model B microprocessor kit operating Raspbian OS with Linux kernel version 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='65 − v7+ along with 1 GB primary memory and ARMv7 processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We primarily utilize the IMU, the GPS, and the video data captured through the front camera (facing towards the front windscreen) as different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For this purpose, we embed one MPU−9250 IMU sensor, one u-blox NEO−6M GPS module, and one Logitech USB camera over the Raspberry Pi board, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We deployed DriCon over three different types of vehicles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', SUV, Sedan, & Hatchback).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We hired 6 different drivers in the age group of [20 − 45] who regularly drive in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Therefore, our whole experimentation ran for more than two months over three cities, resulting in approximately 33 hours of driving over 1000 km distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The drivers drove freely without any specific instructions given, with each trip varying from approximately 20 minutes to 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In addition, each driver drove over five different types of roads (city street, highway, residential, parking & campus road) at three different times of the day (day, dusk & night).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We evaluate DriCon by analyzing how well our proposed model extracts the generative micro-events FGEN (see §V-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For implementing DriCon, we consider δ = 5 seconds, ǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The impact of other hy- perparameters and resource consumption have been discussed later during the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We next discuss the ground-truth annotation procedure used for the evaluation of DriCon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Annotating Micro-events We launched an annotation drive by floating a Google form among a set of recruited annotators, where they had to watch a video of at most 10 seconds and choose the top-3 most influential factors impacting the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We do this annotation over the in-house data (video data collected during the live experiments) and the videos over the BDD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For each video from both the datasets given in the form, we showed only the clipped portion where the score fluctuations had occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, out of the total 15 factors (including driving maneuvers and spatial micro-events) given in a list, they were instructed to choose the top-3 most influential factors responsible for the poor driving behavior based on their visual perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Besides, we also provided the model-generated sentences (§V-B) and asked how relevant and well-structured the sentences are (on a scale of [1−5]) for explaining the change in the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The annotators also had the option to write their own explanation if they perceived a better reason behind the driving behavior change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' As the number of trips is quite large, we need to design a set of Google forms (sample form1), each containing at most 20 videos to ensure the least cognitive load on the annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We also collected annotators’ demographic information such as age, gender, city, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We find that most participants (> 67%) had prior driving skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' At least three independent annotators had annotated each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Upon receiving the annotated factors, we need to find the agreement among the annotators to ensure the received ground truth is unbiased and non- random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' As standard inter-annotator agreement policies (say, Cohen’s kappa index) work on quantitative analysis or one-to- one mapping, we cannot apply such metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Thus, we use the majority voting technique where each listed factor is assigned a percentage, signifying how many times the annotators choose that factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Each factor having a vote of at least 60% is kept in FGT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We observe the minimum and the maximum cardinality of FGT are 3 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' This also indicates that the annotators agreed on selecting the factors that influenced the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' FGT contains the annotated micro-events against which FGEN is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Performance Metric We use the Dice Similarity Coefficient score [32] (N) which computes the similarity between FGT and FGEN as fol- lows: N = 2×|FGT ∩FGEN| |FGT |+|FGEN| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We report the mean N across all the trips to measure the accuracy of DriCon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, we also use Average Treatment Effect [33] (ATE) to report comparatively higher causal features out of the model identified features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Finally, we define Percentage of Error as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' First, we 1https://forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='gle/97N6uk4ujRaZSWbj8 (Accessed: January 16, 2023) Top-3 Top-5 30 50 70 90 Dice Coefficient (in %) (a) Top-3 Top-5 10 30 50 70 90 Dice Coefficient (in %) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 5: (a) Dice Coefficient Similarity (in %) between Human Annotated and Model Generated Features (b) Ablation Study compute the set-difference as {FGT \\FGEN}, and extract the corresponding feature category (say, FM, FS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Once we get the count of each feature category, we compute its percentage out of the total trips as the Percentage of Error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Baseline Implementation As a baseline for extracting FGEN, we implement a super- vised rule-based Random Forest (RF) algorithm with 20 deci- sion trees where each tree is expanded to an unlimited depth over the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We optimize the labels RP U with the intuition that features will contribute differently to each of the predicted scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Although the RF-based model has a feature importance score signifying the contribution of each feature in constructing the model, we need to have an explanation of how each feature contributes to predicting the driving scores on a trip instance basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Therefore, we use LIME [34] in the background of the RF model for generating the explanatory features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' As LIME is a model-agnostic method, it tries to map the relationship between the input features and output scores by tweaking the feature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Thus, it explains the range of values and probability for each feature that contributes to predicting the score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From the generated explanation, we extract the contributing features FGEN along with their values for further generation of textual explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' This pipeline is executed in a similar manner as described in §VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Accuracy of Characterized Context We present the accuracy of DriCon using the SOM and RF+LIME model over the in-house dataset using Dice Coeffi- cient Similarity N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We extract the top-k features from FGEN where k ∈ {3, 5} and compute N between the two sets of fea- tures – FGEN and FGT with top-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 5(a) shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For top-3, we get 69% & 40% similarity on average with SOM and RF+LIME, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Whereas for top-5, we observe 79% & 48% similarity on average with SOM and RF+LIME, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' As the in-house dataset has more complex micro- events, the slight performance drop over the in-house dataset using the top-3 features is tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Intuitively, the model can capture more diversity as perceived by the human annotators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' therefore, the similarity improves as we move from k = 3 to k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, as the RF+LIME considers each time instance of a trip independently, its performance degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' It captures the dominant features responsible for the driving behavior change within the current time window, contrary to inspecting past time windows’ impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' DriCon DriCon-man DriCon-spat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='SOM RF WI LIMETABLE I: Similarity Measure among Human Annotated vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Model Generated Output Instance# Human Annotated FGT Model Generated FGEN Similarity N(%) ATE 1 Poor Weather Conditions (Heavy Rainfall, Fog, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' ), Swerving, Congestion, Overtaking, Taking Abrupt Stop Congestion, Preceding Vehicle Braking, Weaving, Abrupt Stop, Severe Jerkiness 40% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='96 2 Sideslip, Taking Abrupt Stop, Traffic Lights: Red Traffic Lights: Red, Congestion, Abrupt Stop 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='67% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5 3 Crossing Pedestrian, High Speed Variation among Cars, Weaving Severe Jerkiness, Crossing Pedestrian, Weaving 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='67% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='35 To have a glimpse, we present the explanatory features (FGEN) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' human-annotated ones (FGT) in Table I for a sample of three test instances where the similarity (Dice coef- ficient) is comparatively lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Interestingly, when there is a mismatch, we observe that the corresponding features from the model-generated and human-annotated ones are conceptually related for most of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Additionally, a positive high mean ATE value for the model-generated mismatched features signifies that the model perceived those features as more causal than normal human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' It can be noted that an ATE value ≥ 1 indicates high causal relationships between the features and the corresponding effect (changes in the driving behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For example, in test instance #2, the mismatched features are Sideslip (for human generated) and Congestion (for model generated), where Congestion was relatively more causal, affecting the change in the driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' By manu- ally analyzing this instance and interviewing the corresponding driver, we found that he indeed made a minor sideslip on a congested road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Indeed, the driver was not very comfortable in driving a manually-geared car on a congested road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 6: Generated Map from SOM for a 7×7 Network (Scaled Down) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Ablation Study Next, we understand the importance of different feature categories corresponding to the driving maneuvers and on-road spatial events, as described in §IV-A, on the overall perfor- mance of DriCon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' To study the impact of driving maneuvers and spatial features, we implement SOM, excluding each of the above feature classes one at a time, and evaluate N to inspect the importance of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The two variants other than DriCon are constructed in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (a) DriCon- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' : Here, we exclude the driving maneuvers FM and keep FS only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' (b) DriCon-spat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' : Next, we exclude the spatial features FS and keep FM only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We evaluate these two variants over both top-3 and top-5 generated features, along with DriCon containing all the features, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' On excluding the driving maneuvers and spatial features, performance drops to 45% and 31%, respectively, for top-5 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' This drastic drop signifies the crucial importance of spatial features, as these are the frequently changing features responsible for fluctuating driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Model Insight To understand how the spatiotemporal dependency among different features corresponding to the driving maneuvers and various on-road spatial micro-events are derived, we use 49 neurons spread over a 7 × 7 two-dimensional array (a smaller variant of the SOM network originally used to develop the model, as the original model having 147 neurons is difficult to visualize), fitted over 200 trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' This instance produces a Map as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 6, where all the given trips are assigned to each of the neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The scores RP U are used only for visual depiction purpose of how the trips are located on the Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Each trip captures the change in the driving behavior using the feature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The neurons with multi-color are of more importance than the mono-color, as in those, the score fluctuations are most observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' During a stand-alone trip, the features corresponding to each instance of the trip will have a similar value until there is a change in the driving behavior, thus getting assigned to the same neuron (mono- color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, the difference in the driving behavior induces distinct feature values than the previous instances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' thus, it gets assigned to a different neuron in the Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The neurons having multi-color, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 6, map the trip instances where a sudden change of driving behavior has occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Dissecting DriCon We next benchmark the resource consumption behavior of DriCon, followed by an analysis of the model’s significance and sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 1) Edge-device Resource Consumption: We benchmark the CPU & memory usage, processing time, temperature rise, and energy consumption over two cases: when (a) the device is idle, & (b) DriCon is running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 7(a), we observe that in idle mode, on average, 2% of CPU (using “top” command) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In contrary, running DriCon acquires at most 10% of the processor, which is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, the memory usage is a bit high (≈ 500MB) mainly due to video processing overhead as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Next, we show the required processing time starting from data acquisition to output generation on a number of trip basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Idle Live 2 4 6 8 10 CPU Consumption (in %) (a) Idle Live 100 200 300 400 500 600 Memory Consumption (in MB) (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5 Processing Time (in mins) 0 5 10 15 20 25 #Trips (c) Idle Live 43 46 49 52 55 58 61 Temperature Rise (in ∘ C) (d) 0 12 24 36 48 60 Time (in mins) 0 5 10 15 20 25 Energy Consumption (in W-Hr) Nexar Live Idle (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 7: Resource Consumption over the Edge-device (a) CPU Usage (b) Memory Usage (c) Histogram of Processing Time w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=', #Trips (d) Temperature Rise, (e) Energy Consumed Relevance Well-Structured 3 4 5 Annotation Score (a) Spatial Maneuver 0 5 10 15 %age of Error (b) Top-3 Top-5 30 50 70 90 Dice Coefficient (in %) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 8: (a) Significance of DriCon (b) Sensitivity Analysis of DriCon (c) Performance on BDD Dataset DriCon generates the output within ≈ 3 minutes only for majority of the trips, further validating shorter response time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 7(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' To further delve deeper, we also log the temperature hike (from “vcgencmd measure temp” command) and total energy consumption using Monsoon High Voltage Power Monitor [35] while running DriCon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 7(d) & (e), we observe that the temperature hiked at most to 59°C, while on average, 13 Watt-hour energy is consumed, which is nominal for any live system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' To benchmark DriCon, we have also measured the energy consumption of the Nexar dashcam, which consumes 22 Watt-hour on an average, while capturing very few driving maneuvers (say, hard brake) without any context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' This further justifies that DriCon never exhausts the resources on the edge-device and is can accurately detect the micro-events precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 2) Significance of Generated Explanation: Next, we check how significant our generated explanations are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' As reported in §VI-B, we plot the distribution of annotated scores (given by the recruited annotators) for the two fields – “Relevance” and “Well-Structured”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' “Relevance” signifies the generated explanation’s applicability in explaining unexpected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' In contrast, “Well-structured” indicates how well inter- pretative the generated sentences are as per human cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 8(a) depicts a median value of 5 and 4 for “Relevance” and “Well-Structured”, respectively, which further justifies the credibility of DriCon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We also compute the similarity between the human-annotated and model-generated sentences and obtain a minimum, maximum, and mean similarity value as 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='33%, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5% & 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='57%, respectively, using the TF-IDF vectorizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Thus DriCon resembles human cognition level up to an indistinguishable level (between a human and model) of auto-generating a contextual explanation, which further shows its applicability to give feedback to the stakeholders for their decision-making procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 3) Sensitivity of DriCon: Finally, we inspect the micro- events that DriCon fails to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Because, apart from a model’s efficiency, we must also look into its deficiency to analyze how much that might affect the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Especially, this is important in the case where stakeholders are boosting/penalizing the driver’s profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 8(b), incompetence to capture both the spatial and ma- neuvers is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Although this might lead to degraded model performance, as studied in §VI-F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' driving maneuvers (FM) do not contribute superiorly to model performance due to the inter-dependency on spatial features (FS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' But for FS, the Percentage of Error is still ≤ 13%, making the system less sensitive into generating error-prone contextual explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Offline Performance Finally, we report the accuracy of our system over the BDD dataset comprising 17 hours of driving data over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='5k trips using N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' 8(c), DriCon performs quite well on pre-recorded data, with N = {71%, 84%}, for top- 3 and top-5 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' We observe that SOM can identify the micro-events in a better way for offline analysis with a public dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, as running the system live is essential for a realistic driving environment other than offline analysis, this much of slight accuracy drop can be endured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' CONCLUSION This paper developed an intelligent system on the edge- device called DriCon leveraging multi-modalities to detect the micro-events responsible for unexpected fluctuations in driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The human-interpretable explanations generated by DriCon show their relevance and credibility in identifying such context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Further, the spatiotemporal dependency among various features is inspected in an unsupervised manner to capture a diverse set of driving scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Additionally, the resource-friendly deployment over a live testbed further vali- dates DriCon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Although our study captures the context where each feature’s contribution is taken independently, inter-feature dependency is not captured explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' For instance, say, a driver suddenly weaves while taking a turn to avoid colliding with a crossing pedestrian, making the following vehicle’s driver slam the brake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' Here, the first driver’s action is due to the crossing pedestrian, which in turn impacts the second driver’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' The analysis of such complex and collective interactions among the vehicles needs a more sophisticated SOM RF W/ LIMEsystem, possibly a different modality that can connect the inter-vehicle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' However, DriCon provides a simple, in-the-silo solution that can be independently deployed over vehicles with a dashboard-mounted edge-device or dashcam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' REFERENCES [1] “Road traffic injuries, by world health organization (who),” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='who.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='int/news-room/fact-sheets/detail/road-traffic-injuries, 2022, (Online Accessed: January 16, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content=' [2] 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+page_content='msoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} +page_content='com/online-store/High-Voltage-Power-Monitor-p90002590, (Online Accessed: January 16, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE4T4oBgHgl3EQfzw3P/content/2301.05277v1.pdf'} diff --git a/4dFAT4oBgHgl3EQfExzK/content/tmp_files/2301.08424v1.pdf.txt b/4dFAT4oBgHgl3EQfExzK/content/tmp_files/2301.08424v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9b2cf37a1856a1375b4a485a877c0dea61dbf1d --- /dev/null +++ b/4dFAT4oBgHgl3EQfExzK/content/tmp_files/2301.08424v1.pdf.txt @@ -0,0 +1,1016 @@ +Possible new phase transition in the 3D Ising Model +associated with boundary percolation +Michael Grady +Department of Physics +State University of New York at Fredonia +Fredonia NY 14063 USA +grady@fredonia.edu +January 23, 2023 +Abstract +In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by +a boundary of dual plaquettes. As the temperature is raised, these spin clusters become +more numerous, and it is found that eventually their boundaries undergo a percolation +transition when about 13% of spins are minority. Boundary percolation differs from the +more commonly studied site and link percolation, although it is related to an unusual +type of site percolation that includes next to nearest neighbor relationships. Because +the Ising model can be reformulated in terms of the domain boundaries alone, there is +reason to believe boundary percolation should be relevant here. A symmetry-breaking +order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to +undergo a phase transition at a coupling close to that predicted by duality from the +boundary percolation. This transition lies in the disordered phase of the gauge theory +and has the nature of a spin-glass transition. +Its critical exponent ν ∼ 1.3 is seen +to match the finite-size shift exponent of the percolation transition further cementing +their connection. This predicts a very weak specific heat singularity with exponent +α ∼ −1.9. The third energy cumulant fits well to the expected non-infinite critical +behavior in a manner consistent with both the predicted exponent and critical point, +indicating a true thermal phase transition. Unlike random boundary percolation, the +Ising boundary percolation has two different ν exponents, one associated with largest- +cluster scaling and the other with finite-size transition-point shift. This suggests there +are two different correlation lengths present. +PACS: 05.50+q, 05.70.Jk, 64.60.ah, 64.60.F +Keywords: Ising model, Gauge Ising model, spin glass, percolation, phase transition +arXiv:2301.08424v1 [cond-mat.stat-mech] 20 Jan 2023 + +1 +Introduction +The Ising models in two and three dimensions are the most basic spin models which undergo +order-disorder transitions. These have been extremely well studied and, of course, an exact +solution exists in the two-dimensional case. The 3D model has always been a bit more of a +mystery, and in this paper we explore the possibility of a weak secondary phase transition +within the ordered phase. +Presumably this is associated with some geometrical change +in spin-clustering, but the exact nature of this reordering is unknown. The situation is +clearer in the dual theory, the 3D gauge Ising model. Here the suspected transition is in +the disordered phase, and clearly has the nature of a spin-glass transition. In other words +we have identified a symmetry-breaking order parameter in the dual theory, but not in +the Ising model itself. However, the Ising model does exhibit an interesting percolation +phenomenon near the critical point predicted by duality from the spin-glass transition. +This is a percolation of the domain boundaries between + and - spin clusters. As shown +below, boundary percolation can be considered a third type of percolation, beyond site and +link percolation, although it has a close relationship to an unusual type of site percolation. +Of course percolation is not always related to a phase transition, but sometimes it is. +It’s linkage in this case to a symmetry-breaking transition in the dual theory provides +strong evidence that it is associated with a phase transition in this case. The argument is +further strengthened by independent fits of the third energy cumulant to consistent critical +behavior about the suspected critical point. Each investigation, the order-parameter in the +dual theory, the boundary percolation finite-size shift, and the third energy moment, yields +an independent determination of the critical exponent ν, all of which agree to a fairly close +tolerance. +In the following, first the boundary percolation concept is fleshed out and studied in +both the 2D and 3D Ising models, as well as for 3D random percolation. It is found that +the latter has the same critical exponents as for site percolation, and in fact is equivalent to +site percolation if next to nearest neighbors are included in the cluster definition. The 3D +Ising case is particularly interesting in that an analysis of the finite-size shift in percolation +threshold gives a critical exponent very different from the percolation value, even though +the cluster scaling still obeys the percolation exponents. This suggests it is linked to a phase +transition with its own dynamical scaling and correlation length. Then we move on to the +dual theory and introduce the spin-glass order parameter. A Monte Carlo study here shows +clear crossings in the Binder cumulant and second-moment correlation length divided by +lattice size. Correlation-length finite-size scaling is exhibited around the suspected critical +point using scaling collapse plots, which also yield critical exponents. The critical exponent +ν is found to match well with that found from the finite-size shift in percolation threshold. +Finally, we study energy moments, such as specific heat and higher moments. Unlike an +order parameter, these have both critical and non-critical pieces, so fitting can be difficult. +This leads to the selection of the third energy cumulant as the best prospect for finding +critical behavior as it can be fit without a non-critical part other than a constant. A Monte +Carlo study with several times 109 sweeps per point on 303, 403, and 503 lattices yields a +precise determination of this quantity. An independent critical behavior fit in the region of +the suspected critical point gives values for both ν and κc which agree well with the two +2 + +other predictions. A substantial jump in the coefficient of the critical scaling fit across the +transition further cements evidence for a thermal singularity here. +The rather high value of ν ∼ 1.3 gives a highly negative value for the specific heat +exponent α = 2 − dν ∼ −1.9. This means that both the specific heat and third cumulant +have finite singularities. A very weak infinite singularity is expected in the fourth cumulant +and stronger ones in fifth and higher. +In the Ehrenfest classification this transition is +fourth order. +We attempted to measure fourth and fifth cumulants to find evidence of +peaks growing with lattice size as expected from infinite singularities, but even with the +rather large sample size here, these were still largely obscured by random error. However, +finite singularities are just as singular as infinite ones, so perhaps one lesson is that one +should not necessarily obsess over trying to find infinite singularities in transitions of such +high order. +Figure 1: Example of a boundary cluster. Sites marked with dots have opposite orientation to all +surrounding sites. +3 + +2 +Boundary percolation +The standard partition function for the Ising model is +Z = +� +{σ} +exp(κ +� +n.n. +σiσj), +(1) +where the σ’s are classical spins taking values ±1 and the coupling is between nearest +neighbors only. There is a well-known reformulation of the Ising models in terms of the +boundaries themselves[1]. This reformulation even leads to an alternate exact solution in +the 2D case[2]. The partition function can be written +Z = +� +A +N(A) exp(−κA) +(2) +where A is the total area of dual boundary plaquettes (or dual boundary links in 2D) in +a configuration and N(A) are the number of distinct non-intersecting boundary configura- +tions with that area. In this formulation there are no spins or domains. Only the boundary +surfaces need exist, and the entropy associated with these surfaces controls the phase tran- +sition. This is one reason why percolation of the domain boundary might be important for +this model, as opposed to, say, site percolation. For instance, the density of states could +change abruptly when an infinite boundary cluster forms, because for a finite cluster the +area is usually an increasing function of the volume, whereas an infinite cluster can easily +grow in volume without adding much to the area. If this were the case then the free energy +would form a singularity at the boundary percolation point. +We define boundary percolation as follows. Consider the set of all boundary links that +connect + and - sites. +These are each associated with a plaquette on the dual lattice. +These plaquettes form closed surfaces separating clusters of + and - spins. These surfaces +can form clusters themselves, if we define boundary clusters to be made up of boundary +surfaces that share dual-lattice links. For instance, Fig. 1 shows a single boundary cluster. +This same configuration would, however, count as two separate site-clusters, since sites are +clustered only along lattice directions. If the site cluster concept is extended to include +sites connected by face diagonals, i.e next nearest neighbors (NNN) in addition to near- +est neighbors (NN), then these redefined site clusters would appear to coincide with the +boundary cluster concept. Indeed we have verified for thousands of configurations that +those with percolating boundaries also have percolating NN+NNN site-clusters, and vice +versa, so they do appear to measure the same thing. +It seems boundary percolation, which in three dimensions could also be called plaquette +percolation, has only been studied before in the form of the equivalent extended NN+NNN +site percolation problem[3], as a part of surveys of various extended percolation models, but +never applied to the Ising model. The main result of these studies is establishing a threshold +for random NN+NNN percolation at a minority site probability of 0.1372(1). Because the +Ising model is interacting, correlations would be expected to modify this result, but still +it should be kept in mind. +Fig. 2ab shows the evolution of the Ising model boundary +percolation threshold κ∗ +L with lattice size L for both two and three dimensions. These both +4 + +scale well with the finite-size scaling relation +κ∗ +L = κc − cL−1/ν +(3) +where κc is the infinite lattice threshold. The percolation threshold is defined here as the +point where 50% of lattices have a cluster which percolates in all directions. For two dimen- +sions, boundary percolation exists in the random phase, and ceases in the ferromagnetic +phase. The above fit gives κc = 0.4405(5) which agrees well with the known ferromagnetic +transition point 1 +2 ln( +√ +2 + 1) ≃ 0.44069. This is just the opposite of majority-site perco- +lation, which happens only in the magnetized phase. Thus in two dimensions boundary- +percolation and site-percolation seem equally relevant. +The exponent derived from the +finite-size scaling fit to Fig. 2a is ν = 1.261(18). This seems slightly different from the +standard 2D site-percolation exponent ν = 4/3 but of course that is for a non-interacting +system. There could also be a small correction from next to leading order scaling effects. +As far as we know, the critical exponents for random NN+NNN site percolation or ran- +dom boundary-percolation have not been previously measured. +In principle they could +differ from site percolation, however in three dimensions we find below that the critical +exponents for random boundary percolation appear to be the same as for ordinary site +percolation. Probably the same is true in two dimensions, but we did not perform that +measurement. +0.37 +0.38 +0.39 +0.40 +0.41 +0.42 +0.43 +0.44 +0.45 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +k* +L +1/L +0.2475 +0.2480 +0.2485 +0.2490 +0.2495 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +k* +L +1/L +Figure 2: Finite-size shift of the boundary percolation threshold for the 2D (a) and 3D (b) Ising +model. Error bar ranges are 1/10 to 1/20 of symbol size. +In two dimensions minority sites never percolate. In three dimensions there are a lot +more paths. Majority sites always percolate and minority sites percolate in the random +phase and about the first 5% of the ferromagnetic phase, measured by temperature. Even- +tually minority sites get too few and percolation is lost at κ = 0.2346(13) where the mag- +netization is about 0.62[4]. There is no visible effect on other quantities at this point. For +comparison, the ferromagnetic transition is at κ = 0.2216595(26) [5]. Because in boundary +5 + +percolation the clusters are more liberally defined, it persists even further into the ferro- +magnetic phase. From the fit to Fig. 2b we find κc = 0.24781(4) and ν = 1.30(3). Here the +magnetization is about 0.7364. This means that 13.18(4)% of the sites are minority, which +is about 4% lower than the value mentioned above, pc = 0.1372, for random NN+NNN site +percolation (our study of random boundary percolation below also corroborates this value). +The value found here for the exponent ν is particularly interesting. It is not at all close to +the correlation length exponent for random site percolation ν ≃ 0.88[6, 7] measured from +the finite-size scaling of the infinite cluster. For random boundary percolation we find a +similar value below. Even in the 3D Ising case where interactions could change the result +we still find scaling of the largest cluster gives ν ≃ 0.87 (detailed below). We are led to +conclude that a different correlation length is controlling the finite-lattice shift exponent in +this case. This makes the case of boundary percolation in the 3D Ising model of consider- +able theoretical interest, because a system with two different correlation lengths diverging +at the same place is, to say the least, unusual. +0.1360 +0.1362 +0.1364 +0.1366 +0.1368 +0.1370 +0.1372 +0.1374 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +p* +L +1/L +Figure 3: Finite-size shift of boundary percolation threshold for 3D random percolation model. +Error bar range is about 1/8 symbol size. +Now we consider the case of random boundary percolation. This is an interaction-free +model where positive sites are placed at random in the lattice, with the fraction of positive +sites given as p. The remaining sites are, of course, set negative. Fig. 3 shows the finite-size +shift of percolation threshold, p∗ +L. From the scaling relation given above we find the infinite +lattice threshold as pc = 0.13730(4), which is fairly close to the percentage of positive sites +at the 3D Ising boundary percolation threshold (they differ by 4%). However, the exponent +here is quite different from the 3D Ising value of 1.30(3). We find ν = 0.91(5), consistent +with well-known measurements of the site-percolation exponent. One can also determine ν +from scaling of the largest cluster. If one defines P to be the fraction of plaquettes occupied +by the largest cluster, then the same finite-size scaling analysis as is usually applied to the +magnetization in a magnetic system undergoing a thermal phase transition can be applied +[8] to P, its susceptibility +6 + +χ = (< P 2 > − < P >2)Np +(4) +and the corresponding Binder fourth-order cumulant +U = (< P 4 > − < P 2 >2)/(3 < P 2 >2). +(5) +Here Np is the number of plaquettes in the lattice. The correlation-length scaling hypoth- +esis implies that these should collapse onto universal functions if scaled according to their +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +0.136 +0.137 +0.138 +0.139 +0.140 +U +p +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +0.136 +0.137 +0.138 +0.139 +0.140 +c +p +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.136 +0.137 +0.138 +0.139 +0.140 +P +p +Figure 4: Boundary percolation study in the 3D random percolation model. Binder cumulant U (a), +susceptibility χ (b), and fraction of sites occupied by largest cluster P (c) vs. fraction of positive +sites, p. Triangles are 403, boxes 643, ×’s 1003, and open circles 1283. Error bar ranges for U are +about 1/30 the size of plotted points, between 1/4 and 1/10 for χ, and 1/15 for P. +7 + +respective exponents and plotted against the scaling variable +x = (p − pc)L1/ν +(6) +where L is the linear lattice size. Fig. 4abc shows the data for U, χ and P as a function +of the concentration of positive links p for lattices of size 403, 643, 1003, and 1283. All +datapoints are from samples of 100,000 randomly generated lattices. One sees a crossing +in U, similar to the case of a thermal phase transition. Here the crossing point marks +the infinite-lattice percolation threshold. Fig. 5ab shows the scaling collapse plots, where +ν, β, γ and pc are adjusted to give the best collapse. Here the scaled χ is χL−γ/ν and +scaled P is PLβ/ν. Although a good fit can be achieved using all four lattice sizes, a small +systematic shift was seen in exponents toward typical percolation values when the 403 data +were omitted, suggesting a small correction-to-scaling effect of order the random error. For +this fit there are 65 degrees of freedom overall and the fit to the three universal functions (in +this case power laws) has χ2/d.f= 0.77. The fit gives ν = 0.872(4), β/ν = 0.472(3), γ/ν = +2.056(4) and pc = 0.137317(5). The latter agrees with that determined from finite-size shift +above as well as with the threshold previously measured for NN+NNN site percolation, +pc = 0.1372(1)[3], which we believe to be equivalent to boundary percolation. As far as +we know exponents have not been previously measured for these cases. The quantities γ/ν +and β/ν should be related by the hyperscaling relation +γ/ν + 2β/ν = d +(7) +where d is the spatial dimension. +Our values give, for the LHS, 3.001(7). +For ran- +dom site percolation a fairly recent high statistics study gives ν = 0.8764(11) and +β/ν = 0.47705(15)[6]. Comparing with our result leads to the conclusion that all of the +exponents for random boundary percolation likely match those of ordinary site percolation. +For random boundary percolation, finite-size shift and largest cluster scaling give con- +sistent measurements of ν. However that is not the case for boundary percolation in the +3D Ising model. Figs. 6abc and 7ab analyze the largest cluster scaling for boundary perco- +lation in the 3D Ising model in the same way as above. Of course now the abscissa is the +Ising coupling strength κ. The study was similar to the above but with 1,000,000 sweeps +per point, sampled every 10, and 200,000 initial equilibration sweeps on 403, 643 and 1003 +lattices. Again we have a Binder cumulant crossing and excellent scaling collapse plots. +These give κc = 0.247925(6), ν = 0.867(5), β/ν = 0.465(5), and γ/ν = 2.068(20). So there +are no surprises here as these exponents are consistent with the random percolation values. +The fraction of minority sites at κc is 0.1318(4), about 4% lower than for random percola- +tion. However, the finite-size-shift exponent, ν, obtained above from the fit to Fig. 2b was +1.30(3) . This is clearly incompatible with the percolation value just obtained from largest +cluster scaling in the same system. This would seem to indicate that some other dynam- +ics has taken over the scaling of the finite-size shift, driven by another correlation length +which is becoming infinite at a different rate. One possibility for how this could happen +is if the percolation is linked to a thermal phase transition which has its own correlation +8 + +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +Scaled P +U +x +0.05 +0.06 +0.07 +0.08 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Scaled c +x +Figure 5: Scaling collapse plots for boundary percolation in the 3D random percolation model. +Binder cumulant (left graph) and scaled largest-cluster fraction (a), and scaled susceptibility (b). +length controlled by different dynamics. This is a very curious behavior that invites further +investigation because, as previously mentioned, it is quite unusual for a system to have two +different correlation lengths. +3 +Dual order parameter +As is well known, percolations are not necessarily coincident with phase transitions, but +sometimes are. The situation is clearer if a symmetry-breaking order parameter exists. In +that case an energy singularity follows from the hyperscaling relation α = 2 − dν, where +α is the specific heat exponent, d is the number of dimensions and ν is the correlation +length exponent associated with the order parameter near the symmetry-breaking phase +transition. The spontaneous breaking of an exact symmetry is always associated with a +mathematical singularity because the order parameter is exactly zero in the unbroken phase, +and is non-zero in the broken phase[9]. A function which is zero over a range of values can +only become non-zero at a point of non-analyticity. +In order to build further evidence of a phase transition at the point of boundary per- +colation, one can examine the dual theory, the three-dimensional gauge Ising model. This +has action +S = −β +� +p +Up +(8) +where Up is the product of four gauge fields Uµijk around an elementary plaquette. The Uµijk +exist on links with µ a direction index and ijk the site address. The duality relation maps +the coupling of the Ising model κ to β of the dual gauge theory, β = −0.5 ln(tanh(κ))[10]. +The ordered phase of the spin theory maps to the disordered (confining) phase of the +gauge theory. Generally it is not considered that there is a local symmetry-breaking order- +parameter in gauge theories, because Elitzur’s theorem[11] does not allow a local symmetry +9 + +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.244 +0.245 +0.246 +0.247 +0.248 +0.249 +0.25 +U +k +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +0.244 +0.245 +0.246 +0.247 +0.248 +0.249 +0.25 +c +k +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.244 +0.245 +0.246 +0.247 +0.248 +0.249 +0.25 +P +k +Figure 6: Boundary percolation study in the 3D Ising model. Binder cumulant U (a), susceptibility +χ (b), and fraction of sites occupied by largest cluster P (c) vs. coupling κ. Error bar ranges for U +are about 1/30 the size of plotted points, 1/20 for P, and between 1/5 and 1/20 for χ. +to break spontaneously. However, if one transforms configurations to Coulomb gauge then +a symmetry-breaking order parameter may be defined, for which the remnant symmetry +breaks in the deconfined phase[12]. The Coulomb gauge transformation seeks to maximize +the number of positive links in the one and two directions, ignoring the third direction +links. +This leaves a remnant layered Z2 symmetry. +Two-dimensional global symmetry +operations applied to single 1-2 layers do not alter the one and two direction links on +which Coulomb gauge is defined, but flip all third direction links attached to the layer. +For fixed one and two direction links the third direction links have mostly ferromagnetic +interactions from plaquettes with two positive one or two direction links, especially at high +β. If one takes the third direction links in each separate layer as order parameters, it is +found that these magnetize exactly at the dual-reflection of the 3-d Ising critical point[13]. +10 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +-0.4 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +P-scale +U +x +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +c-scale +x +Figure 7: Scaling collapse graphs for percolation in the 3D Ising model. Binder cumulant (left +graph) and scaled largest-cluster fraction (a), and scaled susceptibility (b). +The deconfined phase is magnetized and the confined phase is not. The dual reflection of the +boundary-percolation point lies in the confined phase, ie. the non-magnetized phase of the +gauge theory. If there is a symmetry-breaking phase transition here it must be a spin-glass +transition, which is a symmetry-breaking transition within the unmagnetized phase. A spin +glass has a hidden pattern of order which does not result in an overall magnetization. To +search for such a transition we used a two-real-replica approach[15]. A second set of third- +direction pointing links is equilibrated to a fixed pattern of one and two direction links +from the main simulation. This is similar to the initial equilibration for any Monte-Carlo +simulation. Then the order parameter is defined as +qk = +� +i,j +R3ijkU3ijk. +(9) +Here R3ijk is the replica third-direction link at site ijk and U3ijk is the original one. Note +there is a separate qk for each 2D layer, because the symmetry being broken is only global +in two directions but still local in the third direction. As is usual one needs to take the +absolute value of the order parameter due to tunneling on the finite lattices. We also choose +to take the square root of the order parameter since it is the product of two spins, but this +is not absolutely necessary. Thus the average spin-glass magnetization to be analyzed is +M ≡< +� +|qk| > +(10) +where the average is both over gauge configurations as well as third direction fixed 2D +layers in each gauge configuration. The order parameter M will become non-zero in a phase +with either spin-glass order or ferromagnetic order. Spin-glass order is symmetry breaking +because the symmetry operation applied only to the original U’s but not the replicas will +invert the order parameter. Another way to say this is that tunneling configurations within +11 + +the replica or original, where half of the lattice is flipped, do not exist in the spin-glass phase +in the thermodynamic limit. For systems without a spin-glass phase this order parameter +will simply turn on at the normal ferromagnetic transition (for instance, this is the case +for Landau-gauge Higgs phase transitions in the combined Ising gauge-Higgs theory[13]). +Note also that although its original motivation was from Coulomb gauge, qk itself is gauge +invariant, so it is no longer necessary to fix the gauge. As detailed below, we indeed find a +phase transition in M away from the ferromagnetic transition indicating the presence of a +spin-glass phase. Here we can use all of the finite-size scaling techniques which have been +developed for studying symmetry-breaking phase transitions with a local order parameter. +0.7 +0.71 +0.72 +0.73 +0.74 +0.75 +0.76 +0.77 +0 +20000 +40000 +60000 +80000 +100000 +M +Equilibration sweeps +0.61 +0.615 +0.62 +0.625 +0.63 +0.635 +0.64 +0 +20000 +40000 +60000 +80000 +100000 +U +Equilibration Sweeps +Figure 8: Equilibration of spin-glass order parameter and Binder cumulant on a 303 lattice. +Before studying the spin-glass order parameter M in Monte Carlo simulations, one +must first perform an equilibration study to determine how long the replica must be equili- +brated to obtain a truly independent configuration. One simply simulates at many different +equilibration sweep values and watches the measured quantities approach constant values +exponentially. We then picked equilibration amounts that insure systematic errors are less +than 25% of random errors in the quantities measured. Detailed studies were made at gauge +coupling β = 0.705, near the suspected critical point, for both the 303 and 503 lattices. The +equilibration value for the 403 lattice was determined from these and the volume scaling +suggested by them. Fig. 8ab shows the equilibration of magnetization (order parameter) +and its Binder cumulant for the 303 lattice. Other quantities were similar. The exponential +fits give an equilibration time constant of 14,000 sweeps. By equilibrating with 105,000 +sweeps systematic errors are brought to less than 25% of random in the planned simula- +tions. For 503 this value was a bit surprisingly high at 700,000 sweeps. We used 190,000 +sweeps for the intermediate 403 case. These high equilibration values indicate the standard +heat bath Monte Carlo algorithm is not working particularly well here, but it still gives +good results if one is patient. The high number of sweeps to equilibrate are due to the fact +that 2/3 of the links, those lying in the 1 and 2 directions are being held fixed, which erects +more barriers that a simulation where all links participate. +12 + +0.3 +0.5 +0.7 +0.9 +1.1 +1.3 +0.4 +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +0.66 +0.68 +0.70 +0.72 +0.74 +M +U +b +0.00 +0.50 +1.00 +1.50 +2.00 +2.50 +0.66 +0.68 +0.7 +0.72 +0.74 +x2nd/L +b +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0.66 +0.68 +0.7 +0.72 +0.74 +c +b +Figure 9: Binder cumulant (left graph) and magnetization (a), ξ2nd/L (b), and susceptibility (c) for +the spin-glass order parameter. Error bar spreads for U and M are about 1/15 the size of plotted +points, and 1/2 to 1/5 for ξ2nd/L and χ. +All simulations had 100 ordinary Monte Carlo sweeps between each measurement of +the order parameter to reduce correlations. There were 1000 measurements for each 303 +and 403 lattices and 500 for 503. +Initial equilibration was 200,000 sweeps. +Error bars +were determined from binned fluctuations. Fig. 9abc shows the Binder cumulant U, order +parameter M, susceptibility χ, and second moment correlation length[16] divided by lattice +size, ξ2nd/L, for the three lattices. The latter, as well as the Binder cumulant, should cross +near the infinite lattice transition point (to determine this precisely one must consider +corrections to scaling which we do not do here). One can see a well-defined crossing in both +near βc = 0.715. The crossings are well established. For instance the 503 value exceeds +the 303 value at β = 0.725 by 15σ for U and 10σ for ξ2nd/L, and points above this have +similar significances. The opposite order in the low β region is never in doubt. Indeed, here +points here are separated by even larger amounts, exceeding 30σ. Scaling collapse plots are +13 + +shown in Fig. 10abc. The overall fit has 75 degrees of freedom and has a χ2/d.f.= 1.48 . +This fit gives βc = 0.7174(3), ν = 1.27(3), β/ν = 0.058(2), and γ/ν = 1.86(2). Checking +hyperscaling on the latter give deff = γ/ν + 2β/ν = 1.97(2). Because the order parameter +is defined on 2-d layers, the expected value is 2. +The dual reflection of the boundary +percolation point of the 3D Ising model itself is −0.5 ln tanh(0.247925) = 0.70741(1). This +is close to the βc here, but certainly not an exact match, and not within statistical errors. +However there could be a systematic error present from corrections to scaling. Looking at +the U crossing (Fig. 9a), it is plausible that the crossing on larger lattices could shift to this +point. Corrections to scaling can give a slightly shifting crossing with increasing lattice size. +There is also the possibility of a residual systematic error from insufficient equilibration. +Although we have tried to limit this to 25% of the random error it could still have an +effect. The fact that the ν seen here and the ν from the coupling-shift of the percolation +transition, 1.30(3) agree within 1σ strongly supports these being dual-manifestations of the +same transition. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.5 +0.55 +0.6 +0.65 +0.7 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Scaled M +U +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +x2nd/L +x +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Scaled c +x +Figure 10: Scaling collapse plots for Binder cumulant (left graph) and scaled magnetization (a), +ξ2nd/L (b), and scaled susceptibility (c), for the spin-glass order parameter. +14 + +4 +Energy moments +Since the spin-glass transition in the dual theory is symmetry breaking, Landau theory +connects this to a thermal phase transition through the hyperscaling relation +α = 2 − dν. +(11) +Here α is the specific heat exponent. At the critical point the expected behavior of the spe- +cific heat is |T − Tc|−α. For ν = 1.3, α = −1.9. This means that the specific heat does not +have an infinite singularity, however it does have a finite singularity. Unfortunately, when +rounded by a finite lattice size, these are difficult to spot using finite-size scaling. Never- +theless one can still try to fit to a fractional power, and in some cases more importantly, +a different coefficient on each side of the transition. However, the energy moments also +have non-singular terms. This makes fitting them more difficult than quantities based on +the order parameter which are purely singular. The non-singular part is expected to vary +slowly through the critical region. For this reason it affects higher moments less, and there +is a good chance these can be fit without a non-singular part other than perhaps a constant. +This simplifies fitting to the expected critical behavior. A study of energy moments of the +3D Ising model itself was performed. This study had approximately 7 × 109 sweeps at each +coupling for the 303 lattice and 2 × 109 for the 403 and 503, with measurements performed +every other sweep. With these statistics, rather precise data can be obtained on the third +cumulant (third central moment), defined as < (E − ¯E)3 > (3L3)2. It is this combination +that corresponds to the derivative of the specific heat. The third cumulant is expected to +scale as |κ − κc|−α−1, which is still a non-infinite singularity. This quantity is shown in +Fig. 11, along with a fit to the expected critical behavior, but leaving α and κc as free +parameters. The fit also allows for a different coefficient on the two sides of the transition. +The result is κc = 0.2477(2), and −α−1 = 0.967(12). The coefficient ratio below and above +the critical point is 1.287(25). The predicted ν from this α is ν = (−α + 2)/3 = 1.322(4). +The critical point agrees well with that extracted from percolation(0.24781(4), and the +exponent ν also agrees with those from both percolation finite-size shift (1.30(3)) and the +spin glass transition in the dual gauge theory(1.27(3)). Even though the singularity is non- +infinite, it can still be seen from this fit. It is important to remember that these functions +are singular in two ways - the fractional power and the jump in coefficient. So even if the +power were to end up being exactly unity, that would not erase the singularity due to the +fairly large coefficient jump, verified to 11.5σ, which can be seen in the change of slope. +Higher moments were also measured, but even with these high statistics were somewhat +of a disappointment due to fairly large statistical errors. Fig. 12 shows the fourth cumulant, +(< (E − ¯E)4 > −3 < (E − ¯E)2 >2)(3L3)3. +(12) +This combination of moments tracks the third derivative of the internal energy with respect +to κ. Also shown is a numerical derivative of the fit function to the third cumulant on a +parameter spacing 1/4 of that used for the simulations. This was done instead of an exact +derivative to simulate finite-lattice rounding, so not an exact prediction of the expected +15 + +-210 +-200 +-190 +-180 +-170 +-160 +-150 +-140 +-130 +0.243 +0.245 +0.247 +0.249 +0.251 +0.253 +Third Energy Cumulant +k +Figure 11: Third order energy cumulant, with fit to critical behavior. Here open circles are 303, +open triangles 403, and × 503. +behavior, but one which should be good away from the critical point. The main effect of +the shift in slope in the third cumulant which translates to a shift in level here can be seen. +In principle some finite-size effect could be seen in this quantity since it diverges with a +very small exponent, but the expected ratio in peak heights between 303 and 403 is only +(4/3)((α+2)/ν) = 1.02, much smaller than our statistical errors. A larger effect is predicted +for the fifth cumulant (1.28), but the errors there are magnified even more. This figure +is shown primarily to illustrate how much further one would have to go in statistics to +see an infinite singularity in a high moment. Our program, which was run for about 24 +processor-years on PC’s, does not employ multi-spin coding. Perhaps a study that did or +used specialized hardware could see these effects. +5 +Conclusion +In this paper evidence has been given for a new high-order phase transition within the or- +dered phase of the 3D Ising model. This transition appears to be associated with boundary +percolation. This is the percolation of dual-plaquettes that lie on the domain boundary +between + and - spins, a type of percolation that has not been much studied. Percolation +of domain boundaries occurs when minority sites occupy 13% or more of the lattice. It is, +incidentally, not coincident with the roughening transition which occurs much deeper into +the ordered phase, around κ = 0.408[17]. Because the Ising model has a formulation in +terms of the domain boundary itself, the percolation of the boundary could be important, +16 + +3000 +4000 +5000 +6000 +7000 +8000 +9000 +0.244 +0.246 +0.248 +0.25 +0.252 +Fourth Energy Cumulant +k +Figure 12: Fourth order energy cumulant for 303 and 403 lattices. Line is a plausible rounded critical +behavior based on third cumulant fit (see text). +possibly producing a sudden change in the entropy function expressed in terms of boundary +area. +Random boundary percolation appears to have the same critical exponents as ordinary +site percolation. Boundary percolation in the Ising model seems to model random boundary +percolation as far as the scaling of cluster sizes is concerned, however it differs in the finite- +size shift exponent, which determines how the percolation threshold depends on lattice +size. Whereas random percolation has a shift exponent agreeing with typical values of the +correlation-length exponent from cluster size scaling (ν ∼ 0.88), the shift exponent from +the 3D Ising model boundary percolation is vastly different, ν ∼ 1.3. This surprising result +means that the system has two different correlation lengths, both diverging at the infinite- +lattice percolation threshold. This also suggests that there is more than just percolation +going on here. If percolation is linked to a thermal phase transition, that could explain +the odd shift exponent, since the order parameter of the phase transition may have its own +correlation length. +To find such an order parameter we examined the dual system, the 3D gauge Ising +model. The dual point of the boundary percolation threshold occurs in the random (con- +fining) phase of the gauge theory. An order parameter for the confinement-deconfinement +transition can be obtained in Coulomb gauge, where as many one and two direction links +as possible are made to be positive by gauge transformations. The third direction links on +each lattice layer can be taken to be a spin-like order parameter, which shows spontaneous +magnetization in the ordered phase and is unmagnetized in the random phase. If there is +a phase transition corresponding to boundary percolation in the Ising model itself, it must +occur within the random phase of the gauge theory. This suggests looking for a spin-glass +transition here, a shift from a completely disordered phase to one with a hidden pattern +of order, but still showing no net magnetization. To this end we utilized a two-real-replica +17 + +order parameter, which indeed does show a phase transition near the dual reflection of +boundary percolation, and with the same critical exponent ∼ 1.30. This is significant be- +cause it is a true symmetry-breaking phase transition. The symmetry being broken is the +layered remnant (Z2)L symmetry left over after Coulomb gauge fixing, which is global in +two dimensions but still local in the third. This is “global enough” to avoid Elitzur’s theo- +rem and has sufficient dimensions (2) for a discrete symmetry to break spontaneously at a +finite coupling. Spontaneous symmetry-breaking always results in a phase transition, i.e. a +mathematical singularity in the order parameter, which also results in an energy singularity +except in a few unusual cases [14]. +Finally we examined energy moments in search of this singularity. +Because of the +high value of ν the specific heat exponent is negative, implying a finite singularity, so the +usual finite-size scaling applied to peak heights cannot be used here. We concentrated on +the third energy cumulant, since it could be fit without the addition of an obfuscating +non-singular part, other than a constant. An open fit to the singular form expected for +this quantity based on the hyperscaling relationship, gives κc and ν values consistent with +those determined by boundary percolation and the dual order parameter. There is also +a noticeable jump in coefficient here, another expectation of this sort of singularity. Our +study did not have enough statistics to see the small expected peak scaling in the fourth +cumulant or somewhat larger effect in the fifth, which should have infinite singularities +on the infinite lattice. +Although observing these would be satisfying, still the singular +fit to the third cumulant does match well with the prediction from the order parameter. +This demonstrates that phase transitions as weak as these can be studied by numerical +methods. The existence of an order parameter and associated symmetry breaking is key +in establishing this as a true phase transition. The coincidence of boundary percolation is +also interesting and gives another measure of ν, but cannot by itself be used to imply the +presence of a phase transition. However, it has the advantage of being very easy to measure. +It appears to have the same cluster-size scaling exponents as ordinary site percolation, but a +different threshold. It may be interesting to explore boundary percolation in other systems. +Since percolation has so many practical applications, it’s possible boundary percolation is a +better fit than site or link percolation in some cases. Finally, we note that a previous study +of the Z2 gauge-Higgs system showed a total of four phase transition lines further into the +diagram[13]. The current paper shows there are two phase transitions on each axis, gauge +and Higgs, so also a total of four. It will be interesting to follow these new phase transitions +into the phase diagram to see how they connect with the lines previously found. +The +previous paper showed that the Z2 gauge-Higgs system appears to be more complicated +than previously thought. +The present paper shows that these additional complications +extend to the 3D Ising model itself, and its dual, the 3D gauge Ising model. +It seems +possible that similar weak phase transitions may also be lurking in other well-known spin +and gauge systems. +18 + +References +[1] R.P. Feynman, Statistical mechanics - a set of lectures, Addison-Wesley, Reading MA, +1998, ch.5. +[2] M. Kac and J.C. Ward, A combinatorial solution of the two dimensional Ising model, +Phys. Rev. 88, 1332-1337 (1952). +[3] L. Kurzawski and K. Malarz, Simpe cubic random-site percolation thresholds for com- +plex neighborhoods, Rep. Math. Phys. 70, 163-169 (2012); C. Domb and N.W. Walton, +Crystal statistics with long-range forces I. The equivalent neighbor model, Proc. Phys. +Soc. 89, 859-871 (1966). +[4] H. M¨uller-Krumbhaar, Percolation in a lattice system with particle interaction, Phys. +Lett. A 50, 27-28 (1974). +[5] A.M. Ferrenberg and D.P. Landau, Critical behavior of the three-dimensional Ising +model: A high-resolution Monte Carlo study, Phys. Rev. B 44, 5081-5091 (1991). +[6] J. Wang, Z. Zohu, W. Zhang, T.M. Garoni, and Y. Deng, Bond and site percolation +in three dimensions, Phys. Rev. E 87, 052107 (2013); Erratum, 89, 069907 (2014). +[7] D. Stauffer and A. Aharony, Introduction to Percolation Theory, Revised 2nd edition, +Taylor and Francis, London, 1994. +[8] K. Binder and D.W.Heermann, Monte Carlo simulation in statistical physics - an +introduction, 6th ed., Springer Nature, Cham Switzerland, 2019. +[9] L.D. Landau and E.M. Lifshitz, Statistical Physics - Vol. 5 of the Course of Theoretical +Physics, Pergamon Press, London, 1958, p452. +[10] H.A. Kramers and G.H. Wannier, Statistics of the two-dimensional ferromagnet Part 1, +Phys. Rev. 60, 252-262 (1941); R. Savit, Duality in field theory and statistical systems, +Rev. Mod. Phys. 52, 453-487 (1980). +[11] S. Elitzur, Impossibility of spontaneously breaking local symmetries, Phys. Rev. D12, +3978-3982 (1975) . +[12] J. Greensite, S. Olejn´ık, and D. Zwanziger, Coulomb energy, remnant symmetry, and +phases of non-Abelian gauge theories, Phys. Rev. D 69, 074506 (2004); D. Zwanziger, +No confinement without Coulomb confinement, Phys. Rev. Lett. 90, 102001 (2003). +[13] M. Grady, Exploring the 3D Ising gauge-Higgs theory in exact Coulomb gauge and +with a gauge-invariant substitute for Landau gauge, arXiv:2109.04560 (2021). +[14] ibid Appendix A. +[15] K. Binder and W. Kob, Glassy Materials and Disordered Solids, World Scientific, New +Jersey, 2005, pp. 248, 261. +19 + +[16] F. Cooper, B. Freedman, and D. Preston, Solving φ4 +1,2 field theory with Monte Carlo, +Nucl. Phys. B 210, 210-228 (1982); D.J. Amit and V. Mart´ın-Mayor, Field Theory, +the Renormalization Group, and Critical Phenomena: Graphs to Computers, 3rd ed., +World Scientific, Singapore, 2005. +[17] K.K. Mon, S. Wansleben, D.P. Landau and K. Binder, Anisotropic surface tension, +step free energy, and interface roughening in the three-dimensional Ising model, Phys. +Rev. Lett. 60, 708-711 (1988); Erratum, 61, 902 (1988); K.K Mon, D.P. Landau, and +D. Stauffer, Interface roughening in the three-dimensional Ising model, Phys. Rev. B +42, 545-547 (1990). +20 + diff --git a/4dFAT4oBgHgl3EQfExzK/content/tmp_files/load_file.txt b/4dFAT4oBgHgl3EQfExzK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8273659c7e35c587dfd91c88829adfaed9cfaff9 --- /dev/null +++ b/4dFAT4oBgHgl3EQfExzK/content/tmp_files/load_file.txt @@ -0,0 +1,819 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf,len=818 +page_content='Possible new phase transition in the 3D Ising Model associated with boundary percolation Michael Grady Department of Physics State University of New York at Fredonia Fredonia NY 14063 USA grady@fredonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='edu January 23, 2023 Abstract In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Its critical exponent ν ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 is seen to match the finite-size shift exponent of the percolation transition further cementing their connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This predicts a very weak specific heat singularity with exponent α ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest- cluster scaling and the other with finite-size transition-point shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This suggests there are two different correlation lengths present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' PACS: 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='50+q, 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='Jk, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='ah, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='F Keywords: Ising model, Gauge Ising model, spin glass, percolation, phase transition arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='08424v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='stat-mech] 20 Jan 2023 1 Introduction The Ising models in two and three dimensions are the most basic spin models which undergo order-disorder transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' These have been extremely well studied and, of course, an exact solution exists in the two-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The 3D model has always been a bit more of a mystery, and in this paper we explore the possibility of a weak secondary phase transition within the ordered phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Presumably this is associated with some geometrical change in spin-clustering, but the exact nature of this reordering is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The situation is clearer in the dual theory, the 3D gauge Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Here the suspected transition is in the disordered phase, and clearly has the nature of a spin-glass transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In other words we have identified a symmetry-breaking order parameter in the dual theory, but not in the Ising model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However, the Ising model does exhibit an interesting percolation phenomenon near the critical point predicted by duality from the spin-glass transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is a percolation of the domain boundaries between + and - spin clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' As shown below, boundary percolation can be considered a third type of percolation, beyond site and link percolation, although it has a close relationship to an unusual type of site percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Of course percolation is not always related to a phase transition, but sometimes it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It’s linkage in this case to a symmetry-breaking transition in the dual theory provides strong evidence that it is associated with a phase transition in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The argument is further strengthened by independent fits of the third energy cumulant to consistent critical behavior about the suspected critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Each investigation, the order-parameter in the dual theory, the boundary percolation finite-size shift, and the third energy moment, yields an independent determination of the critical exponent ν, all of which agree to a fairly close tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In the following, first the boundary percolation concept is fleshed out and studied in both the 2D and 3D Ising models, as well as for 3D random percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It is found that the latter has the same critical exponents as for site percolation, and in fact is equivalent to site percolation if next to nearest neighbors are included in the cluster definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The 3D Ising case is particularly interesting in that an analysis of the finite-size shift in percolation threshold gives a critical exponent very different from the percolation value, even though the cluster scaling still obeys the percolation exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This suggests it is linked to a phase transition with its own dynamical scaling and correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Then we move on to the dual theory and introduce the spin-glass order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A Monte Carlo study here shows clear crossings in the Binder cumulant and second-moment correlation length divided by lattice size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Correlation-length finite-size scaling is exhibited around the suspected critical point using scaling collapse plots, which also yield critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The critical exponent ν is found to match well with that found from the finite-size shift in percolation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Finally, we study energy moments, such as specific heat and higher moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Unlike an order parameter, these have both critical and non-critical pieces, so fitting can be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This leads to the selection of the third energy cumulant as the best prospect for finding critical behavior as it can be fit without a non-critical part other than a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A Monte Carlo study with several times 109 sweeps per point on 303, 403, and 503 lattices yields a precise determination of this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' An independent critical behavior fit in the region of the suspected critical point gives values for both ν and κc which agree well with the two 2 other predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A substantial jump in the coefficient of the critical scaling fit across the transition further cements evidence for a thermal singularity here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The rather high value of ν ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 gives a highly negative value for the specific heat exponent α = 2 − dν ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This means that both the specific heat and third cumulant have finite singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A very weak infinite singularity is expected in the fourth cumulant and stronger ones in fifth and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In the Ehrenfest classification this transition is fourth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' We attempted to measure fourth and fifth cumulants to find evidence of peaks growing with lattice size as expected from infinite singularities, but even with the rather large sample size here, these were still largely obscured by random error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However, finite singularities are just as singular as infinite ones, so perhaps one lesson is that one should not necessarily obsess over trying to find infinite singularities in transitions of such high order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Figure 1: Example of a boundary cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Sites marked with dots have opposite orientation to all surrounding sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 3 2 Boundary percolation The standard partition function for the Ising model is Z = � {σ} exp(κ � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' σiσj), (1) where the σ’s are classical spins taking values ±1 and the coupling is between nearest neighbors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' There is a well-known reformulation of the Ising models in terms of the boundaries themselves[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This reformulation even leads to an alternate exact solution in the 2D case[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The partition function can be written Z = � A N(A) exp(−κA) (2) where A is the total area of dual boundary plaquettes (or dual boundary links in 2D) in a configuration and N(A) are the number of distinct non-intersecting boundary configura- tions with that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In this formulation there are no spins or domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Only the boundary surfaces need exist, and the entropy associated with these surfaces controls the phase tran- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is one reason why percolation of the domain boundary might be important for this model, as opposed to, say, site percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For instance, the density of states could change abruptly when an infinite boundary cluster forms, because for a finite cluster the area is usually an increasing function of the volume, whereas an infinite cluster can easily grow in volume without adding much to the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' If this were the case then the free energy would form a singularity at the boundary percolation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' We define boundary percolation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Consider the set of all boundary links that connect + and - sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' These are each associated with a plaquette on the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' These plaquettes form closed surfaces separating clusters of + and - spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' These surfaces can form clusters themselves, if we define boundary clusters to be made up of boundary surfaces that share dual-lattice links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For instance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 1 shows a single boundary cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This same configuration would, however, count as two separate site-clusters, since sites are clustered only along lattice directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' If the site cluster concept is extended to include sites connected by face diagonals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='e next nearest neighbors (NNN) in addition to near- est neighbors (NN), then these redefined site clusters would appear to coincide with the boundary cluster concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Indeed we have verified for thousands of configurations that those with percolating boundaries also have percolating NN+NNN site-clusters, and vice versa, so they do appear to measure the same thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It seems boundary percolation, which in three dimensions could also be called plaquette percolation, has only been studied before in the form of the equivalent extended NN+NNN site percolation problem[3], as a part of surveys of various extended percolation models, but never applied to the Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The main result of these studies is establishing a threshold for random NN+NNN percolation at a minority site probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1372(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Because the Ising model is interacting, correlations would be expected to modify this result, but still it should be kept in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 2ab shows the evolution of the Ising model boundary percolation threshold κ∗ L with lattice size L for both two and three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' These both 4 scale well with the finite-size scaling relation κ∗ L = κc − cL−1/ν (3) where κc is the infinite lattice threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The percolation threshold is defined here as the point where 50% of lattices have a cluster which percolates in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For two dimen- sions, boundary percolation exists in the random phase, and ceases in the ferromagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The above fit gives κc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='4405(5) which agrees well with the known ferromagnetic transition point 1 2 ln( √ 2 + 1) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='44069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is just the opposite of majority-site perco- lation, which happens only in the magnetized phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Thus in two dimensions boundary- percolation and site-percolation seem equally relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The exponent derived from the finite-size scaling fit to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 2a is ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='261(18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This seems slightly different from the standard 2D site-percolation exponent ν = 4/3 but of course that is for a non-interacting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' There could also be a small correction from next to leading order scaling effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' As far as we know, the critical exponents for random NN+NNN site percolation or ran- dom boundary-percolation have not been previously measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In principle they could differ from site percolation, however in three dimensions we find below that the critical exponents for random boundary percolation appear to be the same as for ordinary site percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Probably the same is true in two dimensions, but we did not perform that measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='38 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='025 k* L 1/L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2480 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='030 k* L 1/L Figure 2: Finite-size shift of the boundary percolation threshold for the 2D (a) and 3D (b) Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Error bar ranges are 1/10 to 1/20 of symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In two dimensions minority sites never percolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In three dimensions there are a lot more paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Majority sites always percolate and minority sites percolate in the random phase and about the first 5% of the ferromagnetic phase, measured by temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Even- tually minority sites get too few and percolation is lost at κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2346(13) where the mag- netization is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='62[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' There is no visible effect on other quantities at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For comparison, the ferromagnetic transition is at κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2216595(26) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Because in boundary 5 percolation the clusters are more liberally defined, it persists even further into the ferro- magnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' From the fit to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 2b we find κc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='24781(4) and ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='30(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Here the magnetization is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This means that 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='18(4)% of the sites are minority, which is about 4% lower than the value mentioned above, pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1372, for random NN+NNN site percolation (our study of random boundary percolation below also corroborates this value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The value found here for the exponent ν is particularly interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It is not at all close to the correlation length exponent for random site percolation ν ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='88[6, 7] measured from the finite-size scaling of the infinite cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For random boundary percolation we find a similar value below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Even in the 3D Ising case where interactions could change the result we still find scaling of the largest cluster gives ν ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='87 (detailed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' We are led to conclude that a different correlation length is controlling the finite-lattice shift exponent in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This makes the case of boundary percolation in the 3D Ising model of consider- able theoretical interest, because a system with two different correlation lengths diverging at the same place is, to say the least, unusual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1370 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='010 p* L 1/L Figure 3: Finite-size shift of boundary percolation threshold for 3D random percolation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Error bar range is about 1/8 symbol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Now we consider the case of random boundary percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is an interaction-free model where positive sites are placed at random in the lattice, with the fraction of positive sites given as p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The remaining sites are, of course, set negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 3 shows the finite-size shift of percolation threshold, p∗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' From the scaling relation given above we find the infinite lattice threshold as pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='13730(4), which is fairly close to the percentage of positive sites at the 3D Ising boundary percolation threshold (they differ by 4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However, the exponent here is quite different from the 3D Ising value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='30(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' We find ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='91(5), consistent with well-known measurements of the site-percolation exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' One can also determine ν from scaling of the largest cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' If one defines P to be the fraction of plaquettes occupied by the largest cluster, then the same finite-size scaling analysis as is usually applied to the magnetization in a magnetic system undergoing a thermal phase transition can be applied [8] to P, its susceptibility 6 χ = (< P 2 > − < P >2)Np (4) and the corresponding Binder fourth-order cumulant U = (< P 4 > − < P 2 >2)/(3 < P 2 >2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' (5) Here Np is the number of plaquettes in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The correlation-length scaling hypoth- esis implies that these should collapse onto universal functions if scaled according to their 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='140 U p 0 200 400 600 800 1000 1200 1400 1600 1800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='140 c p 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='140 P p Figure 4: Boundary percolation study in the 3D random percolation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Binder cumulant U (a), susceptibility χ (b), and fraction of sites occupied by largest cluster P (c) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' fraction of positive sites, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Triangles are 403, boxes 643, ×’s 1003, and open circles 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Error bar ranges for U are about 1/30 the size of plotted points, between 1/4 and 1/10 for χ, and 1/15 for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 7 respective exponents and plotted against the scaling variable x = (p − pc)L1/ν (6) where L is the linear lattice size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 4abc shows the data for U, χ and P as a function of the concentration of positive links p for lattices of size 403, 643, 1003, and 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' All datapoints are from samples of 100,000 randomly generated lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' One sees a crossing in U, similar to the case of a thermal phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Here the crossing point marks the infinite-lattice percolation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 5ab shows the scaling collapse plots, where ν, β, γ and pc are adjusted to give the best collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Here the scaled χ is χL−γ/ν and scaled P is PLβ/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Although a good fit can be achieved using all four lattice sizes, a small systematic shift was seen in exponents toward typical percolation values when the 403 data were omitted, suggesting a small correction-to-scaling effect of order the random error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For this fit there are 65 degrees of freedom overall and the fit to the three universal functions (in this case power laws) has χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='f= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The fit gives ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='872(4), β/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='472(3), γ/ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='056(4) and pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='137317(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The latter agrees with that determined from finite-size shift above as well as with the threshold previously measured for NN+NNN site percolation, pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1372(1)[3], which we believe to be equivalent to boundary percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' As far as we know exponents have not been previously measured for these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The quantities γ/ν and β/ν should be related by the hyperscaling relation γ/ν + 2β/ν = d (7) where d is the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Our values give, for the LHS, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='001(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For ran- dom site percolation a fairly recent high statistics study gives ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='8764(11) and β/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='47705(15)[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Comparing with our result leads to the conclusion that all of the exponents for random boundary percolation likely match those of ordinary site percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For random boundary percolation, finite-size shift and largest cluster scaling give con- sistent measurements of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However that is not the case for boundary percolation in the 3D Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 6abc and 7ab analyze the largest cluster scaling for boundary perco- lation in the 3D Ising model in the same way as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Of course now the abscissa is the Ising coupling strength κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The study was similar to the above but with 1,000,000 sweeps per point, sampled every 10, and 200,000 initial equilibration sweeps on 403, 643 and 1003 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Again we have a Binder cumulant crossing and excellent scaling collapse plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' These give κc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='247925(6), ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='867(5), β/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='465(5), and γ/ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='068(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' So there are no surprises here as these exponents are consistent with the random percolation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The fraction of minority sites at κc is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1318(4), about 4% lower than for random percola- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However, the finite-size-shift exponent, ν, obtained above from the fit to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 2b was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='30(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is clearly incompatible with the percolation value just obtained from largest cluster scaling in the same system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This would seem to indicate that some other dynam- ics has taken over the scaling of the finite-size shift, driven by another correlation length which is becoming infinite at a different rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' One possibility for how this could happen is if the percolation is linked to a thermal phase transition which has its own correlation 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='4 Scaled P U x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='4 Scaled c x Figure 5: Scaling collapse plots for boundary percolation in the 3D random percolation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Binder cumulant (left graph) and scaled largest-cluster fraction (a), and scaled susceptibility (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' length controlled by different dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is a very curious behavior that invites further investigation because, as previously mentioned, it is quite unusual for a system to have two different correlation lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 3 Dual order parameter As is well known, percolations are not necessarily coincident with phase transitions, but sometimes are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The situation is clearer if a symmetry-breaking order parameter exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In that case an energy singularity follows from the hyperscaling relation α = 2 − dν, where α is the specific heat exponent, d is the number of dimensions and ν is the correlation length exponent associated with the order parameter near the symmetry-breaking phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The spontaneous breaking of an exact symmetry is always associated with a mathematical singularity because the order parameter is exactly zero in the unbroken phase, and is non-zero in the broken phase[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A function which is zero over a range of values can only become non-zero at a point of non-analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In order to build further evidence of a phase transition at the point of boundary per- colation, one can examine the dual theory, the three-dimensional gauge Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This has action S = −β � p Up (8) where Up is the product of four gauge fields Uµijk around an elementary plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The Uµijk exist on links with µ a direction index and ijk the site address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The duality relation maps the coupling of the Ising model κ to β of the dual gauge theory, β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 ln(tanh(κ))[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The ordered phase of the spin theory maps to the disordered (confining) phase of the gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Generally it is not considered that there is a local symmetry-breaking order- parameter in gauge theories, because Elitzur’s theorem[11] does not allow a local symmetry 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='25 U k 0 100 200 300 400 500 600 700 800 900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='244 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='25 P k Figure 6: Boundary percolation study in the 3D Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Binder cumulant U (a), susceptibility χ (b), and fraction of sites occupied by largest cluster P (c) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' coupling κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Error bar ranges for U are about 1/30 the size of plotted points, 1/20 for P, and between 1/5 and 1/20 for χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' to break spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However, if one transforms configurations to Coulomb gauge then a symmetry-breaking order parameter may be defined, for which the remnant symmetry breaks in the deconfined phase[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The Coulomb gauge transformation seeks to maximize the number of positive links in the one and two directions, ignoring the third direction links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This leaves a remnant layered Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Two-dimensional global symmetry operations applied to single 1-2 layers do not alter the one and two direction links on which Coulomb gauge is defined, but flip all third direction links attached to the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For fixed one and two direction links the third direction links have mostly ferromagnetic interactions from plaquettes with two positive one or two direction links, especially at high β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' If one takes the third direction links in each separate layer as order parameters, it is found that these magnetize exactly at the dual-reflection of the 3-d Ising critical point[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 c-scale x Figure 7: Scaling collapse graphs for percolation in the 3D Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Binder cumulant (left graph) and scaled largest-cluster fraction (a), and scaled susceptibility (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The deconfined phase is magnetized and the confined phase is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The dual reflection of the boundary-percolation point lies in the confined phase, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' the non-magnetized phase of the gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' If there is a symmetry-breaking phase transition here it must be a spin-glass transition, which is a symmetry-breaking transition within the unmagnetized phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A spin glass has a hidden pattern of order which does not result in an overall magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' To search for such a transition we used a two-real-replica approach[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A second set of third- direction pointing links is equilibrated to a fixed pattern of one and two direction links from the main simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is similar to the initial equilibration for any Monte-Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Then the order parameter is defined as qk = � i,j R3ijkU3ijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' (9) Here R3ijk is the replica third-direction link at site ijk and U3ijk is the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Note there is a separate qk for each 2D layer, because the symmetry being broken is only global in two directions but still local in the third direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' As is usual one needs to take the absolute value of the order parameter due to tunneling on the finite lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' We also choose to take the square root of the order parameter since it is the product of two spins, but this is not absolutely necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Thus the average spin-glass magnetization to be analyzed is M ≡< � |qk| > (10) where the average is both over gauge configurations as well as third direction fixed 2D layers in each gauge configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The order parameter M will become non-zero in a phase with either spin-glass order or ferromagnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Spin-glass order is symmetry breaking because the symmetry operation applied only to the original U’s but not the replicas will invert the order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Another way to say this is that tunneling configurations within 11 the replica or original, where half of the lattice is flipped, do not exist in the spin-glass phase in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For systems without a spin-glass phase this order parameter will simply turn on at the normal ferromagnetic transition (for instance, this is the case for Landau-gauge Higgs phase transitions in the combined Ising gauge-Higgs theory[13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Note also that although its original motivation was from Coulomb gauge, qk itself is gauge invariant, so it is no longer necessary to fix the gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' As detailed below, we indeed find a phase transition in M away from the ferromagnetic transition indicating the presence of a spin-glass phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Here we can use all of the finite-size scaling techniques which have been developed for studying symmetry-breaking phase transitions with a local order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='77 0 20000 40000 60000 80000 100000 M Equilibration sweeps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='64 0 20000 40000 60000 80000 100000 U Equilibration Sweeps Figure 8: Equilibration of spin-glass order parameter and Binder cumulant on a 303 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Before studying the spin-glass order parameter M in Monte Carlo simulations, one must first perform an equilibration study to determine how long the replica must be equili- brated to obtain a truly independent configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' One simply simulates at many different equilibration sweep values and watches the measured quantities approach constant values exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' We then picked equilibration amounts that insure systematic errors are less than 25% of random errors in the quantities measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Detailed studies were made at gauge coupling β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='705, near the suspected critical point, for both the 303 and 503 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The equilibration value for the 403 lattice was determined from these and the volume scaling suggested by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 8ab shows the equilibration of magnetization (order parameter) and its Binder cumulant for the 303 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Other quantities were similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The exponential fits give an equilibration time constant of 14,000 sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' By equilibrating with 105,000 sweeps systematic errors are brought to less than 25% of random in the planned simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For 503 this value was a bit surprisingly high at 700,000 sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' We used 190,000 sweeps for the intermediate 403 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' These high equilibration values indicate the standard heat bath Monte Carlo algorithm is not working particularly well here, but it still gives good results if one is patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The high number of sweeps to equilibrate are due to the fact that 2/3 of the links, those lying in the 1 and 2 directions are being held fixed, which erects more barriers that a simulation where all links participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='74 M U b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='74 x2nd/L b 0 10 20 30 40 50 60 70 80 90 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='74 c b Figure 9: Binder cumulant (left graph) and magnetization (a), ξ2nd/L (b), and susceptibility (c) for the spin-glass order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Error bar spreads for U and M are about 1/15 the size of plotted points, and 1/2 to 1/5 for ξ2nd/L and χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' All simulations had 100 ordinary Monte Carlo sweeps between each measurement of the order parameter to reduce correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' There were 1000 measurements for each 303 and 403 lattices and 500 for 503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Initial equilibration was 200,000 sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Error bars were determined from binned fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 9abc shows the Binder cumulant U, order parameter M, susceptibility χ, and second moment correlation length[16] divided by lattice size, ξ2nd/L, for the three lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The latter, as well as the Binder cumulant, should cross near the infinite lattice transition point (to determine this precisely one must consider corrections to scaling which we do not do here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' One can see a well-defined crossing in both near βc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The crossings are well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For instance the 503 value exceeds the 303 value at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='725 by 15σ for U and 10σ for ξ2nd/L, and points above this have similar significances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The opposite order in the low β region is never in doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Indeed, here points here are separated by even larger amounts, exceeding 30σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Scaling collapse plots are 13 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 10abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The overall fit has 75 degrees of freedom and has a χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='48 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This fit gives βc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7174(3), ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='27(3), β/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='058(2), and γ/ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='86(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Checking hyperscaling on the latter give deff = γ/ν + 2β/ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='97(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Because the order parameter is defined on 2-d layers, the expected value is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The dual reflection of the boundary percolation point of the 3D Ising model itself is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 ln tanh(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='247925) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='70741(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is close to the βc here, but certainly not an exact match, and not within statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However there could be a systematic error present from corrections to scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Looking at the U crossing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 9a), it is plausible that the crossing on larger lattices could shift to this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Corrections to scaling can give a slightly shifting crossing with increasing lattice size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' There is also the possibility of a residual systematic error from insufficient equilibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Although we have tried to limit this to 25% of the random error it could still have an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The fact that the ν seen here and the ν from the coupling-shift of the percolation transition, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='30(3) agree within 1σ strongly supports these being dual-manifestations of the same transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 Scaled M U x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 x2nd/L x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='0 Scaled c x Figure 10: Scaling collapse plots for Binder cumulant (left graph) and scaled magnetization (a), ξ2nd/L (b), and scaled susceptibility (c), for the spin-glass order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 14 4 Energy moments Since the spin-glass transition in the dual theory is symmetry breaking, Landau theory connects this to a thermal phase transition through the hyperscaling relation α = 2 − dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' (11) Here α is the specific heat exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' At the critical point the expected behavior of the spe- cific heat is |T − Tc|−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3, α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This means that the specific heat does not have an infinite singularity, however it does have a finite singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Unfortunately, when rounded by a finite lattice size, these are difficult to spot using finite-size scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Never- theless one can still try to fit to a fractional power, and in some cases more importantly, a different coefficient on each side of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However, the energy moments also have non-singular terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This makes fitting them more difficult than quantities based on the order parameter which are purely singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The non-singular part is expected to vary slowly through the critical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' For this reason it affects higher moments less, and there is a good chance these can be fit without a non-singular part other than perhaps a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This simplifies fitting to the expected critical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A study of energy moments of the 3D Ising model itself was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This study had approximately 7 × 109 sweeps at each coupling for the 303 lattice and 2 × 109 for the 403 and 503, with measurements performed every other sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' With these statistics, rather precise data can be obtained on the third cumulant (third central moment), defined as < (E − ¯E)3 > (3L3)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It is this combination that corresponds to the derivative of the specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The third cumulant is expected to scale as |κ − κc|−α−1, which is still a non-infinite singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This quantity is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 11, along with a fit to the expected critical behavior, but leaving α and κc as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The fit also allows for a different coefficient on the two sides of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The result is κc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='2477(2), and −α−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='967(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The coefficient ratio below and above the critical point is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='287(25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The predicted ν from this α is ν = (−α + 2)/3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='322(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The critical point agrees well with that extracted from percolation(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='24781(4), and the exponent ν also agrees with those from both percolation finite-size shift (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='30(3)) and the spin glass transition in the dual gauge theory(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='27(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Even though the singularity is non- infinite, it can still be seen from this fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It is important to remember that these functions are singular in two ways - the fractional power and the jump in coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' So even if the power were to end up being exactly unity, that would not erase the singularity due to the fairly large coefficient jump, verified to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='5σ, which can be seen in the change of slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Higher moments were also measured, but even with these high statistics were somewhat of a disappointment due to fairly large statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 12 shows the fourth cumulant, (< (E − ¯E)4 > −3 < (E − ¯E)2 >2)(3L3)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' (12) This combination of moments tracks the third derivative of the internal energy with respect to κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Also shown is a numerical derivative of the fit function to the third cumulant on a parameter spacing 1/4 of that used for the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This was done instead of an exact derivative to simulate finite-lattice rounding, so not an exact prediction of the expected 15 210 200 190 180 170 160 150 140 130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='253 Third Energy Cumulant k Figure 11: Third order energy cumulant, with fit to critical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Here open circles are 303, open triangles 403, and × 503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' behavior, but one which should be good away from the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The main effect of the shift in slope in the third cumulant which translates to a shift in level here can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' In principle some finite-size effect could be seen in this quantity since it diverges with a very small exponent, but the expected ratio in peak heights between 303 and 403 is only (4/3)((α+2)/ν) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='02, much smaller than our statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' A larger effect is predicted for the fifth cumulant (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='28), but the errors there are magnified even more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This figure is shown primarily to illustrate how much further one would have to go in statistics to see an infinite singularity in a high moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Our program, which was run for about 24 processor-years on PC’s, does not employ multi-spin coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Perhaps a study that did or used specialized hardware could see these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' 5 Conclusion In this paper evidence has been given for a new high-order phase transition within the or- dered phase of the 3D Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This transition appears to be associated with boundary percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is the percolation of dual-plaquettes that lie on the domain boundary between + and - spins, a type of percolation that has not been much studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Percolation of domain boundaries occurs when minority sites occupy 13% or more of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It is, incidentally, not coincident with the roughening transition which occurs much deeper into the ordered phase, around κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='408[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Because the Ising model has a formulation in terms of the domain boundary itself, the percolation of the boundary could be important, 16 3000 4000 5000 6000 7000 8000 9000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='252 Fourth Energy Cumulant k Figure 12: Fourth order energy cumulant for 303 and 403 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Line is a plausible rounded critical behavior based on third cumulant fit (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' possibly producing a sudden change in the entropy function expressed in terms of boundary area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Random boundary percolation appears to have the same critical exponents as ordinary site percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Boundary percolation in the Ising model seems to model random boundary percolation as far as the scaling of cluster sizes is concerned, however it differs in the finite- size shift exponent, which determines how the percolation threshold depends on lattice size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Whereas random percolation has a shift exponent agreeing with typical values of the correlation-length exponent from cluster size scaling (ν ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='88), the shift exponent from the 3D Ising model boundary percolation is vastly different, ν ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This surprising result means that the system has two different correlation lengths, both diverging at the infinite- lattice percolation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This also suggests that there is more than just percolation going on here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' If percolation is linked to a thermal phase transition, that could explain the odd shift exponent, since the order parameter of the phase transition may have its own correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' To find such an order parameter we examined the dual system, the 3D gauge Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The dual point of the boundary percolation threshold occurs in the random (con- fining) phase of the gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' An order parameter for the confinement-deconfinement transition can be obtained in Coulomb gauge, where as many one and two direction links as possible are made to be positive by gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The third direction links on each lattice layer can be taken to be a spin-like order parameter, which shows spontaneous magnetization in the ordered phase and is unmagnetized in the random phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' If there is a phase transition corresponding to boundary percolation in the Ising model itself, it must occur within the random phase of the gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This suggests looking for a spin-glass transition here, a shift from a completely disordered phase to one with a hidden pattern of order, but still showing no net magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' To this end we utilized a two-real-replica 17 order parameter, which indeed does show a phase transition near the dual reflection of boundary percolation, and with the same critical exponent ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is significant be- cause it is a true symmetry-breaking phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The symmetry being broken is the layered remnant (Z2)L symmetry left over after Coulomb gauge fixing, which is global in two dimensions but still local in the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This is “global enough” to avoid Elitzur’s theo- rem and has sufficient dimensions (2) for a discrete symmetry to break spontaneously at a finite coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Spontaneous symmetry-breaking always results in a phase transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' a mathematical singularity in the order parameter, which also results in an energy singularity except in a few unusual cases [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Finally we examined energy moments in search of this singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Because of the high value of ν the specific heat exponent is negative, implying a finite singularity, so the usual finite-size scaling applied to peak heights cannot be used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' We concentrated on the third energy cumulant, since it could be fit without the addition of an obfuscating non-singular part, other than a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' An open fit to the singular form expected for this quantity based on the hyperscaling relationship, gives κc and ν values consistent with those determined by boundary percolation and the dual order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' There is also a noticeable jump in coefficient here, another expectation of this sort of singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Our study did not have enough statistics to see the small expected peak scaling in the fourth cumulant or somewhat larger effect in the fifth, which should have infinite singularities on the infinite lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Although observing these would be satisfying, still the singular fit to the third cumulant does match well with the prediction from the order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' This demonstrates that phase transitions as weak as these can be studied by numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The existence of an order parameter and associated symmetry breaking is key in establishing this as a true phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The coincidence of boundary percolation is also interesting and gives another measure of ν, but cannot by itself be used to imply the presence of a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' However, it has the advantage of being very easy to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It appears to have the same cluster-size scaling exponents as ordinary site percolation, but a different threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It may be interesting to explore boundary percolation in other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Since percolation has so many practical applications, it’s possible boundary percolation is a better fit than site or link percolation in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' Finally, we note that a previous study of the Z2 gauge-Higgs system showed a total of four phase transition lines further into the diagram[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The current paper shows there are two phase transitions on each axis, gauge and Higgs, so also a total of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It will be interesting to follow these new phase transitions into the phase diagram to see how they connect with the lines previously found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The previous paper showed that the Z2 gauge-Higgs system appears to be more complicated than previously thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' The present paper shows that these additional complications extend to the 3D Ising model itself, and its dual, the 3D gauge Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFAT4oBgHgl3EQfExzK/content/2301.08424v1.pdf'} +page_content=' It seems possible that similar weak phase transitions may also be lurking in other well-known spin and gauge systems.' metadata={'source': 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b/69E1T4oBgHgl3EQfBgJT/content/tmp_files/2301.02852v1.pdf.txt @@ -0,0 +1,1274 @@ +Coherent control of wave beams via unidirectional evanescent modes excitation +Shuomin Zhong1*,∗ Xuchen Wang2*, and Sergei A. Tretyakov3 +1. School of Information Science and Engineering, Ningbo University, Ningbo 315211, China +2. Institute of Nanotechnology, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany +3. Department of Electronics and Nanoengineering, Aalto University, Finland +Conventional coherent absorption occurs only when two incident beams exhibit mirror symmetry +with respect to the absorbing surface, i.e., the two beams have the same incident angles, phases, +and amplitudes. In this work, we propose a more general metasurface paradigm for coherent perfect +absorption, with impinging waves from arbitrary asymmetric directions. By exploiting excitation of +unidirectional evanescent waves, the output can be fixed at one reflection direction for any amplitude +and phase of the control wave. We show theoretically and confirm experimentally that the relative +amplitude of the reflected wave can be tuned continuously from zero to unity by changing the phase +difference between the two beams, i.e. switching from coherent perfect absorption to full reflection. +We hope that this work will open up promising possibilities for wave manipulation via evanescent +waves engineering with applications in optical switches, one-side sensing, and radar cross section +control. +I. +INTRODUCTION +Coherent control of propagation of a wave beam by +tuning the amplitude and phase of another beam is a very +promising approach to realize ultra fast optical devices +for optical computing, sensing, and other applications [1– +11]. One of the most important effects in coherent control +of light is coherent perfect absorption [12–22]. In these +devices, the level of absorption of one beam illuminating +a thin sheet is controlled by another coherent beam that +illuminates the same sheet. +In earlier works, coherent perfect absorption (CPA) +was achieved only when with illumination from different +sides of a homogeneous lossy layer and for two incident +waves at the same angle [12, 13, 15, 22]. +The mecha- +nism of coherent perfect absorption is destructive cancel- +lation of all scattered beams. For homogeneous coher- +ent perfect absorbers, there are only specular reflection +and non-diffractive transmission, allowing coherent ab- +sorption only with illumination of both sides and at the +same incidence angle. From the theoretical point of view +and for many applications, it is important to achieve co- +herent control of output for illuminations from the same +side of the metasurface sheet at two or more arbitrary +incidence angles. In Refs. [17, 18, 23], coherent perfect +absorption and scattering for two angularly asymmetric +beams are realized by using surface plasmon-polariton +(SPP) excitation at silver-based diffraction groove grat- +ings. However, such plasmonic grating designs have limi- +tations. In particular, the structures are non-planar and +operate only for TM modes at optical frequencies, where +SPP are supported. Moreover, there are always two out- +put beams for different values of the phase of the control +waves, one of which may cause undesired noise to the +useful output signal due to parasitic scattering. This is- +sue is critical in applications such as optical computing +[24]. +∗ Email: zhongshuomin@nbu.edu.cn, xuchen.wang@kit.edu +In this decade, the emergence of gradient metasurfaces +[25–28] and metagratings [29–35] has opened a new av- +enue for manipulation of light for arbitrary incidence an- +gles and versatile functionalities. For periodical metasur- +faces or metagratings with the period larger than half of +the wavelength, the incident plane wave from one direc- +tion will be scattered into multiple directions, and the +power carried by the incident wave can be redistributed +among a number of diffraction modes. +Based on this +concept, several metasurface devices with perfect anoma- +lous reflection working at microwaves [36, 37] and optical +bands [38] have been developed. However, in these previ- +ous works, the functionality of metasurfaces is designed +only for one incident angle and the response for other illu- +minations is actually not considered. To design metasur- +faces with coherent control functions for multiple simul- +taneously incident coherent beams from different direc- +tions, the matching conditions of amplitude, phase, and +wavevector(direction) of the scattering modes between all +incidences are required [35, 39, 40], which is almost an +impossible task using traditional gradient phase methods +[25, 36] and brute-force numerical optimizations [37, 41]. +In this work, we perform inverse designs of CPA meta- +surfaces by solving the surface impedance satisfying the +boundary condition determined by two coherent incident +waves from two arbitrary angles and the desired total +scattered waves. +The engineering of evanescent waves +in the scattered fields without altering the desired far- +field outputs provides significant freedom in the CPA +metasurface design, making another functionality of co- +herent control of reflection with a single direction possi- +ble. It is demonstrated that excitation of unidirectional +evanescent waves propagating along the surface in the +direction of the incident-wave wavevector can be used to +achieve single-direction output in coherently controlled +optical devices. Furthermore, a mathematical optimiza- +tion method based on scattered harmonics analysis [42] +is utilized to find the surface-impedance profile that si- +multaneously ensures the CPA and coherent maximum +reflection (CMR) in a single direction. Thereafter, the +arXiv:2301.02852v1 [physics.app-ph] 8 Jan 2023 + +2 +substrate parameters are invoked as additional degrees of +freedom in the optimization model, realizing reflection ef- +ficiency of 100%. As an example, we experimentally vali- +date the CPA gradient metasurface design in microwaves +for TE-polarized waves by engineering the Indium Tin +Oxide (ITO) film mounted on a grounded dielectric sub- +strate. It is showed that the normalized output power +can be continuously controlled between 0 and 1 by tun- +ing the phase of the control wave. +II. +DESIGN CONCEPT +Dx +x +z +Zs(x) +θ1 +θ2 +I1 +I2 +FIG. 1. General scattering scenario for a periodically modu- +lated impenetrable impedance surface. Two coherent beams +I1 and I2 are simultaneously incident from two angles. +Let us consider an impenetrable reciprocal metasur- +face whose surface is periodically modulated along the +x-direction, with the period Dx. The surface is in the +xy-plane of a Cartesian coordinate system (see Fig. 1). +The metasurface is simultaneously illuminated by two +TE(s)-polarized plane waves I1 and I2 at the incidence +angles θ1 and θ2 (θ1 > θ2). The electric field amplitudes +of the two beams I1 and I2 is E1 = E0 and E2 = αE0, +respectively (α is the amplitude ratio). The phase differ- +ence between them is ∆φ=0, defined at the origin point +(x = 0, z = 0). The electromagnetic properties of the +metasurface can be characterized by the locally-defined +surface impedance that stands for the ratio of the tangen- +tial electric and magnetic field amplitudes at the surface +plane Zs(x) = Et(x)/Ht(x). +The field reflected by a periodically modulated meta- +surface can be interpreted as a sum of Floquet harmonics. +The tangential wavenumber of the n-th harmonic is re- +lated to the period and the incident wavenumber k0 as +krxn = k0 sin θi + 2πni/Dx, where i = 1, 2. The corre- +sponding normal component of the reflected wavenumber +equals krzn = +� +k2 +0 − k2rxn. If |krxn| is greater than the +incident wave number, the wave is evanescent and it does +not contribute to the far field. For the harmonic wave sat- +isfying |krxn| < k0, krzn is real, and this wave is propagat- +ing. The evanescent harmonics will be dissipated by the +lossy surface and the propagating harmonics will propa- +gate into the far-zone at the angles θrn = arcsin(krxn/k0). +In order to achieve coherent perfect absorption, it is nec- +essary (but not sufficient) to ensure that all the diffracted +propagating modes of two beams have the same set of +angles θrn, that allows mutual cancellation, defining the +period Dx = λ0/(sin θ1 −sin θ2) [43], where λ0 stands for +the wavelength. +Our aim is to achieve coherent perfect absorption for +two coherent in-phase waves simultaneously incident on +the metasurface at two different angles θ1 and θ2. First, +let us assume that no evanescent waves are excited for +these two illuminations. In the CPA case, there should +be no reflected field at the surface. Thus, the tangential +components of the total electric field at the plane z = 0 +can be written as Et(x) = E0(e−jk0 sin θ1x+αe−jk0 sin θ2x), +where the time-harmonic dependency in the form ejωt +is assumed and suppressed. +The corresponding total +magnetic field reads Ht(x) = E0(cos θ1e−jk0 sin θ1x + +α cos θ2e−jk0 sin θ2x)/Z0, with Z0 = +� +µ0/ϵ0 being the +free-space wave impedance. The ratio of these electric +and magnetic fields gives the required surface impedance +ℜ(Zs) = Z0 +cos θ1 + α2 cos θ2 + α cos Φ(cos θ1 + cos θ2) +cos2 θ1 + α2 cos2 θ2 + 2α cos θ1 cos θ2 cos Φ , +ℑ(Zs) = Z0 +α(cos θ1 − cos θ2) sin Φ +cos2 θ1 + α2 cos2 θ2 + 2α cos θ1 cos θ2 cos Φ, +(1) +where Φ = k0(sin θ1 − sin θ2)x is the linearly varying +phase. +The real and imaginary parts of the surface +impedance are even and odd functions of x, respectively. +As is seen from Eqs. (1), the periodicity of the surface +impedance is D = λ0/(sin θ1 − sin θ2), in accord with the +above analysis. For passive metasurfaces, the real part +of the surface impedance must be non-negative. +Con- +sequently, the amplitude ratio should satisfy α ≥ 1 or +α ≤ cos θ1/ cos θ2 to ensure passive solution for CPA by +the surface. +As an example, we consider two incident waves with +incidence angles of (θ1, θ2) = (45◦, 0◦) and the same am- +plitude, assuming α = 1 for simplicity. (Other scenarios +with (θ1, θ2) = (60◦, −30◦), (75◦, 15◦) are illustrated in +the Supplemental Materials[43], corresponding to differ- +ent surface impedance profiles.) As is shown in Fig. 2(a), +everywhere on the surface its resistance is non-negative, +demonstrating that passive gradient periodic surfaces can +realize CPA for two asymmetric incident beams. +To analyze the mechanism of CPA by the periodic +impedance surface further, we can determine the ampli- +tudes of all the Floquet scattered harmonics for general +plane-wave illumination, using the method reported in +[42]. The total reflected field can be represented as an +infinite sum of Floquet harmonic modes: +Er = +∞ +� +n=−∞ +Ane−jkrznze−jkrxnx, +(2) +where An is the complex amplitude of the n-th Floquet +harmonic. Because the surface modulation is periodical, +the surface admittance Ys(x) = 1/Zs(x) can be expanded + +3 +0 +0.2 +0.4 +0.6 +0.8 +1 +-200 +0 +200 +400 +(a) +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +(b) +0 +0.2 +0.4 +0.6 +0.8 +1 +-200 +0 +200 +400 +(c) +-6 +-4 +-2 +0 +2 +4 +6 +8 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +(d) +FIG. 2. (a) Analytical surface impedance over one period to realize CPA for two incidence beams with (θ1, θ2) = (45◦, 0◦). +(b) Magnitudes of the complex amplitudes of different Floquet scattered harmonics (normalized by the amlpitude of the +incident electric field E0) when the gradient surface is illuminated by single-beam incidences at 45◦ and 0◦, and for two- +beam incidences in phase and out of phase, respectively. (c) Optimized surface impedance profile over one period to realize +CPA for in-phase incidences and single-direction reflection for out-of-phase incidences. +The optimized Fourier coefficients +of Ys(x) read g0 = 2.654 × 10−3 + j1.724 × 10−11, g1 = −7.770 × 10−4 − j1.045 × 10−10, g2 = −(6.565 + j4.581) × 10−5, +g3 = −9.143×10−8 +j5.720×10−6, g4 = (−1.644+j1.992)×10−5. (d) Amplitudes of scattered harmonics when the optimized +gradient surface in (c) is illuminated by single-beam incidences at 45◦ and 0◦, and for two-beam incidences in phase and out +of phase, respectively. +into Fourier series: +Ys(x) = ++∞ +� +n=−∞ +gne−j2nπx/D. +(3) +A Toeplitz matrix Ys which we call the admittance ma- +trix is determined only by the Fourier coefficients of the +modulation function and filled with Ys(r, c) = gr−c at +the r-th row and c-th column. The reflection matrix is +found as [44] +Γ = (Y0 + Ys)−1 (Y0 − Ys), +(4) +where Y0 = Z−1 +0 +is a diagonal matrix with its main +entry representing the admittance of each space har- +monic, which is Y0(n, n) =krzn/ω0µ0. The amplitudes +An of reflected harmonics for a given m-th order Flo- +quet harmonic of the incident wave can be calculated +as An = Γ(n, m). Note that Γ is a (2N + 1) × (2N + 1) +square matrix and the columns and rows of Γ are indexed +from −N to +N. When the surface is illuminated by two +waves simultaneously, the amplitudes of all the Floquet +harmonics are linear superpositions of all harmonics. +As is seen from Fig. 2(b), when the two incident waves +are in phase, all the harmonics have zero amplitude, +meaning that CPA with no reflected fields occurs. How- +ever, when the two incident waves are out of phase, the +reflected harmonics come out, including both propagat- +ing modes and evanescent ones, proving that the perfect +absorption effect is phase-coherent, different from perfect +absorption for two angles [45]. To understand the mech- +anism of CPA in the metasurface better, the harmonics + +4 +of the reflected field when single beams illuminate the +surface separately are calculated. As shown in Fig. 2(b), +the complex amplitudes of every scattered harmonic are +equal and 180◦ out of phase (the phases are not shown +here) for 45◦ and 0◦ incidences, resulting in destructive +cancellation when the two beams illuminate simultane- +ously in phase. Here, the propagating harmonic of the +order n = 0 is defined at the specular direction of θ1 for +both incidences. By properly designing the metasurface +with the periodicity of D = λ0/(sin θ1 − sin θ2), three +propagating modes corresponding to n = 0, −1, −2 are +created, and all the diffracted modes for both incidences +have the same wave vectors, ensuing coherent interfer- +ence for all corresponding harmonics. In the out-of-phase +incidence case, the amplitudes of all the scattered har- +monics double as compared to the single-beam case, as +shown in Fig. 2(b). +The analytical method to solve the surface impedance +boundaries used above is based on the objective to real- +ize CPA with the amplitudes of both scattered propagat- +ing and evanescent harmonics being zero when two co- +herent beams illuminate the metasurface simultaneously. +Indeed, the amplitudes of evanescent surface modes can +be nonzero without breaking the CPA condition, because +they do not radiate into the far zone and their power +will be dissipated at the lossy surface. +Thus, the so- +lution of the surface impedance to achieve CPA is not +unique if a certain set of evanescent waves with unknown +complex amplitudes is excited. In addition to CPA, we +invoke another functionality of coherent control of reflec- +tion with single direction, i.e. eliminating the unwanted +outgoing beams at n = −1, −2 orders and keeping the +n = 0 order with the maximal amplitude, when the two +coherent incident beams are out-of-phase. In this case, +finding the complex amplitudes of infinite numbers of +evanescent modes for each incidence scenario is difficult +or even impossible. Thus, instead of using the analyti- +cal method of calculating the surface impedance profile +according to the total fields on the boundary, we ap- +ply a mathematical optimization algorithm described in +Ref. [42] and based on the scattering matrix calculation +to find a surface impedance profile that simultaneously +ensures the coherent control capability for absorption and +reflection of the surface. First, the metasurface is mod- +elled as in Eq. (3). To suppress propagating modes at +the negative orders (n = −1, −2) and ensure that only +the reflection channel at 45◦ is open, the Fourier series of +the surface admittance Ys(x) are set to be unilateral as +Ys(x) = �4 +n=0 gne−j2nπx/D with non-negative-order se- +ries coefficients being nonzero (only five coefficients from +g0 to g4 are used for improving optimization efficiency). +This setting is reasonable because the unilateral surface +admittance, making the admittance matrix Ys a lower +triangular matrix, can lead to the reflection matrix Γ +also being a lower triangular matrix, as is seen from +Eq. (4). Consequently, the scattered modes contain only +components of non-negative orders (n ≥ 0). This effect +highlights the role of unidirectional evanescent fields as +a mechanism of suppressing propagating modes at the +negative orders (n = −1, −2). Moreover, to ensure that +the grid is a passive metasurface, we need to impose con- +straints ℜ(Ys) ≥ 0, i.e., ℜ(g0) ≥ |g1| + |g2| + |g3| + |g4|. +Secondly, the optimization goal is formulated as 6 ob- +jectives, including (|A0|, |A−1|, |A−2|) = (0, 0, 0) for the +in-phase scenario, and (|A0|, |A−1|, |A−2|) = (A0max, 0, 0) +for the out-of-phase scenario, where A0max is the maxi- +mum magnitude of reflection in the out-of-phase case. +In each trial of the optimization, an array of gn is as- +sumed, and the value of all the objectives are calcu- +lated using Eq.(4). The sum of errors calculated for all +the objectives is defined as a cost function C. By em- +ploying MultiStart and fmincon optimization algorithms, +the maximum magnitude of the out-of-phase reflection +A0max = 0.34 is searched out, and the minimum value +of C close to zero is achieved, meaning that the solu- +tions of the impedance profile to realize the desired EM +responses including CPA and single-direction-reflection +are obtained. +Figure 2(c) shows a typical optimized solution of the +surface impedance, which exhibits positive resistance ev- +erywhere along the metasurface. The calculated ampli- +tudes of scattered harmonics for single-beam incidences +at 45◦ and 0◦, and for two-beam incidences in phase and +out of phase, for the impedance profile in Fig. 2(c), are +given in Fig. 2(d), revealing the unilateral characteristic +of scattering. We can see that the propagating compo- +nents at n = −1, −2 orders are suppressed successfully +by exciting the unidirectional evanescent wave. The only +remaining propagating reflected channel is n = 0 order +at the outgoing angle of 45◦. When two incoming beams +are in phase, the reflected propagating harmonic (n = 0) +of each beam cancel each other because they have the +same amplitude and π-reflection-phase difference. Dis- +tinct from the zero-amplitude of all the harmonics for the +in-phase CPA scenario in Fig. 2(b), the CPA in Fig. 2(d) +occurs with non-zero-amplitude evanescent modes in the +n ≥ 1 orders. The amplitude of reflected electric field +at 45◦ (n = 0) is doubled into A0max = 0.34 when two +incoming beams are out of phase (∆φ = π). +We can +conclude that the reflected power at 45◦ can be contin- +uously controlled by phase tuning of the control beam. +When the two beams are out of phase, the reflected power +normalized by the incident beam power at 45◦ has the +maximum reflection efficiency of 11.56 %. +III. +OPTIMIZATION AND PRACTICAL +DESIGN +Low efficiency of the above design based on the im- +penetrable impedance model calls for optimization with +the help of additional degrees of freedom. One possibility +can be the use of one or more parameters of the actual +implementation of the metasurface. +In general, the impedance surface in the impenetra- +ble model used above can be realized as a periodic metal + +5 +x +z +q1 +D +h +I1 +I2 +n = 0 +n = -1 +n = -2 +FIG. 3. +Schematics of reflection amplitude modulation for +two coherent waves with the phase difference ∆φ incident on a +periodic sheet over a grounded dielectric slab. The amplitude +of the output beam is modulated continuously by varying ∆φ, +and switched between 0 (coherent perfect absorption) and 1 +(coherent maximum reflection) when ∆φ is switched between +even and odd multiples of π. +pattern on a thin grounded dielectric slab, as shown in +Fig. 3. The structure can be considered as a grid admit- +tance of the top pattern with a shunt admittance of the +grounded substrate. The characteristic admittance ma- +trix Yd of the grounded substrate contains only diagonal +terms Yd(n, n), where Yd(n, n) is the admittance of the +n-th harmonic, and it is expressed as +Yd(n, n) = kd +rzn/[jµ0ω0 tan(kd +rznh)], +(5) +where kd +rzn = +� +ω2 +0ϵ0ϵdµ0 − k2rxn is the normal compo- +nent of the wavevector in the substrate (see Eq.S23 of +the Supplemental Material of [42]), ϵd and h are the +permittivity and thickness of the substrate, respectively. +The reflection matrix is calculated as Γ = (Y0 + Yg + +Yd)−1(Y0−Yg−Yd). When the thickness h is ultra-thin +compared with the wavelength, for low-order harmonics +we have tan(kd +rznh) ≈ kd +rznh. As is seen from Eq. (5), +the admittance for low-order harmonics equals approxi- +mately to 1/(jµ0ω0h), unrelated to the harmonic num- +ber. Thus, we can approximately design the top surface +with the grid admittance Yg(x) = 1/Zs(x) − Yd(0, 0) us- +ing the optimized surface impedance Zs(x) in Fig. 2(c), +similar to Ref. [41]. Due to the lack of freedom in the sub- +strate design, the evanescent fields engineering is quite +limited in the impenetrable model, resulting in a low +reflection efficiency (11.56 %) in the out-of-phase sce- +nario. In order to implement CPA with a high reflec- +tion efficiency, we need to use the substrate parameters +as additional degrees of freedom in the design. Since the +admittance of the grounded substrate with a moderate +thickness strongly depends on the harmonic number, the +need of complicated matrix operations makes it impos- +sible to analytically solve the grid impedance and sub- +strate parameters. Thus, the optimization algorithm is +extended by introducing the admittance matrix Yd of the +grounded substrate, as described in Ref. [42], to search +for an optimum solution for the grid impedance profile +and substrate thickness. +According to the results of the impenetrable model, +the period of the impedance sheet modulation is set +to D = λ0/ sin 45◦, with three propagating channels at +−45◦, 0◦, and 45◦. The Fourier series of the grid admit- +tance is set to be unilateral as Yg(x) = g0 + g1e−j2πx/D, +ensuring that only the reflection channel at 45◦ is open. +In the optimization process, two Fourier terms g0 and +g1 with four unknowns (the real and imaginary parts) +are considered here to reduce complexity. The substrate +thickness h is another unknown, and an available sub- +strate with the permittivity ϵd = 5.8(1 − j0.002) is used. +The optimization goal is formulated as 6 objectives, the +same as the objectives in the impenetrable model above. +The constraints ℜ(Yg) ≥ 0, i.e., ℜ(g0) ≥ |g1| are imposed +to ensure the grid to be a passive metasurface. Addi- +tionally, to make the reactance easier to implement by +patterning a thin conductive surface, another constraint +ℑ(g0) ≥ |g1| is set to ensure that the surface reactance is +always capacitive at all points of the metasurface. +The maximum magnitude of reflection A0max in the +out-of-phase scenario is searched out to be about 1 in +the optimization, meaning that a reflection beam at +45◦ with amplitude equal to the incident beam I1 is +obtained [46]. +It reveals that the invocation of sub- +strate design provides an important additional degree +of freedom in engineering auxiliary evanescent modes to +find a surface impedance that can realize the desired +optimum scattering properties for all incidence scenar- +ios. +The optimized Fourier coefficients of the grid ad- +mittance Yg(x) read g0 = (2.599 + 7.054j) × 10−3 and +g1 = (−0.807 + 2.463j) × 10−3. The optimal substrate +thickness is h = 0.2525λ0. The required grid impedance +which is passive and capacitive along the metasurface is +shown in Fig. 4(a). +Next, we analyse the scattered harmonics for the de- +signed impedance sheet on the metal-backed dielectric +substrate [see Fig. 4(b)]. The reflection coefficient of the +metasurface has the same magnitude of 0.5 at n = 0 +order for 45◦ and 0◦ single-beam incidences, resulting +from destructive interference when these two beams are +in phase. For the out-of-phase scenario, the normalized +magnitude of the reflected field at n = 0 order (45◦) is +about unity, which means that the reflected power effi- +ciency reaches 100% (normalized by the incoming power +of the 45◦ beam). Parasitic reflections into other direc- +tions (n = −1, −2) are seen to be negligible, due to the +unilateral property of the admittance of the surface. The +evanescent harmonics are also unidirectional, but quite +weak with the magnitude of 0.008 at n = 1 order, and +they are absorbed by the lossy structure, ensuring a CPA +state. Figure 4(c) illustrates the phase-controlled modu- +lation of reflections at three propagating orders. The re- +flection coefficient at 45◦ can be continuously controlled +from 0 to 1 by phase tuning, with the other two par- +asitic reflections maintained very close to zero. +This +phase-sensitive modulation between CPA and coherent + +6 +0 +0.2 +0.4 +0.6 +0.8 +1 +-200 +-100 +0 +100 +(a) +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +0 +0.5 +1 +(b) +0 +1 +2 + ( ) +0 +0.2 +0.4 +0.6 +0.8 +1 +Amplitude (|An|/E0) +n=0 +n=-1 +n=-2 +(c) +𝐸𝑠𝑐/𝐸0 +1 +0 +-1 +2 +3 +-2 +-3 +Df = 0 +Df = p +(d) +FIG. 4. +(a) The optimized and discretized grid impedance distribution over one period. (b) Amplitudes of the scattered +harmonics when the optimized gradient metasurface is illuminated by a single beam at 45◦ and 0◦, and for two-beam in-phase +and out-of-phase illuminations, respectively. (c) The normalized amplitudes of three propagating harmonics (n = 0, −1, −2) +with a varying phase difference ∆φ between incidences at 45◦ and 0◦. (d) The scattered electric fields and power density flow +distributions for the metasurface modeled by the discretized grid impedance (step-wise approximation, 6 subcells per period) +on top of a grounded dielectric substrate. Two plane-wave incidences are in phase (left) and out of phase (right). +maximum reflection (CMR) without parasitic reflections +is important in light switching applications where a low- +return-loss characteristic is required. See the Supplemen- +tal Animation [43] for the switch of reflected beam by an +incident phase-controlled wave. +In implementations, the influence of discretization on +the metasurface performance is an important factor (see +detailed analysis of scattered harmonics versus the num- +ber of subcells in Ref. [43]). +We use six subcells over +a period and each discretized impedance value is set at +the central point of each subcell, as shown in Fig. 4(a). +The scattered fields from the ideal impedance sheet on +the metal-backed dielectric slab for both in-phase and +out-of-phase incidences are presented in Fig. 4(d), using +full-wave simulations in Comsol. The reflected field dis- +tribution confirms that the metasurface with six subcells +per period possesses the desired response: nearly per- +fect absorption with reflection amplitude of only 0.023 +for two in-phase illuminations and nearly total reflection +at 45◦ for two out-of-phase illuminations, relative to the +intensity of the 45◦ incidence. +It is seen that the top +lossy sheet and reflective ground separated by the slab +act as a leaky-wave cavity with enhanced fields. For the +in-phase scenario, the direct reflections of the top surface +and leaky wave components of the cavity destructively +cancel out, and all the power is absorbed by the lossy +surface, causing CPA. By changing the initial phase dif- +ference between the two coherent incidences into π, con- +structive interference occurs among these components, +which results in nearly total reflection. Note that in the +out-of-phase case a half of the total incoming power (two +incident beams) is still absorbed by the lossy surface. +IV. +PHYSICAL IMPLEMENTATION AND +EXPERIMENTAL VALIDATION +The theory above is general and applies to any fre- +quency, and we choose the microwave band for a proof +of concept demonstration. The required impedance pro- +file at 15.22 GHz is realized using an ITO film with the +surface resistance of 5.5 Ω/sq supported by a grounded + +7 +f (GHz) +12 +13 +14 +15 +16 +17 +18 +E/ciency +0 +0.2 +0.4 +0.6 +0.8 +1 +90 +9-1 +9-2 +9'0 +9'-1 +9'-2 +Simulated +(a) +Transmitting +antenna +Receiving +antenna +Metasurface +Scanning track +(b) +3r (deg) +-80 -60 -40 -20 +0 +20 40 60 80 +S21;m (dB) +-120 +-110 +-100 +-90 +-80 +-70 +-60 +3i = 0o +3i = 45o +(c) +f (GHz) +12 +13 +14 +15 +16 +17 +18 +E/ciency +0 +0.2 +0.4 +0.6 +0.8 +1 +90 +9-1 +9'0 +Measured +(d) +FIG. 5. +(a) Simulated and (d) measured reflection efficiency spectrum for different diffracted modes of each single beam at +0◦ (solid lines) and 45◦ (dashed lines). (b) Schematic of the experimental setup (top) and photograph of the fabricated sample +(bottom). (c) Signals at 15.22 GHz measured by the receiving antenna at different orientation angles with the transmitting +antenna at 0◦ and 45◦. +dielectric slab with the thickness h = 4.95 mm, as +shown in Fig. 3. +The detailed parameters and struc- +tures of each unit cell are presented in the Supplementary +Material[43]. Due to the resolution limitation of picosec- +ond laser micro-processing, the complex grid impedance +is implemented as six subcells, and each subcell is divided +into four equal sub-subcells in order to make the local +design of the gradient impedance more robust. By struc- +turing the homogeneous resistive ITO film into I-shaped +cells, the required grid resistance and reactance on a sur- +face in Fig. 4(a) can be created. For y-polarization inci- +dent waves, such I-shaped resonators can be modeled as +RLC series circuits. The required resistance is realized by +tailoring the width and length of the ITO strips. Smaller +width and longer length result in higher grid resistance. +The required reactance can be tailored by adjusting ca- +pacitance of the gap, which can be increased by narrow- +ing the gap or increasing the length or width of the bar, +with a small influence on the resistive part. The 5th and +6th subcells degenerate into strips, to implement resistive +parts as close to the theoretical value as possible. How- +ever, there are still deviations of 3.6 Ω and 1.1 Ω from +the theoretical resistances of the 5th and 6th subcells, +respectively. The deviation can be eliminated if an ITO +film with a lower surface resistance is utilized. To sim- +plify the fabrication process, we neglect this deviation. +The impact is analyzed theoretically, showing that the +reflection amplitude in the in-phase scenario increases +from 0.023 to 0.065, which is tolerable in experiments. +Since the two beams with 0◦ and 45◦ incidence angles +illuminate the surface simultaneously, all the elements +should have angle-independent surface impedances. The + +OLMS +EILMS +SWAi +CILAS +S-iD +SWAiN +SWiM +SWi8 +I-shaped resonators have angle-insensitive impedance un- +der TE incidences, satisfying this requirement [47]. In the +strips of the 5th and 6th subcells, narrow slits are cut out +to reduce the angular sensitivity of the impedance. All +the subcells have been optimized with the geometrical +dimensions specified in Ref. [43]. +Figure 5(a) shows the simulated frequency response of +the metasurface for the normal and 45◦ incidences. For +the normal illumination, strong reflections occur at n = +−1 and n = 0 harmonics (denoted as ξ−1 and ξ0), and the +amplitude of the n = −2 scattered propagating mode is +nearly zero in the whole frequency band. The reflection +at the n = −1 mode (specular reflection at 0◦) also has a +near-zero dip at the design frequency of 15.22 GHz, and +the reflection efficiency at the n = 0 mode(anomalous re- +flection at 0◦) is about 13.9% (the relative amplitude is +0.44). Note that for anomalous reflection, the efficiency +is calculated as ξ = (Er/Ei)2cos θr/cos θi [37]. For the +45◦ illumination, the reflections at both n = −1 and +n = −2 modes (ξ′ +−1 and ξ′ +−2) are close to zero, and +the efficiency at the n = 0 mode (ξ′ +0) is about 21% at +15.22 GHz (the relative amplitude is 0.46). Therefore, at +the operating frequency 15.22 GHz, the reflected modes +for both incidences at the outgoing angle of 45◦ are al- +most equal-amplitude, satisfying the condition of CPA. +The scattered electric field distributions of the designed +metasurface illuminated by two beams in the in-phase +and out-of-phase scenarios obtained from full-wave sim- +ulations are presented in Ref. [43]. It can be seen that +when the two illuminations are in phase, the total scat- +tered fields are quite small (0.02), indicating nearly per- +fect coherent absorption. However, when the two illumi- +nations are switched into the out-of-phase state, the rel- +ative amplitude of the scattered fields is about 0.91, and +the coherent maximum reflection is mainly along the 45◦ +direction. +We have fabricated a sample (see Methods) and car- +ried out several experiments to validate the theoretical +results (see Fig. 5(b)). First, the transmitting antenna +is fixed at 0◦, whereas the receiving antenna is moved +along the scanning track with a step of 2.5◦. The signal +reflected from the metasurface is measured by the receiv- +ing antenna at different angles θr. Then, the transmitting +antenna is fixed at 45◦ and the receiving antenna is scan- +ning its position to measure the reflected signal in the +other half space. As shown in Fig. 5(c), the main peaks +of reflections for both two incidences occur at θr = 45◦, +which is an expected result according to the theory and +simulations. There is another reflection peak at θr = 0◦ +for the normal incidence case, which is about −10 dB +lower than the main peak, corresponding to a low spec- +ular reflection at 15.22 GHz. +To estimate the amplitude efficiency of the metasurface +at all three reflection channels, we replaced the metasur- +face by a copper plate of the identical size and measured +the specular reflection signal amplitudes from the refer- +ence uniform metal mirror for θi = 2.5◦ (approximately +normal incidence), 22.5◦, and 45◦ incidence angles. The +specular reflection efficiency of the metasurface for 0◦ and +45◦ illuminations are calculated by normalizing the signal +amplitude by the amplitude of the signal reflected from +the reference plate, illuminated at 2.5◦ and 45◦ angles, re- +spectively. As shown in Fig. 5(d), at the design frequency +of 15.22 GHz, the specular reflection efficiencies at 0◦ and +45◦ (ξ−1 and ξ′ +0) equal 0.8% and 18.6% (the relative am- +plitude is 0.431), respectively. For the anomalous reflec- +tion at the n = 0 mode for the normal incidence, the re- +flection angle is θr = arcsin(15.22/( +√ +2f)), which equals +45◦ at 15.22 GHz and varies from 63.7◦ to 36.7◦ as the +frequency changes from 12 GHz to 18 GHz. Therefore, +we choose the signal data of a different receiving angle θr +calculated according to different frequency band and nor- +malize its signal amplitude by the signal amplitude from +the reference mirror for different θr/2 incidence angles. +Additionally, we divide the obtained value by an esti- +mated correction factor [37] +� +cos(θr)/ cos(θr/2), which +gives the ratio between the theoretically calculated sig- +nal amplitudes from an ideal metasurface (of the same +size and made of lossless materials) and a perfectly con- +ducting plate. +At the design frequency of 15.22 GHz, +the correction factor is equal to 0.91, thus the reflection +efficiency is calculated as 12%(the relative amplitude is +0.412), as shown in Fig. 5(d). The measured efficiency is +in good agreement with the results obtained using numer- +ical simulations (see Fig. 5(a)), except for some ripples in +the ξ0 curve caused by the discrete angular scanning step +in the measurement. The relative amplitudes of reflec- +tions for both incidences at the n = 0 mode are almost +equal in the measurements, verifying the capability for +CPA. +To experimentally verify the phase-controlled reflec- +tion by the metasurface, in the last measurement shown +in Fig. 6(a), two transmitting antennas fed via a power +divider illuminate the metasurface normally and at 45◦. +A receiving antenna is placed at the 45◦ angle to mea- +sure the total power reflected by the metasurface under +two simultaneous illuminations. To avoid severe insertion +loss caused by the use of a phase shifter in one branch, +which may increase the amplitude inequality between two +beams, we mimic the phase-difference-tuning process by +moving the metasurface along the x direction. As seen in +Fig. 6(b), the phase difference between the two beams is +linearly varying when we change the horizontal position +of the metasurface. Therefore, this shift is equivalent to +a phase change between the two beams. To ensure the +effectively-illuminated area of the metasurface to remain +stable during the moving process, we put two pieces of ab- +sorbing foam on top of both sides of the sample. The to- +tal received power, normalized by the maximum power of +reflected wave is changing with varying the distance ∆x. +As is seen in Fig. 6(c), the modulation depths reach 0.15 +and 0.04 at 15.22 GHz and 15.47 GHz, respectively. This +result indicates that coherent enhancement and cancella- +tion near the design frequency can be achieved by tuning +the phase difference of the two incident beams. The pe- +riod of the modulation is about 29 mm, almost equal to + +9 +Receiving +antenna +Transmitting +antenna 1 +Transmitting +antenna 2 +Moving direction +Absorbing + foam +Absorbing +foam + +(a) +∆∅ = 2𝜋∆𝑥/𝐷 +𝜃 +∆∅ +O’ +O +(b) +"x (mm) +0 +10 +20 +30 +40 +Normalized Recieved Power +0 +0.2 +0.4 +0.6 +0.8 +1 +13GHz +15.22GHz +15.47GHz +17GHz +(c) +FIG. 6. +(a) Experimental setup. Two transmitting antennas fed via a power divider illuminate the metasurface normally and +at 45◦. A receiving antenna is placed at 45◦ to measure the total reflected power. Due to the periodicity of the metasurface, +continuously-changing phase difference between the two beams can be emulated by moving the metasurface horizontally along +the impedance variation direction. Two pieces of absorbing foam are put on both sides, ensuring that the effective exposure +area of the metasurface remains fixed when the surface is shifted. (b) The reference point O is the intersection point of the 0◦ +and 45◦ beams on the metasurface when the phase difference is 0. The phase difference at a distance ∆x from the reference +point O is ∆φ = 2π∆x/D, which is linearly varying as a function of the horizontal distance ∆x. (c) The normalized received +power for different metasurface positions at 13, 15.22, 15.47, and 17 GHz. +the period of the metasurface, which validates the theo- +retical analysis. However, at the frequency far from the +designed one, for instance at 13 GHz and 17 GHz, the +coherent phenomenon becomes much weaker, as is seen +in Fig. 6(c), due to a mismatch of the main reflection +angles and the reflection amplitudes of the normally and +obliquely incident waves. +V. +DISCUSSION +We have demonstrated coherent perfect absorption of +two beams incident at arbitrary angles. It has been found +that this effect is possible for relative beam amplitudes +within a certain range using a gradient passive planar +structures. When these two incidences change into out- +of-phase state, reflections at all three propagating chan- +nels come out. To realize coherent control of reflection +with single direction, the other parasitic reflections can +be suppressed by introducing unidirectional evanescent +modes excitation. To realize a larger reflection for out- +of-phase scenario, we use an optimization algorithm to +search for an optimum solution of grid impedance profile +and substrate thickness, which is powerful when many +degrees of freedom are required in multi-channel meta- +surface design. In the other design methodologies such as +non-local metasurface [37] and plasmonic grating [23, 48], +where the interference between all the elements of a unit +cell are important for the device performance, a brute- +force optimization process in full-wave simulations is re- +quired, which is time consuming and even cannot work +when multiple input beams and multi-functionalities for +multiple channels are involved. +Compared with them, +our approach is much more robust and efficient due to +a rigorous theoretical analysis, particularly by introduc- +ing unidirectional evanescent mode in the scattered field +to eliminate parasitic reflections. Moreover, the angle- +dependence of the impedance of substrate is also con- +sidered in our algorithm, which is vital in metasurface +design for multiple-angle incidence scenarios [49, 50]. +We have realized a gradient metasurface with angular- +asymmetric coherent perfect absorption and reflection +functionalities. The concept of wave control via evanes- +cent harmonics engineering and independent control of +the electromagnetic response for multiple illuminations +can be applied for engineering multi-functional wave pro- +cesses. Metasurface-based designs are attractive in prac- +tical applications. +For example, by placing a planar +structure on a metal-grounded dielectric layer, the veloc- +ity or position of the object can be detected by monitor- + +10 +ing the total reflection of such a object under two coher- +ent illuminations. Additionally, we hope that this work +can find promising applications in phased-array anten- +nas, one-side detection and sensing, and optical switches +with low insertion loss. +VI. +METHODS +Design and modeling of the metasurface +The prototype presented in this work was designed +for operation at 15.22 GHz. The grid impedance is dis- +cretized into 6 sub-cells, and each sub-cell is divided into +4 equal sub-sub-cells. +The effective grid impedance of +each sub-sub-cell is retrieved from simulated reflection +coefficient (S11) through the transmission-line method +approach (see the Supplementary Material[43]). Numeri- +cal simulations are carried out using a frequency-domain +solver, implemented by CST MWS. Excitations propa- +gating along the z-direction from port 1 with the electric +field along the y-direction and the magnetic field along +the x-direction are used in the simulations to obtain the +S11 parameter. The dimensions of all the elements in the +unit cells are designed and optimized one by one to fit +the theoretically found required surface impedance. +Once the dimensions of all the elements in the unit +cells are found, we perform numerical simulations of the +unit cell in CST MWS for the normal and 45◦ incidences. +The simulation domain of the complete unit cell was D× +Dy × D (along the x, y, and z directions), the unit cell +boundary condition and the Floquet port were set. The +scattered fields for the normal and 45◦ incidences were +calculated by subtracting the incident waves from the +total fields. Finally, the total scattered fields when the +metasurface is illuminated by two waves silmutaneously +were obtained by adding the scattered field of each single +beam with different phase differences. +Realization and measurement +The ITO pattern of the metasurface was manufactured +using the picosecond laser micromachining technology on +a 0.175-mm-thick ITO/PET film. The sample comprises +10 unit cells along the x axis and 66 unit cells along the +y axis [Fig. 5(b)] and has the size of 14.15λ × 10.04λ = +278.9 mm × 198 mm. The ITO/PET film was adhered +to a 4.95-mm-thick F4BTM substrate with ϵ = 5.8(1 − +j0.01) backed by a copper ground plane. +The operation of the designed metasurface was tested +using a NRL-arc setup [Fig. +5(b)]. In the experiment, +two double-ridged horn antennas with 17 dBi gain at +15.22 GHz are connected to a vector network analyzer +as the transmitter and receiver. The metasurface was lo- +cated at a distance of 2 m (about 101λ) from both the +transmitting and receiving antennas where the radiation +from the antenna can be approximated as a plane wave. +The antennas are moved along the scanning track to mea- +sure the reflection towards different angles. 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This research was also sup- +ported by the Natural Science Foundation of Zhejiang +Province(LY22F010001), the Natural Science Founda- +tion of China (61701268), and the Fundamental Research +Funds for the Provincial Universities of Zhejiang. +IX. +AUTHOR CONTRIBUTIONS +S.M.Z. and X.C.W. conceived the study. S.M.Z. per- +formed the numerical calculations, and designed the sam- + +12 +ples. S.M.Z. conducted the experiment. S.M.Z., X.C.W., +and S.A.T. wrote the paper. +S.A.T. supervised the +project. All authors contributed to scientific discussions +and editing the manuscript. +X. +COMPETING INTERESTS +The authors declare no competing interests. +XI. +ADDITIONAL INFORMATION +Supplementary information The online version +contains supplementary material available at https:xxxx. +Correspondence and requests for materials should +be addressed to Shuomin Zhong or Xuchen Wang. + diff --git a/69E1T4oBgHgl3EQfBgJT/content/tmp_files/load_file.txt b/69E1T4oBgHgl3EQfBgJT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf504f34e34fe659e68f5c82c90fbace7a23a1d0 --- /dev/null +++ b/69E1T4oBgHgl3EQfBgJT/content/tmp_files/load_file.txt @@ -0,0 +1,839 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf,len=838 +page_content='Coherent control of wave beams via unidirectional evanescent modes excitation Shuomin Zhong1*,∗ Xuchen Wang2*, and Sergei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Tretyakov3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' School of Information Science and Engineering, Ningbo University, Ningbo 315211, China 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Institute of Nanotechnology, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Department of Electronics and Nanoengineering, Aalto University, Finland Conventional coherent absorption occurs only when two incident beams exhibit mirror symmetry with respect to the absorbing surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=', the two beams have the same incident angles, phases, and amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In this work, we propose a more general metasurface paradigm for coherent perfect absorption, with impinging waves from arbitrary asymmetric directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' By exploiting excitation of unidirectional evanescent waves, the output can be fixed at one reflection direction for any amplitude and phase of the control wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' We show theoretically and confirm experimentally that the relative amplitude of the reflected wave can be tuned continuously from zero to unity by changing the phase difference between the two beams, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' switching from coherent perfect absorption to full reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' We hope that this work will open up promising possibilities for wave manipulation via evanescent waves engineering with applications in optical switches, one-side sensing, and radar cross section control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' INTRODUCTION Coherent control of propagation of a wave beam by tuning the amplitude and phase of another beam is a very promising approach to realize ultra fast optical devices for optical computing, sensing, and other applications [1– 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' One of the most important effects in coherent control of light is coherent perfect absorption [12–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In these devices, the level of absorption of one beam illuminating a thin sheet is controlled by another coherent beam that illuminates the same sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In earlier works, coherent perfect absorption (CPA) was achieved only when with illumination from different sides of a homogeneous lossy layer and for two incident waves at the same angle [12, 13, 15, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The mecha- nism of coherent perfect absorption is destructive cancel- lation of all scattered beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For homogeneous coher- ent perfect absorbers, there are only specular reflection and non-diffractive transmission, allowing coherent ab- sorption only with illumination of both sides and at the same incidence angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' From the theoretical point of view and for many applications, it is important to achieve co- herent control of output for illuminations from the same side of the metasurface sheet at two or more arbitrary incidence angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' [17, 18, 23], coherent perfect absorption and scattering for two angularly asymmetric beams are realized by using surface plasmon-polariton (SPP) excitation at silver-based diffraction groove grat- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' However, such plasmonic grating designs have limi- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In particular, the structures are non-planar and operate only for TM modes at optical frequencies, where SPP are supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Moreover, there are always two out- put beams for different values of the phase of the control waves, one of which may cause undesired noise to the useful output signal due to parasitic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' This is- sue is critical in applications such as optical computing [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' ∗ Email: zhongshuomin@nbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='cn, xuchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='wang@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='edu In this decade, the emergence of gradient metasurfaces [25–28] and metagratings [29–35] has opened a new av- enue for manipulation of light for arbitrary incidence an- gles and versatile functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For periodical metasur- faces or metagratings with the period larger than half of the wavelength, the incident plane wave from one direc- tion will be scattered into multiple directions, and the power carried by the incident wave can be redistributed among a number of diffraction modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Based on this concept, several metasurface devices with perfect anoma- lous reflection working at microwaves [36, 37] and optical bands [38] have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' However, in these previ- ous works, the functionality of metasurfaces is designed only for one incident angle and the response for other illu- minations is actually not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To design metasur- faces with coherent control functions for multiple simul- taneously incident coherent beams from different direc- tions, the matching conditions of amplitude, phase, and wavevector(direction) of the scattering modes between all incidences are required [35, 39, 40], which is almost an impossible task using traditional gradient phase methods [25, 36] and brute-force numerical optimizations [37, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In this work, we perform inverse designs of CPA meta- surfaces by solving the surface impedance satisfying the boundary condition determined by two coherent incident waves from two arbitrary angles and the desired total scattered waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The engineering of evanescent waves in the scattered fields without altering the desired far- field outputs provides significant freedom in the CPA metasurface design, making another functionality of co- herent control of reflection with a single direction possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' It is demonstrated that excitation of unidirectional evanescent waves propagating along the surface in the direction of the incident-wave wavevector can be used to achieve single-direction output in coherently controlled optical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Furthermore, a mathematical optimiza- tion method based on scattered harmonics analysis [42] is utilized to find the surface-impedance profile that si- multaneously ensures the CPA and coherent maximum reflection (CMR) in a single direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Thereafter, the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='02852v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='app-ph] 8 Jan 2023 2 substrate parameters are invoked as additional degrees of freedom in the optimization model, realizing reflection ef- ficiency of 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As an example, we experimentally vali- date the CPA gradient metasurface design in microwaves for TE-polarized waves by engineering the Indium Tin Oxide (ITO) film mounted on a grounded dielectric sub- strate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' It is showed that the normalized output power can be continuously controlled between 0 and 1 by tun- ing the phase of the control wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' DESIGN CONCEPT Dx x z Zs(x) θ1 θ2 I1 I2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' General scattering scenario for a periodically modu- lated impenetrable impedance surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Two coherent beams I1 and I2 are simultaneously incident from two angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Let us consider an impenetrable reciprocal metasur- face whose surface is periodically modulated along the x-direction, with the period Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The surface is in the xy-plane of a Cartesian coordinate system (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The metasurface is simultaneously illuminated by two TE(s)-polarized plane waves I1 and I2 at the incidence angles θ1 and θ2 (θ1 > θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The electric field amplitudes of the two beams I1 and I2 is E1 = E0 and E2 = αE0, respectively (α is the amplitude ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The phase differ- ence between them is ∆φ=0, defined at the origin point (x = 0, z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The electromagnetic properties of the metasurface can be characterized by the locally-defined surface impedance that stands for the ratio of the tangen- tial electric and magnetic field amplitudes at the surface plane Zs(x) = Et(x)/Ht(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The field reflected by a periodically modulated meta- surface can be interpreted as a sum of Floquet harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The tangential wavenumber of the n-th harmonic is re- lated to the period and the incident wavenumber k0 as krxn = k0 sin θi + 2πni/Dx, where i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The corre- sponding normal component of the reflected wavenumber equals krzn = � k2 0 − k2rxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' If |krxn| is greater than the incident wave number, the wave is evanescent and it does not contribute to the far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For the harmonic wave sat- isfying |krxn| < k0, krzn is real, and this wave is propagat- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The evanescent harmonics will be dissipated by the lossy surface and the propagating harmonics will propa- gate into the far-zone at the angles θrn = arcsin(krxn/k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In order to achieve coherent perfect absorption, it is nec- essary (but not sufficient) to ensure that all the diffracted propagating modes of two beams have the same set of angles θrn, that allows mutual cancellation, defining the period Dx = λ0/(sin θ1 −sin θ2) [43], where λ0 stands for the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Our aim is to achieve coherent perfect absorption for two coherent in-phase waves simultaneously incident on the metasurface at two different angles θ1 and θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' First, let us assume that no evanescent waves are excited for these two illuminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In the CPA case, there should be no reflected field at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Thus, the tangential components of the total electric field at the plane z = 0 can be written as Et(x) = E0(e−jk0 sin θ1x+αe−jk0 sin θ2x), where the time-harmonic dependency in the form ejωt is assumed and suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The corresponding total magnetic field reads Ht(x) = E0(cos θ1e−jk0 sin θ1x + α cos θ2e−jk0 sin θ2x)/Z0, with Z0 = � µ0/ϵ0 being the free-space wave impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The ratio of these electric and magnetic fields gives the required surface impedance ℜ(Zs) = Z0 cos θ1 + α2 cos θ2 + α cos Φ(cos θ1 + cos θ2) cos2 θ1 + α2 cos2 θ2 + 2α cos θ1 cos θ2 cos Φ , ℑ(Zs) = Z0 α(cos θ1 − cos θ2) sin Φ cos2 θ1 + α2 cos2 θ2 + 2α cos θ1 cos θ2 cos Φ, (1) where Φ = k0(sin θ1 − sin θ2)x is the linearly varying phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The real and imaginary parts of the surface impedance are even and odd functions of x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As is seen from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (1), the periodicity of the surface impedance is D = λ0/(sin θ1 − sin θ2), in accord with the above analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For passive metasurfaces, the real part of the surface impedance must be non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Con- sequently, the amplitude ratio should satisfy α ≥ 1 or α ≤ cos θ1/ cos θ2 to ensure passive solution for CPA by the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As an example, we consider two incident waves with incidence angles of (θ1, θ2) = (45◦, 0◦) and the same am- plitude, assuming α = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (Other scenarios with (θ1, θ2) = (60◦, −30◦), (75◦, 15◦) are illustrated in the Supplemental Materials[43], corresponding to differ- ent surface impedance profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=') As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(a), everywhere on the surface its resistance is non-negative, demonstrating that passive gradient periodic surfaces can realize CPA for two asymmetric incident beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To analyze the mechanism of CPA by the periodic impedance surface further, we can determine the ampli- tudes of all the Floquet scattered harmonics for general plane-wave illumination, using the method reported in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The total reflected field can be represented as an infinite sum of Floquet harmonic modes: Er = ∞ � n=−∞ Ane−jkrznze−jkrxnx, (2) where An is the complex amplitude of the n-th Floquet harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Because the surface modulation is periodical, the surface admittance Ys(x) = 1/Zs(x) can be expanded 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='8 1 200 0 200 400 (a) 8 6 4 2 0 2 4 6 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='8 1 200 0 200 400 (c) 6 4 2 0 2 4 6 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5 (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (a) Analytical surface impedance over one period to realize CPA for two incidence beams with (θ1, θ2) = (45◦, 0◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (b) Magnitudes of the complex amplitudes of different Floquet scattered harmonics (normalized by the amlpitude of the incident electric field E0) when the gradient surface is illuminated by single-beam incidences at 45◦ and 0◦, and for two- beam incidences in phase and out of phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (c) Optimized surface impedance profile over one period to realize CPA for in-phase incidences and single-direction reflection for out-of-phase incidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The optimized Fourier coefficients of Ys(x) read g0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='654 × 10−3 + j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='724 × 10−11, g1 = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='770 × 10−4 − j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='045 × 10−10, g2 = −(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='565 + j4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='581) × 10−5, g3 = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='143×10−8 +j5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='720×10−6, g4 = (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='644+j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='992)×10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (d) Amplitudes of scattered harmonics when the optimized gradient surface in (c) is illuminated by single-beam incidences at 45◦ and 0◦, and for two-beam incidences in phase and out of phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' into Fourier series: Ys(x) = +∞ � n=−∞ gne−j2nπx/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (3) A Toeplitz matrix Ys which we call the admittance ma- trix is determined only by the Fourier coefficients of the modulation function and filled with Ys(r, c) = gr−c at the r-th row and c-th column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The reflection matrix is found as [44] Γ = (Y0 + Ys)−1 (Y0 − Ys), (4) where Y0 = Z−1 0 is a diagonal matrix with its main entry representing the admittance of each space har- monic, which is Y0(n, n) =krzn/ω0µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The amplitudes An of reflected harmonics for a given m-th order Flo- quet harmonic of the incident wave can be calculated as An = Γ(n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Note that Γ is a (2N + 1) × (2N + 1) square matrix and the columns and rows of Γ are indexed from −N to +N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' When the surface is illuminated by two waves simultaneously, the amplitudes of all the Floquet harmonics are linear superpositions of all harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As is seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(b), when the two incident waves are in phase, all the harmonics have zero amplitude, meaning that CPA with no reflected fields occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' How- ever, when the two incident waves are out of phase, the reflected harmonics come out, including both propagat- ing modes and evanescent ones, proving that the perfect absorption effect is phase-coherent, different from perfect absorption for two angles [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To understand the mech- anism of CPA in the metasurface better, the harmonics 4 of the reflected field when single beams illuminate the surface separately are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(b), the complex amplitudes of every scattered harmonic are equal and 180◦ out of phase (the phases are not shown here) for 45◦ and 0◦ incidences, resulting in destructive cancellation when the two beams illuminate simultane- ously in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Here, the propagating harmonic of the order n = 0 is defined at the specular direction of θ1 for both incidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' By properly designing the metasurface with the periodicity of D = λ0/(sin θ1 − sin θ2), three propagating modes corresponding to n = 0, −1, −2 are created, and all the diffracted modes for both incidences have the same wave vectors, ensuing coherent interfer- ence for all corresponding harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In the out-of-phase incidence case, the amplitudes of all the scattered har- monics double as compared to the single-beam case, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The analytical method to solve the surface impedance boundaries used above is based on the objective to real- ize CPA with the amplitudes of both scattered propagat- ing and evanescent harmonics being zero when two co- herent beams illuminate the metasurface simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Indeed, the amplitudes of evanescent surface modes can be nonzero without breaking the CPA condition, because they do not radiate into the far zone and their power will be dissipated at the lossy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Thus, the so- lution of the surface impedance to achieve CPA is not unique if a certain set of evanescent waves with unknown complex amplitudes is excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In addition to CPA, we invoke another functionality of coherent control of reflec- tion with single direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' eliminating the unwanted outgoing beams at n = −1, −2 orders and keeping the n = 0 order with the maximal amplitude, when the two coherent incident beams are out-of-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In this case, finding the complex amplitudes of infinite numbers of evanescent modes for each incidence scenario is difficult or even impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Thus, instead of using the analyti- cal method of calculating the surface impedance profile according to the total fields on the boundary, we ap- ply a mathematical optimization algorithm described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' [42] and based on the scattering matrix calculation to find a surface impedance profile that simultaneously ensures the coherent control capability for absorption and reflection of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' First, the metasurface is mod- elled as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To suppress propagating modes at the negative orders (n = −1, −2) and ensure that only the reflection channel at 45◦ is open, the Fourier series of the surface admittance Ys(x) are set to be unilateral as Ys(x) = �4 n=0 gne−j2nπx/D with non-negative-order se- ries coefficients being nonzero (only five coefficients from g0 to g4 are used for improving optimization efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' This setting is reasonable because the unilateral surface admittance, making the admittance matrix Ys a lower triangular matrix, can lead to the reflection matrix Γ also being a lower triangular matrix, as is seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Consequently, the scattered modes contain only components of non-negative orders (n ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' This effect highlights the role of unidirectional evanescent fields as a mechanism of suppressing propagating modes at the negative orders (n = −1, −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Moreover, to ensure that the grid is a passive metasurface, we need to impose con- straints ℜ(Ys) ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=', ℜ(g0) ≥ |g1| + |g2| + |g3| + |g4|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Secondly, the optimization goal is formulated as 6 ob- jectives, including (|A0|, |A−1|, |A−2|) = (0, 0, 0) for the in-phase scenario, and (|A0|, |A−1|, |A−2|) = (A0max, 0, 0) for the out-of-phase scenario, where A0max is the maxi- mum magnitude of reflection in the out-of-phase case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In each trial of the optimization, an array of gn is as- sumed, and the value of all the objectives are calcu- lated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The sum of errors calculated for all the objectives is defined as a cost function C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' By em- ploying MultiStart and fmincon optimization algorithms, the maximum magnitude of the out-of-phase reflection A0max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='34 is searched out, and the minimum value of C close to zero is achieved, meaning that the solu- tions of the impedance profile to realize the desired EM responses including CPA and single-direction-reflection are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Figure 2(c) shows a typical optimized solution of the surface impedance, which exhibits positive resistance ev- erywhere along the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The calculated ampli- tudes of scattered harmonics for single-beam incidences at 45◦ and 0◦, and for two-beam incidences in phase and out of phase, for the impedance profile in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(c), are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(d), revealing the unilateral characteristic of scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' We can see that the propagating compo- nents at n = −1, −2 orders are suppressed successfully by exciting the unidirectional evanescent wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The only remaining propagating reflected channel is n = 0 order at the outgoing angle of 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' When two incoming beams are in phase, the reflected propagating harmonic (n = 0) of each beam cancel each other because they have the same amplitude and π-reflection-phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Dis- tinct from the zero-amplitude of all the harmonics for the in-phase CPA scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(b), the CPA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(d) occurs with non-zero-amplitude evanescent modes in the n ≥ 1 orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The amplitude of reflected electric field at 45◦ (n = 0) is doubled into A0max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='34 when two incoming beams are out of phase (∆φ = π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' We can conclude that the reflected power at 45◦ can be contin- uously controlled by phase tuning of the control beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' When the two beams are out of phase, the reflected power normalized by the incident beam power at 45◦ has the maximum reflection efficiency of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='56 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' OPTIMIZATION AND PRACTICAL DESIGN Low efficiency of the above design based on the im- penetrable impedance model calls for optimization with the help of additional degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' One possibility can be the use of one or more parameters of the actual implementation of the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In general, the impedance surface in the impenetra- ble model used above can be realized as a periodic metal 5 x z q1 D h I1 I2 n = 0 n = -1 n = -2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Schematics of reflection amplitude modulation for two coherent waves with the phase difference ∆φ incident on a periodic sheet over a grounded dielectric slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The amplitude of the output beam is modulated continuously by varying ∆φ, and switched between 0 (coherent perfect absorption) and 1 (coherent maximum reflection) when ∆φ is switched between even and odd multiples of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' pattern on a thin grounded dielectric slab, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The structure can be considered as a grid admit- tance of the top pattern with a shunt admittance of the grounded substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The characteristic admittance ma- trix Yd of the grounded substrate contains only diagonal terms Yd(n, n), where Yd(n, n) is the admittance of the n-th harmonic, and it is expressed as Yd(n, n) = kd rzn/[jµ0ω0 tan(kd rznh)], (5) where kd rzn = � ω2 0ϵ0ϵdµ0 − k2rxn is the normal compo- nent of the wavevector in the substrate (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='S23 of the Supplemental Material of [42]), ϵd and h are the permittivity and thickness of the substrate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The reflection matrix is calculated as Γ = (Y0 + Yg + Yd)−1(Y0−Yg−Yd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' When the thickness h is ultra-thin compared with the wavelength, for low-order harmonics we have tan(kd rznh) ≈ kd rznh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As is seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (5), the admittance for low-order harmonics equals approxi- mately to 1/(jµ0ω0h), unrelated to the harmonic num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Thus, we can approximately design the top surface with the grid admittance Yg(x) = 1/Zs(x) − Yd(0, 0) us- ing the optimized surface impedance Zs(x) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 2(c), similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Due to the lack of freedom in the sub- strate design, the evanescent fields engineering is quite limited in the impenetrable model, resulting in a low reflection efficiency (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='56 %) in the out-of-phase sce- nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In order to implement CPA with a high reflec- tion efficiency, we need to use the substrate parameters as additional degrees of freedom in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Since the admittance of the grounded substrate with a moderate thickness strongly depends on the harmonic number, the need of complicated matrix operations makes it impos- sible to analytically solve the grid impedance and sub- strate parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Thus, the optimization algorithm is extended by introducing the admittance matrix Yd of the grounded substrate, as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' [42], to search for an optimum solution for the grid impedance profile and substrate thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' According to the results of the impenetrable model, the period of the impedance sheet modulation is set to D = λ0/ sin 45◦, with three propagating channels at −45◦, 0◦, and 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The Fourier series of the grid admit- tance is set to be unilateral as Yg(x) = g0 + g1e−j2πx/D, ensuring that only the reflection channel at 45◦ is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In the optimization process, two Fourier terms g0 and g1 with four unknowns (the real and imaginary parts) are considered here to reduce complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The substrate thickness h is another unknown, and an available sub- strate with the permittivity ϵd = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='8(1 − j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='002) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The optimization goal is formulated as 6 objectives, the same as the objectives in the impenetrable model above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The constraints ℜ(Yg) ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=', ℜ(g0) ≥ |g1| are imposed to ensure the grid to be a passive metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Addi- tionally, to make the reactance easier to implement by patterning a thin conductive surface, another constraint ℑ(g0) ≥ |g1| is set to ensure that the surface reactance is always capacitive at all points of the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The maximum magnitude of reflection A0max in the out-of-phase scenario is searched out to be about 1 in the optimization, meaning that a reflection beam at 45◦ with amplitude equal to the incident beam I1 is obtained [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' It reveals that the invocation of sub- strate design provides an important additional degree of freedom in engineering auxiliary evanescent modes to find a surface impedance that can realize the desired optimum scattering properties for all incidence scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The optimized Fourier coefficients of the grid ad- mittance Yg(x) read g0 = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='599 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='054j) × 10−3 and g1 = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='807 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='463j) × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The optimal substrate thickness is h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2525λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The required grid impedance which is passive and capacitive along the metasurface is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Next, we analyse the scattered harmonics for the de- signed impedance sheet on the metal-backed dielectric substrate [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The reflection coefficient of the metasurface has the same magnitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5 at n = 0 order for 45◦ and 0◦ single-beam incidences, resulting from destructive interference when these two beams are in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For the out-of-phase scenario, the normalized magnitude of the reflected field at n = 0 order (45◦) is about unity, which means that the reflected power effi- ciency reaches 100% (normalized by the incoming power of the 45◦ beam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Parasitic reflections into other direc- tions (n = −1, −2) are seen to be negligible, due to the unilateral property of the admittance of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The evanescent harmonics are also unidirectional, but quite weak with the magnitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='008 at n = 1 order, and they are absorbed by the lossy structure, ensuring a CPA state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Figure 4(c) illustrates the phase-controlled modu- lation of reflections at three propagating orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The re- flection coefficient at 45◦ can be continuously controlled from 0 to 1 by phase tuning, with the other two par- asitic reflections maintained very close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' This phase-sensitive modulation between CPA and coherent 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='8 1 200 100 0 100 (a) 8 6 4 2 0 2 4 6 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5 1 (b) 0 1 2 ( ) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='8 1 Amplitude (|An|/E0) n=0 n=-1 n=-2 (c) 𝐸𝑠𝑐/𝐸0 1 0 1 2 3 2 3 Df = 0 Df = p (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (a) The optimized and discretized grid impedance distribution over one period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (b) Amplitudes of the scattered harmonics when the optimized gradient metasurface is illuminated by a single beam at 45◦ and 0◦, and for two-beam in-phase and out-of-phase illuminations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (c) The normalized amplitudes of three propagating harmonics (n = 0, −1, −2) with a varying phase difference ∆φ between incidences at 45◦ and 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (d) The scattered electric fields and power density flow distributions for the metasurface modeled by the discretized grid impedance (step-wise approximation, 6 subcells per period) on top of a grounded dielectric substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Two plane-wave incidences are in phase (left) and out of phase (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' maximum reflection (CMR) without parasitic reflections is important in light switching applications where a low- return-loss characteristic is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' See the Supplemen- tal Animation [43] for the switch of reflected beam by an incident phase-controlled wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In implementations, the influence of discretization on the metasurface performance is an important factor (see detailed analysis of scattered harmonics versus the num- ber of subcells in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' We use six subcells over a period and each discretized impedance value is set at the central point of each subcell, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The scattered fields from the ideal impedance sheet on the metal-backed dielectric slab for both in-phase and out-of-phase incidences are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 4(d), using full-wave simulations in Comsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The reflected field dis- tribution confirms that the metasurface with six subcells per period possesses the desired response: nearly per- fect absorption with reflection amplitude of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='023 for two in-phase illuminations and nearly total reflection at 45◦ for two out-of-phase illuminations, relative to the intensity of the 45◦ incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' It is seen that the top lossy sheet and reflective ground separated by the slab act as a leaky-wave cavity with enhanced fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For the in-phase scenario, the direct reflections of the top surface and leaky wave components of the cavity destructively cancel out, and all the power is absorbed by the lossy surface, causing CPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' By changing the initial phase dif- ference between the two coherent incidences into π, con- structive interference occurs among these components, which results in nearly total reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Note that in the out-of-phase case a half of the total incoming power (two incident beams) is still absorbed by the lossy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' PHYSICAL IMPLEMENTATION AND EXPERIMENTAL VALIDATION The theory above is general and applies to any fre- quency, and we choose the microwave band for a proof of concept demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The required impedance pro- file at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz is realized using an ITO film with the surface resistance of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5 Ω/sq supported by a grounded 7 f (GHz) 12 13 14 15 16 17 18 E/ciency 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content="8 1 90 9-1 9-2 9'0 9'-1 9'-2 Simulated (a) Transmitting antenna Receiving antenna Metasurface Scanning track (b) 3r (deg) 80 -60 -40 -20 0 20 40 60 80 S21;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='m (dB) 120 110 100 90 80 70 60 3i = 0o 3i = 45o (c) f (GHz) 12 13 14 15 16 17 18 E/ciency 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content="8 1 90 9-1 9'0 Measured (d) FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (a) Simulated and (d) measured reflection efficiency spectrum for different diffracted modes of each single beam at 0◦ (solid lines) and 45◦ (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (b) Schematic of the experimental setup (top) and photograph of the fabricated sample (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (c) Signals at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz measured by the receiving antenna at different orientation angles with the transmitting antenna at 0◦ and 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' dielectric slab with the thickness h = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='95 mm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The detailed parameters and struc- tures of each unit cell are presented in the Supplementary Material[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Due to the resolution limitation of picosec- ond laser micro-processing, the complex grid impedance is implemented as six subcells, and each subcell is divided into four equal sub-subcells in order to make the local design of the gradient impedance more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' By struc- turing the homogeneous resistive ITO film into I-shaped cells, the required grid resistance and reactance on a sur- face in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 4(a) can be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For y-polarization inci- dent waves, such I-shaped resonators can be modeled as RLC series circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The required resistance is realized by tailoring the width and length of the ITO strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Smaller width and longer length result in higher grid resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The required reactance can be tailored by adjusting ca- pacitance of the gap, which can be increased by narrow- ing the gap or increasing the length or width of the bar, with a small influence on the resistive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The 5th and 6th subcells degenerate into strips, to implement resistive parts as close to the theoretical value as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' How- ever, there are still deviations of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6 Ω and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='1 Ω from the theoretical resistances of the 5th and 6th subcells, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The deviation can be eliminated if an ITO film with a lower surface resistance is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To sim- plify the fabrication process, we neglect this deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The impact is analyzed theoretically, showing that the reflection amplitude in the in-phase scenario increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='023 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='065, which is tolerable in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Since the two beams with 0◦ and 45◦ incidence angles illuminate the surface simultaneously, all the elements should have angle-independent surface impedances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The OLMS EILMS SWAi CILAS S-iD SWAiN SWiM SWi8 I-shaped resonators have angle-insensitive impedance un- der TE incidences, satisfying this requirement [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In the strips of the 5th and 6th subcells, narrow slits are cut out to reduce the angular sensitivity of the impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' All the subcells have been optimized with the geometrical dimensions specified in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Figure 5(a) shows the simulated frequency response of the metasurface for the normal and 45◦ incidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For the normal illumination, strong reflections occur at n = −1 and n = 0 harmonics (denoted as ξ−1 and ξ0), and the amplitude of the n = −2 scattered propagating mode is nearly zero in the whole frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The reflection at the n = −1 mode (specular reflection at 0◦) also has a near-zero dip at the design frequency of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz, and the reflection efficiency at the n = 0 mode(anomalous re- flection at 0◦) is about 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='9% (the relative amplitude is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Note that for anomalous reflection, the efficiency is calculated as ξ = (Er/Ei)2cos θr/cos θi [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For the 45◦ illumination, the reflections at both n = −1 and n = −2 modes (ξ′ −1 and ξ′ −2) are close to zero, and the efficiency at the n = 0 mode (ξ′ 0) is about 21% at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz (the relative amplitude is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Therefore, at the operating frequency 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz, the reflected modes for both incidences at the outgoing angle of 45◦ are al- most equal-amplitude, satisfying the condition of CPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The scattered electric field distributions of the designed metasurface illuminated by two beams in the in-phase and out-of-phase scenarios obtained from full-wave sim- ulations are presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' It can be seen that when the two illuminations are in phase, the total scat- tered fields are quite small (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='02), indicating nearly per- fect coherent absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' However, when the two illumi- nations are switched into the out-of-phase state, the rel- ative amplitude of the scattered fields is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='91, and the coherent maximum reflection is mainly along the 45◦ direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' We have fabricated a sample (see Methods) and car- ried out several experiments to validate the theoretical results (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' First, the transmitting antenna is fixed at 0◦, whereas the receiving antenna is moved along the scanning track with a step of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The signal reflected from the metasurface is measured by the receiv- ing antenna at different angles θr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Then, the transmitting antenna is fixed at 45◦ and the receiving antenna is scan- ning its position to measure the reflected signal in the other half space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 5(c), the main peaks of reflections for both two incidences occur at θr = 45◦, which is an expected result according to the theory and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' There is another reflection peak at θr = 0◦ for the normal incidence case, which is about −10 dB lower than the main peak, corresponding to a low spec- ular reflection at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To estimate the amplitude efficiency of the metasurface at all three reflection channels, we replaced the metasur- face by a copper plate of the identical size and measured the specular reflection signal amplitudes from the refer- ence uniform metal mirror for θi = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5◦ (approximately normal incidence), 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5◦, and 45◦ incidence angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The specular reflection efficiency of the metasurface for 0◦ and 45◦ illuminations are calculated by normalizing the signal amplitude by the amplitude of the signal reflected from the reference plate, illuminated at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='5◦ and 45◦ angles, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 5(d), at the design frequency of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz, the specular reflection efficiencies at 0◦ and 45◦ (ξ−1 and ξ′ 0) equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='8% and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6% (the relative am- plitude is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='431), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For the anomalous reflec- tion at the n = 0 mode for the normal incidence, the re- flection angle is θr = arcsin(15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22/( √ 2f)), which equals 45◦ at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz and varies from 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='7◦ to 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='7◦ as the frequency changes from 12 GHz to 18 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Therefore, we choose the signal data of a different receiving angle θr calculated according to different frequency band and nor- malize its signal amplitude by the signal amplitude from the reference mirror for different θr/2 incidence angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Additionally, we divide the obtained value by an esti- mated correction factor [37] � cos(θr)/ cos(θr/2), which gives the ratio between the theoretically calculated sig- nal amplitudes from an ideal metasurface (of the same size and made of lossless materials) and a perfectly con- ducting plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' At the design frequency of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz, the correction factor is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='91, thus the reflection efficiency is calculated as 12%(the relative amplitude is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='412), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The measured efficiency is in good agreement with the results obtained using numer- ical simulations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 5(a)), except for some ripples in the ξ0 curve caused by the discrete angular scanning step in the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The relative amplitudes of reflec- tions for both incidences at the n = 0 mode are almost equal in the measurements, verifying the capability for CPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To experimentally verify the phase-controlled reflec- tion by the metasurface, in the last measurement shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 6(a), two transmitting antennas fed via a power divider illuminate the metasurface normally and at 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' A receiving antenna is placed at the 45◦ angle to mea- sure the total power reflected by the metasurface under two simultaneous illuminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To avoid severe insertion loss caused by the use of a phase shifter in one branch, which may increase the amplitude inequality between two beams, we mimic the phase-difference-tuning process by moving the metasurface along the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 6(b), the phase difference between the two beams is linearly varying when we change the horizontal position of the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Therefore, this shift is equivalent to a phase change between the two beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To ensure the effectively-illuminated area of the metasurface to remain stable during the moving process, we put two pieces of ab- sorbing foam on top of both sides of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The to- tal received power, normalized by the maximum power of reflected wave is changing with varying the distance ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' As is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 6(c), the modulation depths reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='04 at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='47 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' This result indicates that coherent enhancement and cancella- tion near the design frequency can be achieved by tuning the phase difference of the two incident beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The pe- riod of the modulation is about 29 mm, almost equal to 9 Receiving antenna Transmitting antenna 1 Transmitting antenna 2 Moving direction Absorbing foam Absorbing foam (a) ∆∅ = 2𝜋∆𝑥/𝐷 𝜃 ∆∅ O’ O (b) "x (mm) 0 10 20 30 40 Normalized Recieved Power 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='8 1 13GHz 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22GHz 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='47GHz 17GHz (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (a) Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Two transmitting antennas fed via a power divider illuminate the metasurface normally and at 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' A receiving antenna is placed at 45◦ to measure the total reflected power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Due to the periodicity of the metasurface, continuously-changing phase difference between the two beams can be emulated by moving the metasurface horizontally along the impedance variation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Two pieces of absorbing foam are put on both sides, ensuring that the effective exposure area of the metasurface remains fixed when the surface is shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (b) The reference point O is the intersection point of the 0◦ and 45◦ beams on the metasurface when the phase difference is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The phase difference at a distance ∆x from the reference point O is ∆φ = 2π∆x/D, which is linearly varying as a function of the horizontal distance ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' (c) The normalized received power for different metasurface positions at 13, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='47, and 17 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' the period of the metasurface, which validates the theo- retical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' However, at the frequency far from the designed one, for instance at 13 GHz and 17 GHz, the coherent phenomenon becomes much weaker, as is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 6(c), due to a mismatch of the main reflection angles and the reflection amplitudes of the normally and obliquely incident waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' DISCUSSION We have demonstrated coherent perfect absorption of two beams incident at arbitrary angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' It has been found that this effect is possible for relative beam amplitudes within a certain range using a gradient passive planar structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' When these two incidences change into out- of-phase state, reflections at all three propagating chan- nels come out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To realize coherent control of reflection with single direction, the other parasitic reflections can be suppressed by introducing unidirectional evanescent modes excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' To realize a larger reflection for out- of-phase scenario, we use an optimization algorithm to search for an optimum solution of grid impedance profile and substrate thickness, which is powerful when many degrees of freedom are required in multi-channel meta- surface design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In the other design methodologies such as non-local metasurface [37] and plasmonic grating [23, 48], where the interference between all the elements of a unit cell are important for the device performance, a brute- force optimization process in full-wave simulations is re- quired, which is time consuming and even cannot work when multiple input beams and multi-functionalities for multiple channels are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Compared with them, our approach is much more robust and efficient due to a rigorous theoretical analysis, particularly by introduc- ing unidirectional evanescent mode in the scattered field to eliminate parasitic reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Moreover, the angle- dependence of the impedance of substrate is also con- sidered in our algorithm, which is vital in metasurface design for multiple-angle incidence scenarios [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' We have realized a gradient metasurface with angular- asymmetric coherent perfect absorption and reflection functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The concept of wave control via evanes- cent harmonics engineering and independent control of the electromagnetic response for multiple illuminations can be applied for engineering multi-functional wave pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Metasurface-based designs are attractive in prac- tical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' For example, by placing a planar structure on a metal-grounded dielectric layer, the veloc- ity or position of the object can be detected by monitor- 10 ing the total reflection of such a object under two coher- ent illuminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Additionally, we hope that this work can find promising applications in phased-array anten- nas, one-side detection and sensing, and optical switches with low insertion loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' METHODS Design and modeling of the metasurface The prototype presented in this work was designed for operation at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The grid impedance is dis- cretized into 6 sub-cells, and each sub-cell is divided into 4 equal sub-sub-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The effective grid impedance of each sub-sub-cell is retrieved from simulated reflection coefficient (S11) through the transmission-line method approach (see the Supplementary Material[43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Numeri- cal simulations are carried out using a frequency-domain solver, implemented by CST MWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Excitations propa- gating along the z-direction from port 1 with the electric field along the y-direction and the magnetic field along the x-direction are used in the simulations to obtain the S11 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The dimensions of all the elements in the unit cells are designed and optimized one by one to fit the theoretically found required surface impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Once the dimensions of all the elements in the unit cells are found, we perform numerical simulations of the unit cell in CST MWS for the normal and 45◦ incidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The simulation domain of the complete unit cell was D× Dy × D (along the x, y, and z directions), the unit cell boundary condition and the Floquet port were set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The scattered fields for the normal and 45◦ incidences were calculated by subtracting the incident waves from the total fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Finally, the total scattered fields when the metasurface is illuminated by two waves silmutaneously were obtained by adding the scattered field of each single beam with different phase differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Realization and measurement The ITO pattern of the metasurface was manufactured using the picosecond laser micromachining technology on a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='175-mm-thick ITO/PET film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The sample comprises 10 unit cells along the x axis and 66 unit cells along the y axis [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 5(b)] and has the size of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='15λ × 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='04λ = 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='9 mm × 198 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The ITO/PET film was adhered to a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='95-mm-thick F4BTM substrate with ϵ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='8(1 − j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='01) backed by a copper ground plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The operation of the designed metasurface was tested using a NRL-arc setup [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 5(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' In the experiment, two double-ridged horn antennas with 17 dBi gain at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='22 GHz are connected to a vector network analyzer as the transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The metasurface was lo- cated at a distance of 2 m (about 101λ) from both the transmitting and receiving antennas where the radiation from the antenna can be approximated as a plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' The antennas are moved along the scanning track to mea- sure the reflection towards different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Time gating is employed to filter out all the multiple scattering noise signals received by the antenna [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reason- able request.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Photonics Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' 9, 2190–2195 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors are grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Viktar S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Asadchy for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' acknowledges support from China Scholarship Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' This research was also sup- ported by the Natural Science Foundation of Zhejiang Province(LY22F010001), the Natural Science Founda- tion of China (61701268), and the Fundamental Research Funds for the Provincial Universities of Zhejiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' conceived the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' per- formed the numerical calculations, and designed the sam- 12 ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' conducted the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=', X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' All authors contributed to scientific discussions and editing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' COMPETING INTERESTS The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' ADDITIONAL INFORMATION Supplementary information The online version contains supplementary material available at https:xxxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} +page_content=' Correspondence and requests for materials should be addressed to Shuomin Zhong or Xuchen Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfBgJT/content/2301.02852v1.pdf'} diff --git a/6NAyT4oBgHgl3EQfpfik/content/tmp_files/2301.00527v1.pdf.txt b/6NAyT4oBgHgl3EQfpfik/content/tmp_files/2301.00527v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5828976a078a86d7042c1ec0d23c6364b24ebc28 --- /dev/null +++ b/6NAyT4oBgHgl3EQfpfik/content/tmp_files/2301.00527v1.pdf.txt @@ -0,0 +1,775 @@ +Diffusion Probabilistic Models for Scene-Scale 3D Categorical Data +Jumin Lee +Woobin Im +Sebin Lee +Sung-Eui Yoon +Korea Advanced Institute of Science and Technology (KAIST) +{jmlee,iwbn,seb.lee,sungeui}@kaist.ac.kr +Abstract +In this paper, we learn a diffusion model to generate +3D data on a scene-scale. Specifically, our model crafts a +3D scene consisting of multiple objects, while recent diffu- +sion research has focused on a single object. To realize our +goal, we represent a scene with discrete class labels, i.e., +categorical distribution, to assign multiple objects into se- +mantic categories. Thus, we extend discrete diffusion mod- +els to learn scene-scale categorical distributions. In addi- +tion, we validate that a latent diffusion model can reduce +computation costs for training and deploying. To the best +of our knowledge, our work is the first to apply discrete +and latent diffusion for 3D categorical data on a scene- +scale. We further propose to perform semantic scene com- +pletion (SSC) by learning a conditional distribution using +our diffusion model, where the condition is a partial ob- +servation in a sparse point cloud. In experiments, we em- +pirically show that our diffusion models not only generate +reasonable scenes, but also perform the scene completion +task better than a discriminative model. Our code and mod- +els are available at https://github.com/zoomin- +lee/scene-scale-diffusion. +1. Introduction +Learning to generate 3D data has received much atten- +tion thanks to its high performance and promising down- +stream tasks. For instance, a 3D generative model with a +diffusion probabilistic model [2] has shown its effectiveness +in 3D completion [2] and text-to-3D generation [1,3]. +While recent models have focused on 3D object gener- +ation, we aim beyond a single object by generating a 3D +scene with multiple objects. In Fig. 1b, we show a sam- +ple scene from our generative model, where we observe the +plausible placement of the objects, as well as their correct +shapes. Compared to the existing object-scale model [1] +(Fig. 1a), our scene-scale model can be used in a broader +application, such as semantic scene completion (Sec. 4.3), +where we complete a scene given a sparse LiDAR point +6 +Pedestrian +Building +Vegetation +Vehicle +Diffusion +Model +(a) Object-scale generation +6 +Diffusion +Model +6 +(b) Scene-scale generation (ours) +Figure 1. Comparison of object-scale and scene scale generation +(ours). Our result includes multiple objects in a generated scene, +while the object-scale generation crafts one object at a time. (a) is +obtained by Point-E [1]. +cloud. +We base our scene-scale 3D generation method on a dif- +fusion model, which has shown remarkable performance in +modeling complex real-world data, such as realistic 2D im- +ages [4–6] and 3D objects [1–3]. We develop and evaluate +diffusion models learning a scene-scale 3D categorical dis- +tribution. +First, we utilize categorical data for a voxel entity since +we have multiple objects in contrast to the existing work [1– +3], so each category tells each voxel belongs to which cat- +egory. Thus, we extend discrete diffusion models for 2D +categorical data [7, 8] into 3D categorical data (Sec. 3.1). +Second, we validate the latent diffusion model for the 3D +scene-scale generation, which can reduce training and test- +ing computational cost (Sec. 3.2). Third, we propose to per- +form semantic scene completion (SSC) by learning a con- +ditional distribution using our generative models, where the +condition is a partial observation of the scene (Sec. 3.1). +That is, we demonstrate that our model can complete a rea- +arXiv:2301.00527v1 [cs.CV] 2 Jan 2023 + +Building +Barrier +Other +Pedestrian +Pole +Road +Ground +Sidewalk +Vegetation +Vehiclessonable scene in a realistic scenario with a sparse and partial +observation. +Lastly, we show the effectiveness of our method in terms +of the unconditional and conditional (SSC) generation tasks +on the CarlaSC dataset [9] (Sec. 4). Especially, we show +that our generative model can outperform a discriminative +model in the SSC task. +2. Related Work +2.1. Semantic Scene Completion +Leveraging 3D data for semantic segmentation has been +studied from different perspectives. Vision sensors (e.g., +RGB-D camera and LiDAR) provide depth information +from a single viewpoint, giving more information about the +world. One of the early approaches is using an RGB-D (i.e., +color and depth) image with a 2D segmentation map [10]. +In addition, using data in a 3D coordinate system has been +extensively studied. 3D semantic segmentation is the exten- +sion of 2D segmentation, where a classifier is applied to +point clouds or voxel data in 3D coordinates [11,12]. +One of the recent advances in 3D semantic segmentation +is semantic scene completion (SSC), where a partially ob- +servable space – observed via RGB-D image or point clouds +– should be densely filled with class labels [13–16]. In SSC, +a model gets the point cloud obtained in one viewpoint; +thus, it contains multiple partial objects (e.g., one side of a +car). Then, the model not only reconstructs the unobserved +shape of the car but also labels it as a car. Here, the predic- +tion about the occupancy and the semantic labels can mutu- +ally benefit [17]. +Due to the partial observation, filling in occluded and +sparse areas is the biggest hurdle. Thus, a generative model +is effective for 3D scene completion as 2D completion +tasks [18, 19]. Chen et al. [20] demonstrate that generative +adversarial networks (GANs) can be used to improve the +plausibility of a completion result. However, a diffusion- +based generative model has yet to be explored in terms of +a 3D semantic segmentation map. We speculate that us- +ing a diffusion model has good prospects, thanks to the +larger size of the latent and the capability to deal with high- +dimensional data. +In this work, we explore a diffusion model in the context +of 3D semantic scene completion. Diffusion models have +been rapidly growing and they perform remarkably well on +real-world 2D images [21]. Thus, we would like to delve +into the diffusion to generate 3D semantic segmentation +maps; thus, we hope to provide the research community a +useful road map towards generating the 3D semantic scene +maps. +2.2. Diffusion Models +Recent advances in diffusion models have shown that a +deep model can learn more diverse data distribution by a +diffusion process [5]. A diffusion process is introduced to +adopt a simple distribution (e.g., Gaussian) to learn a com- +plex distribution [4]. Especially, diffusion models show im- +pressive results for image generation [6] and conditional +generation [22, 23] on high resolution compared to GANs. +GANs are known to suffer from the mode collapse prob- +lem and struggle to capture complex scenes with multiple +objects [24]. On the other hand, diffusion models have a ca- +pacity to escape mode collapse [6] and generate complex +scenes [23,25] since likelihood-based methods achieve bet- +ter coverage of full data distribution. +Diffusion models have been studied to a large extent in +high-dimensional continuous data. However, they often lack +the capacity to deal with discrete data (e.g., text and seg- +mentation maps) since the discreteness of data is not fully +covered by continuous representations. To tackle such dis- +creteness, discrete diffusion models have been studied for +various applications, such as text generation [7,8] and low- +dimensional segmentation maps generation [7]. +Since both continuous and discrete diffusion models es- +timate the density of image pixels, a higher image res- +olution means higher computation. To address this issue, +latent diffusion models [23, 26] operate a diffusion pro- +cess on the latent space of a lower dimension. To work +on the compressed latent space, Vector-Quantized Varia- +tional Auto-Encoder (VQ-VAE) [27] is employed. Latent +diffusion models consist of two stages: VQ-VAE and dif- +fusion. VQ-VAE trains an encoder to compress the image +into a latent space. Equipped with VQ-VAE, autoregressive +models [28, 29] have shown impressive performance. Re- +cent advances in latent diffusion models further improve +the generative performance by ameliorating the unidirec- +tional bias and accumulated prediction error in existing +models [23,26]. +Our work introduces an extension of discrete diffu- +sion models for high-resolution 3D categorical voxel data. +Specifically, we show the effectiveness of a diffusion model +in terms of unconditional and conditional generation tasks, +where the condition is a partial observation of a scene (i.e., +SSC). Further, we propose a latent diffusion models for 3D +categorical data to reduce the computation load caused by +high-resolution segmentation maps. +2.3. Diffusion Models for 3D Data +Diffusion models have been used for 3D data. Until re- +cently, research has been mainly conducted for 3D point +clouds with xyz-coordinates. PVD [2] applies continuous +diffusion on point-voxel representations for object shape +generation and completion without additional shape en- +coders. LION [3] uses latent diffusion for object shape com- + +Forward Process +Reverse Process +(a) Discrete Diffusion Models +Segmentation Map +Segmentation Map +Reverse Process +Codebook +Stage1:VQ-VAE +Stage2: Latent Diffusion +Forward Process +(b) Latent Diffusion Models +Figure 2. Overview of (a) Discrete Diffusion Models and (b) La- +tent Diffusion Models. Discrete diffusion models conduct diffu- +sion process on voxel space, whereas latent diffusion models op- +erate diffusion process on latent space. +pletion (i.e., conditional generation) with additional shape +encoders. +In this paper, we aim to learn 3D categorical data (i.e., +3D semantic segmentation maps) with a diffusion model. +The study of object generation has shown promising re- +sults, but as far as we know, our work is the first to generate +a 3D scene with multiple objects using a diffusion model. +Concretely, our work explores discrete and latent diffusion +models to learn a distribution of volumetric semantic scene +segmentation maps. We develop the models in an uncon- +ditional and conditional generation; the latter can be used +directly for the SSC task. +3. Method +Our goal is to learn a data distribution p(x) using dif- +fusion models, where each data x ∼ p(x) represents a +3D segmentation map described with the one-hot repre- +sentation. 3D segmentation maps are samples from the +data distribution p(x), which is the categorical distribution +Cat(k0, k1, · · · , kM) with M +1 probabilities of the free la- +bel k0 and M main categories. The discrete diffusion mod- +els could learn data distribution by recovering the noised +data, which is destroyed through the successive transition +of the label [8]. +Our method aims to learn a distribution of voxelized +3D segmentation maps with discrete diffusion (Sec. 3.1). +Specifically, it includes unconditional and conditional gen- +eration, where the latter corresponds to the SSC task. In ad- +dition, we explore a latent diffusion model for 3D segmen- +tation maps (Sec. 3.2). +3.1. Discrete Diffusion Models +Fig. 2a summarizes the overall process of discrete diffu- +sion, consisting of a forward process and a reverse process; +the former gradually adds noise to the data and the latter +learns to denoise the noised data. +In the forward process in the discrete diffusion, an origi- +nal segmentation map x0 is gradually corrupted into a t-step +noised segmentation map xt with 1 ≤ t ≤ T. Each forward +step can be defined by a Markov uniform transition matrix +Qt [8] as xt = xt−1Qt. Based on the Markov property, we +can derive the t-step noised segmentation map xt straight +from the original segmentation map x0, q(xt|x0), with a +cumulative transition matrix ¯Qt = Q1Q2 · · · Qt: +q(xt|x0) = Cat(xt; p = x0 ¯Qt). +(1) +In the reverse process parametrized by θ, a learn- +able model is used to reverse a noised segmentation map +by pθ(xt−1|xt). Specifically, we use a reparametrization +trick [5] to make the model predict a denoised map ˜x0 and +subsequently get the reverse process pθ(xt−1|xt): +pθ(xt−1|xt) = q(xt−1|xt, ˜x0)pθ(˜x0|xt), +(2) +q(xt−1|xt, ˜x0) = q(xt|xt−1, ˜x0)q(xt−1|˜x0) +q(xt|˜x0) +. +(3) +We optimize a joint loss that consists of the KL di- +vergence of the forward process q(xt−1|xt, x0) from the +reverse process pθ(xt−1|xt); of the original segmentation +map q(x0) from the reconstructed one pθ(xt−1|xt) for an +auxiliary loss: +L = DKL( q(xt−1|xt, x0) ∥ pθ(xt−1|xt) ) ++ w0DKL( q(x0) ∥ pθ(˜x0|xt) ), +(4) +where w0 is an auxiliary loss weight. +Unlike existing discrete diffusion models [7,8], our goal +is to learn the distribution of 3D data. Thus, to better handle +3D data, we use a point cloud segmentation network [30] +with modifications for discrete data and time embedding. +Conditional generation. +We propose discrete diffusion +for Semantic Scene Completion (SSC) with conditional +generation. SSC jointly estimates a scene’s complete geom- +etry and semantics, given a sparse occupancy map s. Thus, +it introduces a condition into Eq. 2, resulting in: +pθ(xt−1|xt, s) = q(xt−1|xt, ˜x0)pθ(˜x0|xt, s), +(5) + +where s is a sparse occupancy map. We give the condition +by concatenating a sparse occupancy map s with a corrupted +input xt. +3.2. Latent Diffusion Models +Fig. 2b provides an overview of latent diffusion on 3D +segmentation maps. Latent diffusion models project the 3D +segmentation maps into a smaller latent space and operate +a diffusion process on the latent space instead of the high- +dimensional input space. A latent diffusion takes advantage +of a lower training computational cost and a faster inference +by processing diffusion on a lower dimensional space. +To encode a 3D segmentation map into a latent rep- +resentation, we use Vector Quantized Variational AutoEn- +coder (VQ-VAE) [27]. VQ-VAE extends the VAE by adding +a discrete learnable codebook E = {en}N +n=1 ∈ RN×d, +where N is the size of the codebook and d is the dimension +of the codes. The encoder E encodes 3D segmentation maps +x into a latent z = E(x), and the quantizer V Q(·) maps +the latent z into a quantized latent zq, which is the closest +codebook entry en. Note that the latent z ∈ Rh×w×z×d has +a smaller spatial resolution than the segmentation map x. +Then the decoder D reconstructs the 3D segmentation maps +from the quantized latent, ˜x = D(V Q(E(x))). The encoder +E, the decoder D, and the codebook E can be trained end- +to-end using the following loss function: +LV QV AE = − +� +k +wkxk log(˜xk) + ∥sg(z) − zq∥2 +2 ++ ∥z − sg(zq)∥2 +2, +(6) +where wk is a class weight and sg(·) is the stop-gradient +operation. Training the latent diffusion model is similar to +that of discrete diffusion. Discrete diffusion models diffuse +between labels, but latent diffusion models diffuse between +codebook indexes using Markov Uniform transition matrix +Qt [8]. +4. Experiments +In this section, we empirically study the effectiveness of +the diffusion models on 3D voxel segmentation maps. We +divide the following sub-sections into the learning of the +unconditional data distribution p(x) (Sec. 4.2) and the con- +ditional data distribution p(x|s) given a sparse occupancy +map s (Sec. 4.3); note that the latter corresponds to seman- +tic scene completion (SSC). +4.1. Implementation Details +Dataset. Following prior work [9], we employ the CarlaSC +dataset – a synthetic outdoor driving dataset – for training +and evaluation. The dataset consists of 24 scenes in 8 dy- +namic maps under low, medium, and high traffic conditions. +Model +Resolution +Training +(time/epoch) +Sampling +(time/img) +D-Diffusion +128×128×8 +19m 48s +0.883s +L-Diffusion +32×32×2 +7m 37s +0.499s +16×16×2 +4m 41s +0.230s +8×8×2 +4m 40s +0.202s +Table 1. Computation time comparison between discrete diffu- +sion models and latent diffusion models for 3D segmentation maps +generation. ‘D-Diffusion’ and ‘L-Diffusion’ denote discrete diffu- +sion models and latent diffusion models, respectively. ‘Resolution’ +means the resolution of the space in which diffusion process op- +erates. A latent diffusion models process diffusion on a lower di- +mensional latent space, as a result, it shows advantage of a faster +training and sampling time. +The splits of the dataset contain 18 training, 3 validation, +and 3 test scenes, which are annotated with 10 semantic +classes and a free label. Each scene with a resolution of +128 × 128 × 8 covers a range of 25.6 m ahead and behind +the car, 25.6 m to each side, and 3 m in height. +Metrics. Since SSC requires predicting the semantic label +of a voxel and an occupancy state together, we use mIoU +and IoU as SSC and VQ-VAE metrics. The mIoU measures +the intersection over union averaged over all classes, and +the IoU evaluates scene completion quality, regardless of +the predicted semantic labels. +Experimental settings. Experiments are deployed on two +NVIDIA GTX 3090 GPUs with a batch size of 8 for dif- +fusion models and 4 for VQ-VAE. Our models follow the +same training strategy as multinomial diffusion [7]. We set +the hyper-parameters of the diffusion models with the num- +ber of time steps T = 100 timesteps. And for VQ-VAE, +we set the codebook E = {en}N +n=1 ∈ RN×d where the +codebook size N = 1100, dimension of codes d = 11 and +en ∈ R32×32×2×d. For diffusion architecture, we slightly +modify the encoder–decoder structure in Cylinder3D [30] +for time embedding and discreteness of the data. And for +VQ-VAE architecture, we also use encoder–decoder struc- +ture in Cylinder3D [30], but with the vector quantizer mod- +ule. +4.2. 3D Segmentation Maps Generation +We use the discrete and the latent diffusion models for +3D segmentation map generation. Fig. 3 shows the quali- +tative results of the generation. As seen in the figure, both +the discrete and latent models learn the categorical distri- +bution as they produce a variety of reasonable scenes. Note +that our models are learned on a large-scale data distribution +like the 3D scene with multiple objects; this is worth noting +since recent 3D diffusion models for point clouds have been +performed on an object scale [2,3,31,32]. +In Tab. 1, we compare training and sampling time mod- + +Codebook size +(N) +Resolution +(h × w × z) +IoU +mIoU +220 +8×8×2 +72.5 +27.3 +16×16×2 +78.7 +36.9 +32×32×2 +84.6 +56.5 +550 +8×8×2 +67.7 +25.7 +16×16×2 +79.4 +39.7 +32×32×2 +85.8 +58.4 +1,100 +8×8×2 +70.3 +25.7 +16×16×2 +79.3 +35.0 +32×32×2 +89.1 +65.1 +2,200 +8×8×2 +70.2 +26.5 +16×16×2 +77.7 +37.9 +32×32×2 +89.2 +64.2 +Table 2. Ablation study on VQ-VAE hyper-parameters. We +compare different sizes of codebook N and resolutions of the la- +tent space h×w×z. +els for different resolutions on which each diffusion model +operates. Compared to the discrete diffusion, the latent dif- +fusion tends to show shorter training and inference time. +This is because the latent diffusion models compress the +data into a smaller latent so that the time decreases as the +compression rate increases. In particular, compared to dis- +crete diffusion, which performs a diffusion process in voxel +space, 32 × 32 × 32 latent diffusion has 2.6 times faster +training time for one epoch and 1.8 times faster sampling +time for generating one image. +Ablation study on VQ-VAE. +Latent diffusion models +consist of two stages. The VQ-VAE compresses 3D seg- +mentation maps to latent space, and then discrete diffusion +models apply on the codebook index of latent. Therefore, +the performance of VQ-VAE may set the upper bound for +the final generation quality. So we conduct an ablation study +about VQ-VAE while adjusting the resolution of the latent +space h×w×z and the codebook capacities N while keep- +ing the code dimension d fixed. Concretely, we compress +the 3D segmentation maps from 128×128×8 to 32×32×2, +16×16×2, and 8×8×2 with four different codebook size +N ∈ {220, 550, 1100, 2200}. +The quantitative comparison is shown in Tab. 2. The big- +ger the codebook size is, the higher the performance is, but +it saturates around 1,100. That is because most of the codes +are not updated, and the update of the codebook can lapse +into a local optimum [33]. +The resolution of latent space has a significant impact on +performance. As the resolution of the latent space becomes +smaller, it cannot contain all the information of the 3D seg- +mentation map. Setting the resolution to 32 × 32 × 2 with +a 1,100 codebook size strike a good balance between effi- +ciency and fidelity. +Methods +IoU +mIoU +LMSCNet SS [16] +85.98 +42.53 +SSCNet Full [17] +80.69 +41.91 +MotionSC (T=1) [9] +86.46 +46.31 +Our network w/o Diffusion +80.70 +39.94 +Discrete Diffusion (Ours) +80.61 +45.83 +Table 3. Semantic Scene Completion results on test set of CarlaSC +4.3. Semantic Scene Completion +We use a discrete diffusion model for conditional 3D +segmentation map generation (i.e., SSC). As a baseline +model against the diffusion model, we train a network with +an identical architecture by discriminative learning without +a diffusion process. We optimize the baseline with a loss +term L = − � +k wkxk log(˜xk), where wk is a weight for +each semantic class. We visualize results from the baseline +and our discrete diffusion model in Fig. 4. Despite the com- +plexities of the networks being identical, our discrete dif- +fusion model improves mIoU (i.e., class-wise IoU) up to +5.89%p than the baseline model as shown in Tab. 4. Es- +pecially, our method achieves outstanding results in small +objects and fewer frequency categories like ‘pedestrian’, +‘pole’, ‘vehicles,’ and ‘other’. The qualitative results in +Fig. 4 better demonstrate the improvement. +In Tab. 3, we compare our model with existing SSC mod- +els whose network architectures and training strategies are +specifically built for the SSC task. Nonetheless, our diffu- +sion model outperforms LMSCNet [16] and SSCNet [17], +in spite of the simpler architecture and training strategies. +Although MotionSC [9] shows a slightly better result, we +speculate that the diffusion probabilistic model can be im- +proved by extensive future research dedicated to this field. +5. Conclusion +In this work, we demonstrate the extension of the diffu- +sion model to scene-scale 3D categorical data beyond gen- +erating a single object. We empirically show that our mod- +els have impressive generative power to craft various scenes +through a discrete and latent diffusion process. Addition- +ally, our method provides an alternative view for the SSC +task, showing superior performance compared to a discrim- +inative counterpart. We believe that our work can be a useful +road map for generating 3D data with a diffusion model. +References +[1] A. Nichol, H. Jun, P. Dhariwal, P. Mishkin, and +M. Chen, “Point-e: A system for generating 3d +point clouds from complex prompts,” 2022. [Online]. +Available: https://arxiv.org/abs/2212.08751 1 + +Training Datasets +Latent Diffusion Models +Discrete Diffusion Models +Figure 3. Samples from our unconditional diffusion models. The first column shows samples from training datasets. From the second +column, we show samples from our discrete diffusion and latent diffusion models. We can observe our diffusion models learn the 3D +categorical distribution well, so that it is capable to generate a variety of plausible maps. Color assignment for each class is available in +Tab. 4. +Class IoU +mIoU +Free +Building +Barrier +Other +Pedestrian +Pole +Road +Ground +Sidewalk +Vegetation +Vehicles +IoU +w/o Diffusion +39.94 +96.40 +27.72 +3.15 +8.77 +22.15 +37.14 +89.02 +18.22 +59.25 +29.74 +47.72 +80.70 +Discrete Diffusion (Ours) +45.83 +96.00 +31.75 +3.42 +25.43 +46.22 +43.32 +84.57 +13.01 +67.50 +37.45 +55.46 +80.61 +Table 4. Semantic scene completion results on test set of CarlaSC. The discriminative learning result with the diffusion model architecture +is denoted as ‘w/o Diffusion’. Values with a difference equal to or greater than 0.5%p are bold. +Ground +Truth +Discrete +Diffusion +(ours) +Input +w/o +Diffusion +Figure 4. Qualitative comparison of a deterministic model (w/o diffusion) and ours (discrete diffusion) on the test split of CarlaSC. +The first row shows the sparse inputs for the scene completion task, and the last row shows the corresponding ground-truth. 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Sug- +anthan, “Global context with discrete diffusion in vec- +tor quantised modelling for image generation,” in Pro- +ceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, 2022, pp. 11 502– +11 511. 5 + diff --git a/6NAyT4oBgHgl3EQfpfik/content/tmp_files/load_file.txt b/6NAyT4oBgHgl3EQfpfik/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd1a618c6f98fabeba65139ad1d48b78a352a21f --- /dev/null +++ b/6NAyT4oBgHgl3EQfpfik/content/tmp_files/load_file.txt @@ -0,0 +1,600 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf,len=599 +page_content='Diffusion Probabilistic Models for Scene-Scale 3D Categorical Data Jumin Lee Woobin Im Sebin Lee Sung-Eui Yoon Korea Advanced Institute of Science and Technology (KAIST) {jmlee,iwbn,seb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='lee,sungeui}@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='kr Abstract In this paper, we learn a diffusion model to generate 3D data on a scene-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Specifically, our model crafts a 3D scene consisting of multiple objects, while recent diffu- sion research has focused on a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' To realize our goal, we represent a scene with discrete class labels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', categorical distribution, to assign multiple objects into se- mantic categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Thus, we extend discrete diffusion mod- els to learn scene-scale categorical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In addi- tion, we validate that a latent diffusion model can reduce computation costs for training and deploying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' To the best of our knowledge, our work is the first to apply discrete and latent diffusion for 3D categorical data on a scene- scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We further propose to perform semantic scene com- pletion (SSC) by learning a conditional distribution using our diffusion model, where the condition is a partial ob- servation in a sparse point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In experiments, we em- pirically show that our diffusion models not only generate reasonable scenes, but also perform the scene completion task better than a discriminative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Our code and mod- els are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='com/zoomin- lee/scene-scale-diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Introduction Learning to generate 3D data has received much atten- tion thanks to its high performance and promising down- stream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' For instance, a 3D generative model with a diffusion probabilistic model [2] has shown its effectiveness in 3D completion [2] and text-to-3D generation [1,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' While recent models have focused on 3D object gener- ation, we aim beyond a single object by generating a 3D scene with multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 1b, we show a sam- ple scene from our generative model, where we observe the plausible placement of the objects, as well as their correct shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Compared to the existing object-scale model [1] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 1a), our scene-scale model can be used in a broader application, such as semantic scene completion (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='3), where we complete a scene given a sparse LiDAR point 6 Pedestrian Building Vegetation Vehicle Diffusion Model (a) Object-scale generation 6 Diffusion Model 6 (b) Scene-scale generation (ours) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Comparison of object-scale and scene scale generation (ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Our result includes multiple objects in a generated scene, while the object-scale generation crafts one object at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' (a) is obtained by Point-E [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We base our scene-scale 3D generation method on a dif- fusion model, which has shown remarkable performance in modeling complex real-world data, such as realistic 2D im- ages [4–6] and 3D objects [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We develop and evaluate diffusion models learning a scene-scale 3D categorical dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' First, we utilize categorical data for a voxel entity since we have multiple objects in contrast to the existing work [1– 3], so each category tells each voxel belongs to which cat- egory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Thus, we extend discrete diffusion models for 2D categorical data [7, 8] into 3D categorical data (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Second, we validate the latent diffusion model for the 3D scene-scale generation, which can reduce training and test- ing computational cost (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Third, we propose to per- form semantic scene completion (SSC) by learning a con- ditional distribution using our generative models, where the condition is a partial observation of the scene (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' That is, we demonstrate that our model can complete a rea- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='00527v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='CV] 2 Jan 2023 Building Barrier Other Pedestrian Pole Road Ground Sidewalk Vegetation Vehiclessonable scene in a realistic scenario with a sparse and partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Lastly, we show the effectiveness of our method in terms of the unconditional and conditional (SSC) generation tasks on the CarlaSC dataset [9] (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Especially, we show that our generative model can outperform a discriminative model in the SSC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Semantic Scene Completion Leveraging 3D data for semantic segmentation has been studied from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Vision sensors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', RGB-D camera and LiDAR) provide depth information from a single viewpoint, giving more information about the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' One of the early approaches is using an RGB-D (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', color and depth) image with a 2D segmentation map [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In addition, using data in a 3D coordinate system has been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3D semantic segmentation is the exten- sion of 2D segmentation, where a classifier is applied to point clouds or voxel data in 3D coordinates [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' One of the recent advances in 3D semantic segmentation is semantic scene completion (SSC), where a partially ob- servable space – observed via RGB-D image or point clouds – should be densely filled with class labels [13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In SSC, a model gets the point cloud obtained in one viewpoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' thus, it contains multiple partial objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', one side of a car).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Then, the model not only reconstructs the unobserved shape of the car but also labels it as a car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Here, the predic- tion about the occupancy and the semantic labels can mutu- ally benefit [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Due to the partial observation, filling in occluded and sparse areas is the biggest hurdle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Thus, a generative model is effective for 3D scene completion as 2D completion tasks [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' [20] demonstrate that generative adversarial networks (GANs) can be used to improve the plausibility of a completion result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' However, a diffusion- based generative model has yet to be explored in terms of a 3D semantic segmentation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We speculate that us- ing a diffusion model has good prospects, thanks to the larger size of the latent and the capability to deal with high- dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In this work, we explore a diffusion model in the context of 3D semantic scene completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Diffusion models have been rapidly growing and they perform remarkably well on real-world 2D images [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Thus, we would like to delve into the diffusion to generate 3D semantic segmentation maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' thus, we hope to provide the research community a useful road map towards generating the 3D semantic scene maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Diffusion Models Recent advances in diffusion models have shown that a deep model can learn more diverse data distribution by a diffusion process [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' A diffusion process is introduced to adopt a simple distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', Gaussian) to learn a com- plex distribution [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Especially, diffusion models show im- pressive results for image generation [6] and conditional generation [22, 23] on high resolution compared to GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' GANs are known to suffer from the mode collapse prob- lem and struggle to capture complex scenes with multiple objects [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' On the other hand, diffusion models have a ca- pacity to escape mode collapse [6] and generate complex scenes [23,25] since likelihood-based methods achieve bet- ter coverage of full data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Diffusion models have been studied to a large extent in high-dimensional continuous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' However, they often lack the capacity to deal with discrete data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', text and seg- mentation maps) since the discreteness of data is not fully covered by continuous representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' To tackle such dis- creteness, discrete diffusion models have been studied for various applications, such as text generation [7,8] and low- dimensional segmentation maps generation [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Since both continuous and discrete diffusion models es- timate the density of image pixels, a higher image res- olution means higher computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' To address this issue, latent diffusion models [23, 26] operate a diffusion pro- cess on the latent space of a lower dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' To work on the compressed latent space, Vector-Quantized Varia- tional Auto-Encoder (VQ-VAE) [27] is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Latent diffusion models consist of two stages: VQ-VAE and dif- fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' VQ-VAE trains an encoder to compress the image into a latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Equipped with VQ-VAE, autoregressive models [28, 29] have shown impressive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Re- cent advances in latent diffusion models further improve the generative performance by ameliorating the unidirec- tional bias and accumulated prediction error in existing models [23,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Our work introduces an extension of discrete diffu- sion models for high-resolution 3D categorical voxel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Specifically, we show the effectiveness of a diffusion model in terms of unconditional and conditional generation tasks, where the condition is a partial observation of a scene (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', SSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Further, we propose a latent diffusion models for 3D categorical data to reduce the computation load caused by high-resolution segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Diffusion Models for 3D Data Diffusion models have been used for 3D data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Until re- cently, research has been mainly conducted for 3D point clouds with xyz-coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' PVD [2] applies continuous diffusion on point-voxel representations for object shape generation and completion without additional shape en- coders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' LION [3] uses latent diffusion for object shape com- Forward Process Reverse Process (a) Discrete Diffusion Models Segmentation Map Segmentation Map Reverse Process Codebook Stage1:VQ-VAE Stage2: Latent Diffusion Forward Process (b) Latent Diffusion Models Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Overview of (a) Discrete Diffusion Models and (b) La- tent Diffusion Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Discrete diffusion models conduct diffu- sion process on voxel space, whereas latent diffusion models op- erate diffusion process on latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' pletion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', conditional generation) with additional shape encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In this paper, we aim to learn 3D categorical data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', 3D semantic segmentation maps) with a diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The study of object generation has shown promising re- sults, but as far as we know, our work is the first to generate a 3D scene with multiple objects using a diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Concretely, our work explores discrete and latent diffusion models to learn a distribution of volumetric semantic scene segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We develop the models in an uncon- ditional and conditional generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' the latter can be used directly for the SSC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Method Our goal is to learn a data distribution p(x) using dif- fusion models, where each data x ∼ p(x) represents a 3D segmentation map described with the one-hot repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3D segmentation maps are samples from the data distribution p(x), which is the categorical distribution Cat(k0, k1, · · · , kM) with M +1 probabilities of the free la- bel k0 and M main categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The discrete diffusion mod- els could learn data distribution by recovering the noised data, which is destroyed through the successive transition of the label [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Our method aims to learn a distribution of voxelized 3D segmentation maps with discrete diffusion (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Specifically, it includes unconditional and conditional gen- eration, where the latter corresponds to the SSC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In ad- dition, we explore a latent diffusion model for 3D segmen- tation maps (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Discrete Diffusion Models Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 2a summarizes the overall process of discrete diffu- sion, consisting of a forward process and a reverse process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' the former gradually adds noise to the data and the latter learns to denoise the noised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In the forward process in the discrete diffusion, an origi- nal segmentation map x0 is gradually corrupted into a t-step noised segmentation map xt with 1 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Each forward step can be defined by a Markov uniform transition matrix Qt [8] as xt = xt−1Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Based on the Markov property, we can derive the t-step noised segmentation map xt straight from the original segmentation map x0, q(xt|x0), with a cumulative transition matrix ¯Qt = Q1Q2 · · · Qt: q(xt|x0) = Cat(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' p = x0 ¯Qt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' (1) In the reverse process parametrized by θ, a learn- able model is used to reverse a noised segmentation map by pθ(xt−1|xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Specifically, we use a reparametrization trick [5] to make the model predict a denoised map ˜x0 and subsequently get the reverse process pθ(xt−1|xt): pθ(xt−1|xt) = q(xt−1|xt, ˜x0)pθ(˜x0|xt), (2) q(xt−1|xt, ˜x0) = q(xt|xt−1, ˜x0)q(xt−1|˜x0) q(xt|˜x0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' (3) We optimize a joint loss that consists of the KL di- vergence of the forward process q(xt−1|xt, x0) from the reverse process pθ(xt−1|xt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' of the original segmentation map q(x0) from the reconstructed one pθ(xt−1|xt) for an auxiliary loss: L = DKL( q(xt−1|xt, x0) ∥ pθ(xt−1|xt) ) + w0DKL( q(x0) ∥ pθ(˜x0|xt) ), (4) where w0 is an auxiliary loss weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Unlike existing discrete diffusion models [7,8], our goal is to learn the distribution of 3D data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Thus, to better handle 3D data, we use a point cloud segmentation network [30] with modifications for discrete data and time embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Conditional generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We propose discrete diffusion for Semantic Scene Completion (SSC) with conditional generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' SSC jointly estimates a scene’s complete geom- etry and semantics, given a sparse occupancy map s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Thus, it introduces a condition into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 2, resulting in: pθ(xt−1|xt, s) = q(xt−1|xt, ˜x0)pθ(˜x0|xt, s), (5) where s is a sparse occupancy map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We give the condition by concatenating a sparse occupancy map s with a corrupted input xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Latent Diffusion Models Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 2b provides an overview of latent diffusion on 3D segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Latent diffusion models project the 3D segmentation maps into a smaller latent space and operate a diffusion process on the latent space instead of the high- dimensional input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' A latent diffusion takes advantage of a lower training computational cost and a faster inference by processing diffusion on a lower dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' To encode a 3D segmentation map into a latent rep- resentation, we use Vector Quantized Variational AutoEn- coder (VQ-VAE) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' VQ-VAE extends the VAE by adding a discrete learnable codebook E = {en}N n=1 ∈ RN×d, where N is the size of the codebook and d is the dimension of the codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The encoder E encodes 3D segmentation maps x into a latent z = E(x), and the quantizer V Q(·) maps the latent z into a quantized latent zq, which is the closest codebook entry en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Note that the latent z ∈ Rh×w×z×d has a smaller spatial resolution than the segmentation map x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Then the decoder D reconstructs the 3D segmentation maps from the quantized latent, ˜x = D(V Q(E(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The encoder E, the decoder D, and the codebook E can be trained end- to-end using the following loss function: LV QV AE = − � k wkxk log(˜xk) + ∥sg(z) − zq∥2 2 + ∥z − sg(zq)∥2 2, (6) where wk is a class weight and sg(·) is the stop-gradient operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Training the latent diffusion model is similar to that of discrete diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Discrete diffusion models diffuse between labels, but latent diffusion models diffuse between codebook indexes using Markov Uniform transition matrix Qt [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Experiments In this section, we empirically study the effectiveness of the diffusion models on 3D voxel segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We divide the following sub-sections into the learning of the unconditional data distribution p(x) (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2) and the con- ditional data distribution p(x|s) given a sparse occupancy map s (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' note that the latter corresponds to seman- tic scene completion (SSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Implementation Details Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Following prior work [9], we employ the CarlaSC dataset – a synthetic outdoor driving dataset – for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The dataset consists of 24 scenes in 8 dy- namic maps under low, medium, and high traffic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Model Resolution Training (time/epoch) Sampling (time/img) D-Diffusion 128×128×8 19m 48s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='883s L-Diffusion 32×32×2 7m 37s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='499s 16×16×2 4m 41s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='230s 8×8×2 4m 40s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='202s Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Computation time comparison between discrete diffu- sion models and latent diffusion models for 3D segmentation maps generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' ‘D-Diffusion’ and ‘L-Diffusion’ denote discrete diffu- sion models and latent diffusion models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' ‘Resolution’ means the resolution of the space in which diffusion process op- erates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' A latent diffusion models process diffusion on a lower di- mensional latent space, as a result, it shows advantage of a faster training and sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The splits of the dataset contain 18 training, 3 validation, and 3 test scenes, which are annotated with 10 semantic classes and a free label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Each scene with a resolution of 128 × 128 × 8 covers a range of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='6 m ahead and behind the car, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='6 m to each side, and 3 m in height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Since SSC requires predicting the semantic label of a voxel and an occupancy state together, we use mIoU and IoU as SSC and VQ-VAE metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The mIoU measures the intersection over union averaged over all classes, and the IoU evaluates scene completion quality, regardless of the predicted semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Experiments are deployed on two NVIDIA GTX 3090 GPUs with a batch size of 8 for dif- fusion models and 4 for VQ-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Our models follow the same training strategy as multinomial diffusion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We set the hyper-parameters of the diffusion models with the num- ber of time steps T = 100 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' And for VQ-VAE, we set the codebook E = {en}N n=1 ∈ RN×d where the codebook size N = 1100, dimension of codes d = 11 and en ∈ R32×32×2×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' For diffusion architecture, we slightly modify the encoder–decoder structure in Cylinder3D [30] for time embedding and discreteness of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' And for VQ-VAE architecture, we also use encoder–decoder struc- ture in Cylinder3D [30], but with the vector quantizer mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3D Segmentation Maps Generation We use the discrete and the latent diffusion models for 3D segmentation map generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3 shows the quali- tative results of the generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' As seen in the figure, both the discrete and latent models learn the categorical distri- bution as they produce a variety of reasonable scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Note that our models are learned on a large-scale data distribution like the 3D scene with multiple objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' this is worth noting since recent 3D diffusion models for point clouds have been performed on an object scale [2,3,31,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 1, we compare training and sampling time mod- Codebook size (N) Resolution (h × w × z) IoU mIoU 220 8×8×2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='3 16×16×2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='9 32×32×2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='5 550 8×8×2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='7 16×16×2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='7 32×32×2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='4 1,100 8×8×2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='7 16×16×2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='0 32×32×2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='1 2,200 8×8×2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='5 16×16×2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='9 32×32×2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='2 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Ablation study on VQ-VAE hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We compare different sizes of codebook N and resolutions of the la- tent space h×w×z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' els for different resolutions on which each diffusion model operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Compared to the discrete diffusion, the latent dif- fusion tends to show shorter training and inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' This is because the latent diffusion models compress the data into a smaller latent so that the time decreases as the compression rate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In particular, compared to dis- crete diffusion, which performs a diffusion process in voxel space, 32 × 32 × 32 latent diffusion has 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='6 times faster training time for one epoch and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='8 times faster sampling time for generating one image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Ablation study on VQ-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Latent diffusion models consist of two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The VQ-VAE compresses 3D seg- mentation maps to latent space, and then discrete diffusion models apply on the codebook index of latent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Therefore, the performance of VQ-VAE may set the upper bound for the final generation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' So we conduct an ablation study about VQ-VAE while adjusting the resolution of the latent space h×w×z and the codebook capacities N while keep- ing the code dimension d fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Concretely, we compress the 3D segmentation maps from 128×128×8 to 32×32×2, 16×16×2, and 8×8×2 with four different codebook size N ∈ {220, 550, 1100, 2200}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The quantitative comparison is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The big- ger the codebook size is, the higher the performance is, but it saturates around 1,100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' That is because most of the codes are not updated, and the update of the codebook can lapse into a local optimum [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The resolution of latent space has a significant impact on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' As the resolution of the latent space becomes smaller, it cannot contain all the information of the 3D seg- mentation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Setting the resolution to 32 × 32 × 2 with a 1,100 codebook size strike a good balance between effi- ciency and fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Methods IoU mIoU LMSCNet SS [16] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='98 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='53 SSCNet Full [17] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='69 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='91 MotionSC (T=1) [9] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='46 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='31 Our network w/o Diffusion 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='70 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='94 Discrete Diffusion (Ours) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='61 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='83 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Semantic Scene Completion results on test set of CarlaSC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Semantic Scene Completion We use a discrete diffusion model for conditional 3D segmentation map generation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', SSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' As a baseline model against the diffusion model, we train a network with an identical architecture by discriminative learning without a diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We optimize the baseline with a loss term L = − � k wkxk log(˜xk), where wk is a weight for each semantic class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We visualize results from the baseline and our discrete diffusion model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Despite the com- plexities of the networks being identical, our discrete dif- fusion model improves mIoU (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=', class-wise IoU) up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='89%p than the baseline model as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Es- pecially, our method achieves outstanding results in small objects and fewer frequency categories like ‘pedestrian’, ‘pole’, ‘vehicles,’ and ‘other’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The qualitative results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4 better demonstrate the improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 3, we compare our model with existing SSC mod- els whose network architectures and training strategies are specifically built for the SSC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Nonetheless, our diffu- sion model outperforms LMSCNet [16] and SSCNet [17], in spite of the simpler architecture and training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Although MotionSC [9] shows a slightly better result, we speculate that the diffusion probabilistic model can be im- proved by extensive future research dedicated to this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Conclusion In this work, we demonstrate the extension of the diffu- sion model to scene-scale 3D categorical data beyond gen- erating a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We empirically show that our mod- els have impressive generative power to craft various scenes through a discrete and latent diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Addition- ally, our method provides an alternative view for the SSC task, showing superior performance compared to a discrim- inative counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We believe that our work can be a useful road map for generating 3D data with a diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Nichol, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Jun, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Dhariwal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Mishkin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Chen, “Point-e: A system for generating 3d point clouds from complex prompts,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='org/abs/2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='08751 1 Training Datasets Latent Diffusion Models Discrete Diffusion Models Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Samples from our unconditional diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The first column shows samples from training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' From the second column, we show samples from our discrete diffusion and latent diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' We can observe our diffusion models learn the 3D categorical distribution well, so that it is capable to generate a variety of plausible maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Color assignment for each class is available in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Class IoU mIoU Free Building Barrier Other Pedestrian Pole Road Ground Sidewalk Vegetation Vehicles IoU w/o Diffusion 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='94 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='40 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='77 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='15 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='14 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='02 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='22 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='25 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='74 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='72 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='70 Discrete Diffusion (Ours) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='83 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='00 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='42 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='43 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='22 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='32 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='57 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='01 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='50 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='45 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='46 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='61 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Semantic scene completion results on test set of CarlaSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The discriminative learning result with the diffusion model architecture is denoted as ‘w/o Diffusion’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Values with a difference equal to or greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content='5%p are bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Ground Truth Discrete Diffusion (ours) Input w/o Diffusion Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Qualitative comparison of a deterministic model (w/o diffusion) and ours (discrete diffusion) on the test split of CarlaSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' The first row shows the sparse inputs for the scene completion task, and the last row shows the corresponding ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Compared to the deterministic model, our probabilistic model produces more plausible shape and class inference, as highlighted by the red circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Note that the both models (w/o diffusion and discrete diffusion) use the same network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQfpfik/content/2301.00527v1.pdf'} +page_content=' Color assignment for each class is available in Tab.' metadata={'source': 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Camacho1, José Cruz2 +1 PhD, vitor.camacho@syone.com, R&D Data Science, Syone. +2 MSc, jose.cruz@syone.com, R&D Data Science, Syone. + +Abstract +In this work a novel recommender system (RS) for Tourism is presented. The RS is context aware +as is now the rule in the state-of-the-art for recommender systems and works on top of a tourism +ontology which is used to group the different items being offered. The presented RS mixes +different types of recommenders creating an ensemble which changes on the basis of the RS’s +maturity. Starting from simple content-based recommendations and iteratively adding popularity, +demographic and collaborative filtering methods as rating density and user cardinality increases. +The result is a RS that mutates during its lifetime and uses a tourism ontology and natural +language processing (NLP) to correctly bin the items to specific item categories and meta +categories in the ontology. This item classification facilitates the association between user +preferences and items, as well as allowing to better classify and group the items being offered, +which in turn is particularly useful for context-aware filtering. + +Keywords: recommender system, CARS, ontology, tourism, content-based, collaborative +filtering, demographic-based. + + +1 +Introduction +This work presents a novel recommender system (RS) approach, which builds on context +awareness, domain ontology and different types of recommenders that enter the process at +different stages of maturity. From simple recommenders that are less prone to cold-start issues +to more complex and powerful recommenders which struggle quite a bit with initial lack of data. +At the final stage of maturity, when all the recommenders are already deployed in the +recommender pool, the different recommenders analyze different aspects of the data, from +demographic features to ratings, and provide an ensemble of recommendations to the users, +based on different approaches and with varying degrees of personalization. The approach is novel + +2 + +in how it uses several techniques, from domain ontology to bin the items using NLP to achieve +concept similarity, and then from there applies content-based, demographic-based, popularity- +based and collaborative filtering approaches to attain the recommended items. The collaborative +filtering employed are field-aware factorization machines which are the state-of-the-art in matrix +factorization, which can easily include context-awareness. The aim is to provide a powerful and +adaptable recommender system framework which can adapt to any domain, given the respective +domain ontology, and can overcome cold-start issues by using an approach with 4 stages of +maturity, which are subsequently entered when given thresholds are reached. In the following the +structure of the paper is presented, with an explanation of every section. In the present section, +the Introduction, an overview of the presented recommender system framework is provided as +well as a literature review of the relevant works on the subject. In section 2, the framework and +all its components are presented, from adopted technologies to used algorithms and techniques. +A presentation of the architecture is given as well as a mock-up of the designed UI to provide the +link between user and recommender system. In section 3, the technologies and techniques, +mainly the ones central to the recommender system are better explained with some formulas +being provided. In section 4, the recommender system is tested with a synthetic dataset with +varying stages of maturity, to show how the recommender system evolves as the data changes. +In section 5, conclusions are given as well as a brief discussion on future works. + +1.1 +Literature review +Recommender systems (RS) have been the focus of research for many years now, both on the +algorithm side and on the applied side. The study of RS started in the beginning of the 1990s but +it was in the last 15 years that research and the number of publications on the topic surged. +Concerning our application, tourism, RS have been the focus of studies since, at least, the start +of the 2000s with many publications having been made since then [1]–[15]. As for works that +concern more an algorithmic approach without an explicit thematic application, several studies +have been published on the different types of RS, from content-based approaches to collaborative +filtering, as well as context-aware solutions, so called CARS [16]–[64]. +Going into more detail regarding the tourism themed recommenders, it is relevant to give +particular attention to ontology-based approaches. One of the more important examples +concerning the present work is Moreno, A. et al. [8]. In this work, an ontology-based approach +(SigTur/E-destination) is developed to get recommendations for tourism in the region of +Tarragona. The developed approach begins with the definition of a tourism domain ontology, +which describes the tourist activities in a hierarchy, and bins the activities according to a given +taxonomy. The ontology is thus used to explicitly classify the activities to recommend among a +predefined set of distinctive main concepts, which are used by the intelligent recommender +system in its reasoning processes. The recommender than applies collaborative and content- +based techniques to provide the recommendation. Another relevant work is that of García- +Crespo, A. et al. [11], which proposes a semantic based expert system to provide + +3 + +recommendations in the tourist domain (Sem-Fit). The proposed system works based on the +consumer’s experience about recommendations provided by the system. Sem-Fit uses the +experience point of view in order to apply fuzzy logic techniques to relating customer and hotel +characteristics, represented by means of domain ontologies and affect grids. An early and +interesting work that applies Bayesian networks to attain personalized recommendations for +tourist attractions by Huang, Y. and Bian, L. [15] is also worth mentioning. This work is from 2009 +and uses ontologies to classify different types of tourist attractions. It then uses a Bayesian +network, to calculate the posterior probabilities of a given tourist’s preferred activities and the +traveler category he fits into. Other works on recommender system tourism applications could +also be mentioned but instead one can mention three surveys done on this topic. First, one from +2014, Borràs, J. et al. [2] present a survey entitled “Intelligent tourism recommender systems”. In +this survey the various works in the state-of-the-art are analyzed and their different approaches +concerning user interface, functionalities, recommendation techniques and use of AI techniques +are presented. The second work that gives an overview on the topic is from Kzaz, L. et al. [3] from +2018. In this overview, the focus is essentially on recommender approaches and employed user +and item data models. A third survey on this topic is given to us by Renjith, S. et al. [60] in a work +titled “An extensive study on the evolution of context-aware personalized travel recommender +systems”. Herein, the authors start by defining the different recommender approaches that can +be employed: content-based, collaborative, demographic-based, knowledge-based, hybrid, +personalized and context-aware. The authors also go into detail on the different machine learning +algorithms that are commonly employed, as well as the different employed metrics to evaluate +the quality of the predictions. Finally, they present a table with many different works with the +identification of whether or not they employ the previously mentioned techniques. +One of the aspects of the present work is that, as happens with some of the examples given +above, it employs ontologies to organize and classify the items to be recommended in some way. +Two works can also be mentioned concerning tourism domain ontologies, but in this case their +formulation rather than their use. These works are by Ruíz-Martinez, J. et al. [65] and Barta, R. +et al. [66] and they present different approaches to integrate and define tourism domain +ontologies. In the latter work an approach is presented that shows how to cover the semantic +space of tourism and be able to integrate different modularized ontologies. In the former, a +strategy to automatically instantiate and populate a domain ontology by extracting semantic +content from textual web documents. This work deals essentially with natural language +processing and named entity recognition, which are some of the techniques also employed in this +paper in terms of ontology population or, in other words, the classification of the different items to +recommend according to the ontology. +Many other works should also be referenced, this time not necessarily linked to the tourism theme, +but instead due to their focus on the algorithmic aspect or rather the recommendation strategy +regardless of its field of application. One particular type of recommender system that is very much +dominant in the literature in recent times is the context aware recommender system (CARS). The +work by Kulkarni, S. et al. [32] gives us a review on the state-of-the-art techniques employed in + +4 + +context aware recommender systems. In this work the authors list the most common algorithmic +approaches from bio-inspired algorithms to other common and less common machine learning +algorithms and then enumerate the works that employed each type of solution. Another review +study on context aware recommender systems is authored by Haruna, K. et al. [67]. In this work, +the authors particularly emphasize the manner in which the contextual filtration is applied, for +which there are three variants, pre-filtering, post-filtering and context modelling. The difference +between each approach has to do with how context filtering is applied together with the process +of recommendation. Hence, in pre-filtering the recommender filters the items prior to +recommendation, while in post-filtering the opposite happens. In context modelling there is a more +complex integration of the context filtering and the recommendations. The authors then go on to +classify the different works in the literature according to this and other topics such as employed +algorithms, etc. A third overview paper on the topic of CARS is the work by Raza, S. et al. [44]. +In this work, the authors focus on the type of algorithms, the dimensionality reduction techniques, +user modelling techniques and finally the evaluation metrics and datasets employed. Still focusing +on CARS, a context-aware knowledge-based recommender system for movie showtimes called +RecomMetz is presented in the work by Colombo-Mendoza, L. et al. [58]. In this work, the CARS +developed has time awareness, crowd awareness and location awareness, as part of its context +awareness composition. It is interesting to verify that its location awareness employs an +exponential distance decay that discards items that are far away from the user. This sort of +mechanism is also employed in the current work but with other goals. A last example on CARS is +a genetic algorithm (GA) approach based on spatio-temporal aspects [68] by Linda, S. et al. Here, +the interesting aspect is the inclusion of a GA to optimize the temporal weights of each individual +while employing collaborative filtering for the recommendations. +Lately, one of the most studied techniques for recommender systems have been Factorization +Machines (FM) [69]. In the present work, a field-aware version of this technique is employed, also +known as an FFM. This technique is a kind of collaborative filtering method that gained some +notoriety for solving click-through prediction rates [64], among other problems. Several versions +exist of these FMs in the literature, with ensembles with deep neural networks [45], for example, +being one of such versions. The value of FM is that they are more powerful than traditional matrix +factorization techniques, being able to incorporate features and information such as implicit +feedback. For these reasons, an FM, more specifically an FFM, is one of the recommenders +employed in the proposed recommender system, constituting the collaborative filtering +component of the proposed RS. + + + + +5 + +1.2 +Description of the RS and field of application +The proposed RS in this work is to be applied in the tourism industry. More specifically, the project +entails the creation of a recommender system to be used by hotel companies to recommend to +their guests their vast lists of partners in the region. It is very common that large hotel companies +have hundreds of partners offering products and most hotel guests are unaware of most of them. +The partners usually offer a wide array of products, which need an ontology to be organized and +better recommended. The proposed RS starts by having a Partner Management Platform (PMP) +for the hotel’s partners where they can manually introduce the items they want to be +recommended in the RS. The PMP, which is essentially an interface of the Item DB, feeds the +Domain Ontology which exists in a graph DB. The users are clients of the hotel that have checked- +in, and they exist in the User DB, which houses not only demographic information but also user +preferences which are collected and inferred by the RS. The RS interface is a web-app which is +presented in a further section of the paper. In the following sections more detail is provided +concerning the various components of the RS, starting with the presentation of the RS +architecture in the following section. + +2 +Architecture and frameworks of the recommender system +The architecture of the RS can be essentially divided into 4 parts, the data repository, the context- +aware subsystem, the recommender system per se and the user interface. In the following figure +the architecture is presented with each of its subcomponents. An overview of each of the +subcomponents is given in the following subsections. + +Figure 1 Architecture of the RS. + +2.1 +Data repository +The first element of the recommender system is its data repository, in the sense that this is where +it starts, particularly with the Partner Management Platform (PMP). It is through this PMP that we + +Location-aware +Weather-aware +Repetition-aware +Userprofile +manager +Context-awareSubsystem +Preference +UserInterface +manager +ItemDB +UserDB +Domain +Ontology +Recommender +pool +Partner +Management +Recommender +Platform +Data Repository +System6 + +have the introduction of the items, by the partners, to be recommended by the RS. In the PMP, +the partners introduce the items alongside necessary description and keywords. This information +introduced in the PMP is organized into an Item DB and later inserted into the domain ontology, +which is later explained in detail. +Other than the PMP with its Item DB and the mentioned domain ontology, the data repository also +has a User DB. This DB has both the demographic information collected from the users that +check-in to the hotel, but also the preference vectors that are inferred and managed by the RS. +The RS uses these two components of the user info to make predictions and to build different +recommendation models based on demographic, content, and collaborative filtering techniques. + +2.1.1 Domain ontology - Neo4j and automatic population of ontology +As for the domain ontology, the initial approach was to adopt the ontology presented in SigTur +[8]. In addition, Neo4j (www.neo4j.com), which is a graph DB, was chosen to house the ontology +and to facilitate the automatic ontological extension with the items from the PMP. In the following +figures, the original ontology is shown already inserted in a Neo4j graph. + +Figure 2 Ontology inserted in Neo4j. + +Shopping +Wine +SCO +Wine_E.. +Popular. +SCO +Music_F. +NightLife +Sco +SCO +SCo +sCO +BookF +Gastron.. +Leisure +SCO +Leisure. +SCO +Oos +SCo +Gastron. +SCO +Events +SCO +Sco +Health +SCO +Tradifion. +Relaxafi.. +Gastron. +Sport_ E. +Sco +TownR. +Relaxafi.. +$Co +Arts_An. +Sco +ScO +Towns +SCO +Sco +SCO +SCC +NonAqu. +sCO +Sport_R.. +SCO +Air_Spor. +SCo +Culture +SCo +Tradition.. +Sco +SCo - + SCO +Sco +Sports +Culture +SCO +Tradition. +Driving. +MotorSp. +Sco +Aquatic_ +Nature +Sco +Climbing +Sco +Monume. +SCO +Ethnogr +OOS +Saiting +Culture. +SCo +Surfing +Nature +UnderW. +ViewPoi. +8 +SCO +Archeol. +sco +8 +$CO +Protecte. +Art_Mus. +LandSc. +Nature +History. +ichitect +SCo +SCO +SCO +Mountai. +Coastal. +Inland +Rural_A.7 + + + +Figure 3 Sample of the ontology (highlighted section in previous figure). + +The advantage of using the Neo4j framework is that it facilitates the automation of ontological +extension. This ontological extension is achieved through the use of NLP techniques, such as +named entity recognition and cosine similarity between semantic concepts, using the spaCy +Python library integrated with Neo4j methods. These processes start with the insertion of the +items from the PMP or the Item DB. These items are parsed and tokenized, using both the item +descriptions and/or keywords. These parsed and tokenized items are then linked to the ontology +by means of semantic similarity between its keywords and description with each of the ontological +subclasses. The similarity scores above a given threshold originate a link between the item and +that specific ontological subclass. This process that ends with concept similarity and starts with +parsing, removal of stopwords and tokenization is performed with methods in the spaCy library. +The concept similarity is performed using spaCy’s vast pretrained word vectors. In addition, +named entity recognition is also performed on the items, automatically linking a Wikipedia entry, +if such entry exists. In Figure 4, a representation of the ontology after being extended with some +items, via the described process. One can see the original nodes in orange, that belong to the +ontology classes, some of which are now linked to grey nodes representing the items. The green +nodes represent the Wikipedia page object when such an object was found. In Figure 5 a zoomed +view of the highlighted zone in Figure 4 is shown. One can see two instances in which a Wikipedia +page object was found from the Named Entity Recognition procedure. The items were linked to +the ontology subclasses and one can observe that the links make sense in these cases, with +driving an F1 racecar linked to “Motor Sports”, and golf lessons and discounts on clubs linked to +“Golf”. + + ie oi + useums + istory +Culture +A uatic + usic +Sailing + nder +Rural A + onume + ature +Surfing + ight ife + ealth + o ns + radition + ood + eaches +Culture +Gastron + radition + ine + ature +Air Spor + ature +Golf + vents + otorSp + oo +Culture + riving +Shopping +Gastron + eisure +Relaxati + eisure + ine +Adventu +Climbing +Arts An + andSc +Relaxati +Gastron +Sport R +Sport +Architect +Routes +Archeol + radition +Art us +Inland +Sports +Coastal + thnogr + rotecte + o n R + ance + onA u + ountai + opular + +8 + + +Figure 4 Ontology extended with the addition of items. + + +9 + + +Figure 5 Sample of the extended ontology (highlighted section in previous figure). + +The recommender system module then imports the extended ontology, both the classes and the +items. It will use the extended ontology to give content-based recommendations. + +2.2 +Context-aware subsystem module +The context-aware subsystem module does item pre-filtering on the basis of three context +submodules: location-aware, weather-aware and repetition-aware. In the case of the location- +aware submodule, the objective is to filter out the hotel partners that are not located close by to a +specific instance of the hotel. Since the hotel company can have a wide array of partners that +may, in many cases, be close to one specific hotel but not to other hotels in other locations, such +as local or regional partners that only provide services to the hotels in the area, a first contextual +filtering phase is to apply location pre-filtering. Then we go on to the weather-aware submodule, +where the ontological sub-classes are associated with a given fuzzy definition of when they make +sense to be recommended, for example the beach ontology class or the outdoor sports ontology +class would tend to be penalized with bad weather. Finally, a third module, which is very much +novel, which is the repetition-aware module. Here, each ontological class would have a different +elapsed time parameter that affects an inverse exponential penalization factor to mimic the +repeatability of a given item. For example, one would probably be more adept to repeat a +restaurant than a museum in the same week. So, different ontological classes have different +factors that affect the inverse exponential function, that we may call the unwillingness to repeat +function, which defines how soon a user may be willing to repeat a given item. + ie oi + useums + istory +Culture +A uatic + usic +Sailing + nder +Rural A + onume + ature +Surfing + ight ife + ealth + o ns + radition + ood + eaches +Culture +Gastron + radition + ine + ature +A +service +that +offers +A +tavern +that +serv + ne +of the +main +nigh +Surfing +lessons +Ancient +history +muse +Great +meals +that are +tasty +Rest +and +relaxa +visiting + isneyl + atch a +live +football + edieval +fair + atch a +motogp +race +drive a + +racecar + ry +spearfis + a e +a trip in a +hot air +ball +Go +shopping +in our +ne + ry +scubadi + ne +day +snor +Golf +lessons +go to the +spa + ry +go arts + ith your +frien +Get a +free pint +at the +pub + atch a +Sporting +C m + atch a +S + enfica + atch a + C orto +match +Get a +voucher +for +Sep +Get a +free +pi a at + i a + iscount +for Call + atch a +live +concert +Get a +discount +for +Co +Air Spor + ature +Golf + vents + otorSp + oo +Culture + riving +Shopping +Gastron + eisure +Relaxati + eisure + ine +Adventu +Climbing +Arts An + andSc +Relaxati +Gastron +Sport R +Sport +Architect +Routes +Archeol + radition +Art us +Inland +Sports +Coastal + thnogr + rotecte + o n R + ance + onA u + ountai + opular + +10 + + +2.3 +Recommender system module +The recommender system module is the main module as the name entails. This module is +constituted by a user profile manager and a preference manager, besides the recommender pool. +Concerning the recommender pool and the models that compose it, that is addressed in depth in +Section 3 of this work. Here it suffices to say that the recommender pool is the set of different +recommender models that provide user recommendations. The models create an ensemble, +when more than one is active, that provides recommendations using different techniques and +approaches. +As for the remainder of the recommender system module, the user profile and the preference +manager, these two sub-modules manage the user related information, such as item ratings and +other user feedback in the case of the former, while the latter manages the user preference +vectors and propagates the user feedback on items to update the user preference vectors +accordingly. The way this is done will become clearer in the next sections. + +2.4 +User interface – web app +The last component is the user interface, which in this case is a web app that connects to the +recommender system module and other modules through a real-time and batch inference +endpoints that connect to ML pipelines defined in Azure. + +11 + + +Figure 6 App mockup showing the four main screens: welcome, preference definition, home and user profile. + +In the previous figure one can observe the four different screens the user sees during his App +experience. The FILTER screen is only presented to the user on the first time he logs in and is, +in essence, a series of check boxes where the user defines his preferences. These check boxes +are used to give a first estimate on the user’s preferences concerning the ontology classes. The +user’s choices define his preference vectors which then are used to make content-based +recommendations. As for the HOME screen, it shows the different recommendations made to the +user by the RS, here the user can bookmark items, book items or mar an item as “uninteresting”. +Finally, in the PROFILE screen, the user can observe his profile in terms preferences collected +and inferred by the RS as well as demographic information, such as date of birth, nationality, etc. +The different interactions the user can have with the App and the consequent interactions +between the App and the RS and back to the user are shown in Figure 7. In this figure one can +see how these interactions cascade and what the user gets back from each action he undertakes. +One can summarize the actions the user can take in the following: +• +Logging in +• +Preference input +• +Viewing recommendations + +viser +Adviser +see +syone +syone +YOU'RESTAYINGHERE +Ldviser +PAULAESTEVES +Whatareyouinterested in? +Hello +HotelGoldenCrown +MUSO +Noture +Paulo Esteves +Vewpoinitsv +HOTEL +TMYPROFILE +Concerte +Le'sure +Sports +Walks +Utorciaugue,faucibusatioculisid +Nnrnec +Pauio Esteves +efficitursagittis diam.Etiom eget nunc +RestourantsV +Foutes +Cinema +DOB: +10/03/1983 +acus +Adcress: Ruo Efficitur:sogittis dion,#3,1c Lisboc +Finess +Beatchv +Top:5 +Foryou +Jeb: +Sorior BIAnclyst +PhasellusacportatellusVivamus +EatProhik +tempormattisultrces.Proinvitae +Tellmemore +conseouortortorguispnoretotortor +A +WHATWE'VELEARNEDABOUTYOU +SKYDIVErush +50% +Ipere +Foucbusoticcuisidetficit +LEISURE +ROUTES +EVENTS +FooFightersy +tiverpoolFo +22, +gogittis Ciam.Etiam pgetnont +locus. +Francesinho +Guzado +START +TOWNS +CULTURE +NATURE +Dive classes +VEWPOINTS +SPORTS +50% +oucibuaticcuisi,eficitur +2 +18. +pogittisdigmEtiomegetmunt +I'MALSOINTO: +WELCOME +Search fortopics you/reinterestedin +Shortintroduction +回 +T +Woles +Andeboly +FILTER Screen +HOME +ACTMITES +HISTORY +MYPROTRE +... +Hke +Guizodo +Collectthefirstlayer +ofinformationfrom +HOMEScreen +HONE +ACTMTES +ISTORY +MYPRORLE +the user +Present the activities +andpositions the user +PROFILEScreen +Consult andeditthe user's +information12 + +• +Item feedback +• +Item booking +• +Item rating + + +Figure 7 User-App-RS interaction. User’s various possible actions and respective interactions between the +App and the RS. + +3 +Recommenders and stages in RS +The recommender system module mentioned in the previous section is composed by three +components: user profile manager, preference manager and recommender pool. The two former +ones have already been covered, and in this Section, the latter will be explained in depth. The +recommender pool is composed by four recommenders of different types: content-based, +popularity-based, demographic-based and collaborative. These four recommenders are modeled +with specific algorithms or employ specific techniques and they come into play in different phases +of maturity of the RS. These phases of maturity concern amount of data, that is, number of users + +Yo +RecSys +User +App +1 User's first log in +2 Asks preferences +3 Inputspreferences +4 Sendspreferences +5Returnsrecommendations +6Views recommendations +7 Gives feedback and/or +makesabooking +- +8Sendsfeedback +9 Returns updated +recommendations +10Ratesbooked item +11Sends itemrating +(First rating in system) +12Hybrid Recommender initiated +13 Returns updated +recommendations13 + +and rating density. Only after certain pre-specified values of users and rating density have been +reached are some of these methods activated, or in other words, are some of the phases reached. +In the following, the different phases and algorithms used are explained. + +3.1 +Phase 1 +At the beginning, the RS is void of any ratings or users, and only items exist in the RS. When a +new user logs in for the first time, in order for the RS to make any meaningful recommendation, +some information has to be provided in the form of user preferences. This is, at this stage, the +only way to overcome cold-start issues. he user’s preferences, which are associated to the +predetermined ontology are given and used to give content-based recommendations to the user. +The user then will provide explicit and implicit feedback, in the form of booking items, bookmarking +items or explicitly indicating they don’t li e the item. This feedback is then received by the RS who +then uses the said feedbac to update the user’s preference vectors. This update originates new +recommendations to the user. + +3.1.1 Preference vectors +At the core of phase 1 are the user preference vectors. These preference vectors are ontology +related and they are used to make content-based recommendations. There are three preference +vectors per user: +• +High-level preferences +• +Low-level preferences +• +Specific preferences +The high-level preferences are the ones the user identifies in the beginning and are associated +with the ontological super-classes. These classes are the most abstract classes and lower in +number. They are the first layer of ontological classes and are the ones that don’t have a parent +class and only child classes. Observing Figure 4, the Sports ontological class is an example of a +high-level preference since there is no ontology class above it. +The low-level preferences are associated to the ontological classes that link directly to the items. +These ontological classes are more specific, less abstract and in larger number. Observing Figure +4 and Figure 5, Golf is an example of a low-level preference, because two items link to it. +Finally, the specific preferences relate directly to the items, and is a vector that results from the +other two higher-level preference vectors and the user’s feedbac on the items. +The way these vectors interact is explained in the following: + +14 + +1. The user identifies the high-level preferences when he logs in for the first time. These +preferences are propagated by way of vector multiplication with the low-level ontological +preferences. +2. The low-level preferences are then propagated to the item level by way of vector +multiplication as well, originating the specific preference vector. The items are ranked, +and a subset of the highest ranked items are recommended to the user. +3. The user gives feedback on the recommendations by either bookmarking items, booking +items or dismissing items. The feedback is propagated upwards to the higher-level +preference vectors with different intensities. The low-level preference vector is strongly +affected, while the high-level preference vector is less affected because it is higher +upstream. This sort of “trickle-up” propagation of user feedback alters both high-level and +low-level preference vectors with different magnitude. +4. New item recommendations are calculated, this time using both the high-level and low- +level preference vectors to predict whether an item should be recommended or not. The +prediction by each vector is weighed and aggregated originating an ensemble prediction +using both high and low preference vectors. The items are ranked, and a subset of the +highest ranked items are recommended to the user. +5. Repeat step 3. + +3.1.2 Ontological content-based recommender +The content-based recommender is essentially vector multiplication between preference vectors +and content vectors. Content vectors are binary vectors which map one preference level to the +items content or to another preference vector content, while preference vectors show the intensity +levels of preference for each ontological category. +In step 4, the high and low preference vectors multiply with their corresponding item content vector +originating a content-based prediction. Both predictions are weighed and aggregated, and a +subset of the highest ran ed items is recommended to the user. After the user’s feedbac both +preference vectors are updated according to the “tric le-up” propagation concept introduced +above. Then, new recommendations are calculated with the new preference vectors. + +3.2 +Phase 2 +If the user booked and used an item, he can then rate said item, which will kickstart the hybrid +recommender composed by the initial content-based recommender and the new popularity-based +appendix. This popularity-based recommender uses a so-called damped mean on every item so +that little cardinality of ratings doesn’t give an exaggerated edge of an item over another, such as +an item with a single 5-star rating having a 5-star average. + +15 + +𝐷𝑎𝑚𝑝𝑒𝑑 𝑀𝑒𝑎𝑛𝑗 = +∑ +𝑟𝑗𝑖 + 𝑘 ∙ 𝑟̿𝐺 +𝑛 +𝑖=1 +𝑛 + 𝑘 + +Where 𝑟𝑗𝑖 is item j’s rating i, 𝑘 is the damping coefficient, 𝑟̿𝐺 is the global mean rating or some +other default value, and 𝑛 is the number of reviews of item j. + +3.2.1 Hybrid recommender (content-based + popularity-based) +The start of the hybrid recommender marks the start of phase 2. At this point in the RS, there +aren’t many users and there aren’t many ratings. he lac in both mean that popularity-based, +demographic-based or collaborative approaches are still of little use. As more users join and more +ratings are given, other recommenders can become increasingly useful. As we reach a given +threshold of user and rating numbers we can initiate the demographic-based recommender. +The way in which the hybrid recommender uses both recommenders is by cascading ensemble. +That is, the popularity recommender pre-filters the items according to a rating threshold and then +the content-based recommender recommends items that were not eliminated by the popularity +recommender. + +3.3 +Phase 3 +As more users are added to the RS, and as these users give feedback on recommended items, +other types of recommenders can enter the recommender pool. A first set of threshold values for +number of users and rating density is defined. When these thresholds are reached, phase 3 is +initiated with yet another recommender being added: the demographic-based recommender. + +3.3.1 Demographic-based recommender +The demographic-based recommender is composed by two ML algorithms. One clustering +algorithm and one classification algorithm. The clustering algorithm has the purpose of identifying +clusters of similar users based on their demographic features. he user’s demographic features +can be age, region/country, group composition, budget, academic degree, etc. These features +can be a mix of numerical, ordinal and nominal features and so a clustering algorithm that can +handle different data types is necessary. After the clustering has been performed, and the users +are all organized in clusters, a classification algorithm is used to predict whether a user will enjoy +each item based on the item feedback of other users in the same cluster. +For clustering, the algorithm employed was K-Prototypes, which works similarly to K-Means but +can deal with mixed data types, particularly ordinal and nominal data. To define the clustering +model, a knee region identifier is employed to automatically identify the optimal (or close to + +16 + +optimal) number of clusters. The clustering model is retrained from time to time when sufficient +new users have been added since the last model fitting. +For classification a k-Nearest Neighbor algorithm, or kNN, was employed. Here, the users from +the same cluster are used to predict whether a given user will enjoy the items, based on those +users’ feedbac . he uses a custom distance metric that ta es into account both Jaccard +and Manhattan distance metrics for the ordinal and nominal features. The kNN than weighs the +opinion of the other users inversely proportional to their distance to the user to whom the +predictions are being made. The predictions given by this algorithm are weighed and added to +the predictions made by the hybrid recommender. + +3.4 +Phase 4 +In phase 4, collaborative filtering is added to the pool. As it happens with phase 3, the entry into +phase 4 takes place when thresholds of user cardinality and rating density are reached. Once this +happens the collaborative filtering model is fitted and starts giving recommendations. The +algorithm used for collaborative filtering is a Field-Aware Factorization Machine (FFM), which has +already been introduced in Section 1. In the following sub-section, the FFM application is +explained in more detail. + +3.4.1 Collaborative filtering with Field-Aware Factorization Machines (FFM) +To use FFMs, a specific Python library (xLearn) is used and the data also has to be transformed +into a specific format. A sample of a dataset in said format is shown in the following table. + +Table 1 Dataset in the FFM format where each column represents a feature, except for column 0 which +represents the labels. + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +0 +0 +0:1:1 +1:2:1 +2:3:1 +3:4:1 +4:5:1 +5:6:1 +6:7:1 +7:8:1 +8:9:1 +1 +1 +0:10:1 +1:2:1 +2:11:1 +3:4:1 +4:5:1 +5:6:1 +6:12:1 +7:13:1 +8:14:1 +2 +0 +0:15:1 +1:16:1 +2:3:1 +3:4:1 +4:17:1 +5:6:1 +6:18:1 +7:19:1 +8:20:1 +3 +1 +0:15:1 +1:2:1 +2:21:1 +3:22:1 +4:17:1 +5:6:1 +6:23:1 +7:8:1 +8:24:1 +4 +1 +0:10:1 +1:16:1 +2:3:1 +3:4:1 +4:17:1 +5:25:1 +6:23:1 +7:26:1 +8:27:1 +... +... +... +... +... +... +... +... +... +... +... +686422 +1 +0:1:1 +1:2:1 +2:3:1 +3:4:1 +4:17:1 +5:25:1 +6:23:1 +7:8:1 +8:37:1 +686423 +1 +0:34:1 +1:2:1 +2:21:1 +3:4:1 +4:5:1 +5:25:1 +6:35:1 +7:8:1 +8:36:1 +686424 +1 +0:10:1 +1:16:1 +2:3:1 +3:4:1 +4:17:1 +5:25:1 +6:18:1 +7:8:1 +8:24:1 +686425 +1 +0:34:1 +1:16:1 +2:21:1 +3:22:1 +4:17:1 +5:25:1 +6:50:1 +7:13:1 +8:49:1 +686426 +1 +0:15:1 +1:2:1 +2:3:1 +3:4:1 +4:17:1 +5:6:1 +6:23:1 +7:8:1 +8:44:1 + + +17 + +This format is more complex than that for the Standard FM. This is due to the more complex +information that is ingested by the FFM which uses information about the fields to define the latent +vectors. That is, while in FMs each feature (field) has one latent vector, in FFMs this single +representation is broken down into multiple latent vectors, one to represent each other field. +𝑦̂(𝑥) ∶= 𝜔0 + ∑ 𝜔𝑖𝑥𝑖 + ∑ ∑ 〈𝕧𝑖, 𝕧𝑗〉𝑥𝑖𝑥𝑗 +𝑛 +𝑗=𝑖+1 +𝑛 +𝑖=1 +𝑛 +𝑖=1 + +In the equation that represents the FM, which is shown above, the feature interactions +represented by 〈𝕧𝑖, 𝕧𝑗〉 would correspond to the following in our case scenario (user +demographic features): +𝑣𝑚𝑎𝑙𝑒 ∙ 𝑣𝑏𝑙𝑢𝑒𝑐𝑜𝑙𝑙𝑎𝑟 + 𝑣𝑚𝑎𝑙𝑒 ∙ 𝑣𝑙𝑜𝑤𝑏𝑢𝑑𝑔𝑒𝑡 + 𝑣𝑚𝑎𝑙𝑒 ∙ 𝑣𝑛𝑜𝑟𝑡ℎ𝑒𝑢𝑟𝑜𝑝𝑒 + ⋯ +That is, the male latent vector that multiplies with each other latent vector is the same. The idea +behind FFM is that the weight of the male latent vector might not be the same when multiplying +with the job latent vectors as they are with the budget latent vectors, and so on. Thus, in the FFM, +the latent vectors are field-aware, which results in the following: +𝑣𝑚𝑎𝑙𝑒,𝑗𝑜𝑏 ∙ 𝑣𝑏𝑙𝑢𝑒𝑐𝑜𝑙𝑙𝑎𝑟,𝑔𝑒𝑛𝑑𝑒𝑟 + 𝑣𝑚𝑎𝑙𝑒,𝑏𝑢𝑑𝑔𝑒𝑡 ∙ 𝑣𝑙𝑜𝑤𝑏𝑢𝑑𝑔𝑒𝑡,𝑔𝑒𝑛𝑑𝑒𝑟 + 𝑣𝑚𝑎𝑙𝑒,𝑟𝑒𝑔𝑖𝑜𝑛 ∙ 𝑣𝑛𝑜𝑟𝑡ℎ𝑒𝑢𝑟𝑜𝑝𝑒,𝑔𝑒𝑛𝑑𝑒𝑟 + ⋯ + +Besides demographic features, as is shown in this example, the latent-vectors can also easily +incorporate item features as well as contextual features and can thus integrate context-awareness +in a deeper sense than simple contextual pre-filtering or post-filtering. +The FFM model represents the last phase addition to the recommender pool. The predictions +attained from it are weighed and then aggregated with the predictions given by the other two, the +hybrid and the demographic recommender. The weighs given to each recommender may be set +to change over time so that it accompanies the maturity and complexity of each of the +recommenders in the pool, thus giving progressively larger weight to the FFM as more users and +more ratings are added to the system. + +18 + + +Figure 8 Diagram of the various RS phases and interactions between RS and Data Repository (DB) +components. + +V1.0 +4 +Phase1 +11 +Phase2 +Phase3 +5 +Phase4 +8&9 +Newuserlogsn +Feedback +Seneretes reApp +Gets user from UserDB +receivedfromAp +toApp +12 +Hybridt Rec ir +Updates ontology +Step4.5.8.9811repealedh +AftermanyStep +A4.5.8.9&11 +AnerSlep4 +589811 +GeneratestreAs +Serenerates recs +Phase3beqins. +all threemodeis. +Shokds for FFM reiraining +en +Rec initiated +Recalculate clusters +Demog ec cotnue +Recaculalecstrs +Clusters defined +Collab +FFMiniliated +RetrainFFM. +Sends ue lo ecomnender +Ses ies to GraoDB. +GraphDB +Sennsertsilemontology +SsnsertsnewiteminOntoloy19 + +4 +Recommender system - Case study (CS) with synthetic data +One of the main challenges in designing the recommender system proposed in this work was the +lack of data to perform any type of experiment or even just to aid and inspire in the definition of +the algorithms to employ. The lack of data was absolute, both on the side of the items as on the +side of the users and preferences. The main issue is the non-existence of a dataset with user +demographic features and user preferences, since such a dataset would allow to overcome some +of the cold-start issues as well as give some idea of the data schema to be adopted. +As a result, and since no public datasets were found that could overcome this hinderance, the +decision was made to generate a synthetic dataset. The generated dataset was done so by using +many different techniques from gaussian copulas to fuzzy logic. Further information on that work +will be available in another paper by the author Camacho, VT. In the following sub-section, the +synthetic data employed in this or ’s case study is presented. +Besides the synthetic data, a set of metrics was chosen to get an idea about the quality of the +results from the recommenders. Traditional ML metrics are not always adequate for RS, mainly +because, by principle, the objective of an RS is not to emulate exactly the choices of a given user +since, if that were the case, there ouldn’t be a need for an RS in the first place. In the metrics +sub-section, the set of used metrics is presented. +The remainder of this section is applying the recommenders introduced in the previous section +and testing them with different amounts of data which will attempt to emulate the data present at +the different phases. + +4.1 +Synthetic data +In the work mentioned above, a methodology for the generation of synthetic datasets for +recommender systems is presented, thus allowing to overcome the obstacle of not having quality +data in sufficient amount (or even at all) readily available. The difficulties that are associated with +this task are essentially the definition of a dataset with multiple datatypes, such as numerical +(continuous), ordinal and nominal, and with different levels of correlation among the data, as well +as the definition of user-ratings based on well-defined latent user preferences. To overcome this, +a methodology was devised where several different techniques are employed in sequence to +create the datasets concerning user characteristics, item properties, item categories and latent +user preferences associated to user and item features, and as a result, a user-item sparse ratings +matrix. The output of the methodology is: +1) Item dataset with item names and categories. +2) User dataset with user characteristics (demographic features). +3) User-item sparse ratings matrix. + +20 + +4) Latent preferences and Multinomial Logit model to compare with the outputs of the +Recommender System. + +4.1.1 Data Schema +From the output presented above, we can see 4 DataFrames with different information. These +DataFrames each have their own schema and have features from different data types. In the +following, the created DataFrames are introduced: +• +Demographic Features +• +Preferences +• +Item Features +• +User Ratings +Going into more detail regarding the user demographic features DataFrame: +• +Demographic Features: +o +User ID +o +Age +o +Gender +o +Job +o +Academic Degree +o +Budget +o +Country/Region +o +Group Composition +o +Accommodation +Concerning the type of feature, they can be divided essentially into three groups: numerical, +categorical ordinal and categorical nominal. Concerning numerical and categorical ordinal +features, we have the following: +• +Numerical +o +Age – numerical (can be transformed into age bins) +• +Ordinal: +o +Age bins = ['18-30','31-40', '41-50', '51-60', '60+'] +o +Academic Degree = ['None', 'High School', 'Some College', 'College Degree'] +o +Budget = ['Low', 'Mid', 'High'] +o +Accommodation = ['Single', 'Double', 'Suite', 'Villa'] +As for categorical nominal features, the following were modelled: +• +Gender = ['Male', 'Female'] +• +Job = ['Blue Collar', 'White Collar'] + +21 + +• +Country/Region = ['South Europe', 'North Europe', 'East Europe', 'North America', 'South +America', 'Asia', 'Africa', 'Middle East'] +• +Group Composition = ['1 Adult', '2 Adults', '2 Adults + Child', 'Group of Friends'] + +4.1.2 Samples of the generated DataFrames +The resulting DataFrames (DF) can be used to train and test RS. In the case of the present work, +they are used to simulate the different phases of data availability, thus testing the recommenders +employed in each of the four phases. In the following, samples of the generated DFs are +presented. The first sample shown is the User DF in Table 2. This DF is composed by the user +demographic features and UserID. The demographic features are ordinal (Age, AcDeg, Budget, +Accom) and nominal (Gender, Job, Region, GroupComp). The entire set of users created has +cardinality of 100,000. + +Table 2 User DF composed by the demographic features of the users. +UserID +Age +AcDeg +Budget +Accom +Gender +Job +Region +GroupComp +0 +4 +2 +1 +2 +Female +blue collar +North Europe +2Adlt +1 +5 +4 +2 +3 +Male +white collar +North Europe +GrpFriends +2 +3 +3 +2 +2 +Female +blue collar +North Europe +2Adlt+Child +3 +4 +4 +2 +2 +Female +white collar +North Europe +2Adlt+Child +4 +3 +3 +2 +3 +Female +white collar +South Europe +2Adlt +... +... +... +... +... +... +... +... +... +99995 +4 +4 +2 +2 +Female +white collar +North Europe +2Adlt+Child +99996 +3 +4 +3 +2 +Male +white collar +Asia +2Adlt+Child +99997 +1 +1 +1 +1 +Female +blue collar +South Europe +2Adlt +99998 +1 +3 +1 +2 +Female +blue collar +South Europe +2Adlt+Child +99999 +4 +3 +2 +2 +Male +blue collar +North America +2Adlt+Child + +The second DF is the User-Preference DF which contains the latent preferences and is presented +in Table 3. These latent preferences are related to the ontology classes. The latent preferences +of each user were modeled through a multinomial logit model based on their demographic +features. This DF shows the relative interest of a given user in a given preference category versus +any other preference category. The values between different users are not comparable. + +Table 3 User-Preference DF containing the latent preferences from the Multinomial Logit model. +UserID +Beach +Relax +Shop +Nightlife +Theme park +Gastro +Sports +Culture +Nature +Events + +22 + +0 +0 .408 +0 .026 +0 .020 +0 .041 +0 .002 +0 .002 +0 .004 +0 .009 +0 .487 +0 .002 +1 +0 .002 +0 .077 +0 .017 +0 .015 +0 .009 +0 .457 +0 .041 +0 .271 +0 .107 +0 .002 +2 +0 .554 +0 .156 +0 .039 +0 .041 +0 .027 +0 .010 +0 .021 +0 .015 +0 .135 +0 .003 +3 +0 .005 +0 .038 +0 .012 +0 .000 +0 .003 +0 .252 +0 .003 +0 .674 +0 .009 +0 .002 +4 +0 .002 +0 .229 +0 .003 +0 .001 +0 .000 +0 .137 +0 .001 +0 .623 +0 .000 +0 .002 +... +. . . +. . . +. . . +. . . +. . . +. . . +. . . +. . . +. . . +. . . +99995 +0 .003 +0 .106 +0 .202 +0 .000 +0 .020 +0 .115 +0 .005 +0 .202 +0 .337 +0 .010 +99996 +0 .001 +0 .127 +0 .064 +0 .000 +0 .002 +0 .034 +0 .001 +0 .750 +0 .016 +0 .005 +99997 +0 .050 +0 .285 +0 .030 +0 .337 +0 .110 +0 .006 +0 .091 +0 .019 +0 .015 +0 .057 +99998 +0 .031 +0 .712 +0 .007 +0 .083 +0 .103 +0 .004 +0 .021 +0 .027 +0 .006 +0 .007 +99999 +0 .005 +0 .880 +0 .064 +0 .000 +0 .035 +0 .000 +0 .009 +0 .003 +0 .002 +0 .003 + +The third DF sample presented is the Item DF in Table 4. Here a set of 29 items were included +belonging to different categories which are the user latent preferences presented in the previous +table. + +Table 4 Item DF with corresponding item category (ontology and latent preferences). +itemID +Item Name +Category +0 +A service that offers you the opportunity to +do bungee-jumping +['Leisure', 'Sports', 'Routes', 'Events', +'Nature'] +1 +A tavern that serves traditional food +['Leisure', 'Events', 'Culture', 'Towns'] +2 +Ancient history museum +['Culture', 'ViewPoints', 'Events', +'Nature', 'Routes', 'Towns'] +3 +Discount for Callaway clubs +['Sports'] +4 +Get a discount for Comic-Con +['Sports'] +5 +Get a free pint at the pub +['Events', 'Leisure'] +6 +Get a free pizza at Pizza Hut +['Leisure'] +7 +Get a voucher for Sephora +['Leisure'] +8 +Go shopping in our new mall +['Leisure'] +9 +Golf lessons +['Sports', 'Leisure', 'Events'] + +23 + +10 +Great meals that are tasty +['Leisure', 'Events'] +11 +Medieval fair +['Culture', 'Events', 'Nature', 'Towns'] +12 +One day snorkeling with the fish +['Sports', 'Leisure', 'Nature'] +13 +One of the main nightclubs in the city +['Culture', 'Events', 'Nature', 'Leisure', +'Routes', 'Towns'] +14 +Rest and relaxation at the spa +['Leisure', 'Routes'] +15 +Surfing lessons +['Sports'] +16 +Take a trip in a hot-air balloon +['Sports'] +17 +Try go-karts with your friends +['Sports'] +18 +Try scubadiving +['Sports'] +19 +Try spearfishing with a pro +['Sports'] +20 +Watch a FC Porto match +['Events', 'Sports'] +21 +Watch a SL Benfica match +['Events', 'Sports'] +22 +Watch a Sporting CP match +['Sports', 'Events'] +23 +Watch a live concert of Mastodon +['Events'] +24 +Watch a live football match +['Sports', 'Events'] +25 +Watch a motogp race +['Events', 'Sports'] +26 +drive a F1 racecar +['Sports'] +27 +go to the spa +['Leisure'] +28 +visiting Disneyland +['Leisure'] + +The last data sample is the result of an external product between the user preferences from the +multinomial logit model and the item DF. The result is the input of a Fuzzy Inference System, +which along with other implicit information on user and items returns the User-Item ratings DF, a +sample of which is shown in Table 5. + +Table 5 User-Item ratings DF. + +0 +1 +2 +3 +4 +5 +… +23 +24 +25 +26 +27 +28 + +userId + + + + + + + + + + + + + + +0 +1.41 +0.00 +1.87 +0.00 +3.21 +0.00 +… +0.00 +1.79 +0.00 +1.79 +2.96 +0.00 + +1 +0.00 +4.63 +1.77 +1.26 +0.00 +0.00 +… +0.00 +0.00 +4.06 +0.00 +2.21 +1.77 +2 +0.00 +0.00 +0.00 +2.10 +3.20 +2.38 +… +3.48 +0.00 +0.00 +0.00 +0.00 +0.00 + +3 +0.00 +3.12 +0.00 +0.00 +3.28 +2.89 +… +0.00 +2.22 +0.00 +0.00 +0.00 +0.00 + +4 +1.37 +0.00 +2.31 +1.63 +0.00 +0.00 +… +3.31 +2.30 +0.00 +0.00 +0.00 +0.00 + +… +… +… +… +… +… +… +… + +… +… +… +… +… +… +99995 +0.00 +0.00 +0.00 +1.21 +3.42 +0.00 +… +0.00 +3.84 +3.79 +0.00 +3.36 +0.00 + +99996 +1.46 +0.00 +0.00 +0.00 +2.31 +0.00 +… +2.31 +0.00 +0.00 +0.00 +0.00 +1.39 +99997 +1.47 +0.00 +0.00 +1.32 +2.74 +0.00 +…. +0.00 +0.00 +2.29 +0.00 +0.00 +0.00 + +99998 +0.00 +4.64 +4.11 +1.78 +0.00 +2.94 +… +3.43 +2.65 +3.80 +0.00 +4.65 +4.33 +99999 +0.00 +3.54 +3.06 +0.00 +4.07 +2.65 +… +0.00 +3.07 +3.51 +2.46 +3.50 +2.61 + + +24 + + + +4.2 +Metrics +The metrics for a RS are not a trivial issue. Many works tend to use common ML metrics, such +as classification metrics like precision, recall, accuracy, or regression metrics such as RMSE or +MAE when the goal is to perform a regression on 1-5 ratings, for example. However, these metrics +imply that the data available to us about user behavior is perfect, that is, users are aware of all +the items they li e and the ones they haven’t tried aren’t as relevant. If this were the case, no RS +would be needed in the first place. The drawback of using this type of metrics is that it can +encourage the recommender to make obvious recommendations in some cases, by penalizing +wrong recommendations too much. In addition, these metrics do nothing to the tune of comparing +recommenders based on how personalized its recommendations are, or how diversified. +Other metrics have been developed for RS in recent years that try to address these issues, some +of which are presented in the following. + +1. Mean Average Precision @ K and Mean Average Recall @ K +As in more traditional machine learning, the dataset is split into training and test sets, and +the test set is comprised of cases the learner did not train on and thus it is used to +measure the model’s ability to generali e ith ne data. In recommender systems, the +same is done, and the output of a recommender system is usually a list of K +recommendations for each user in the test set, and to produce those recommendations +the recommender only trained on the items that user enjoyed in the training set. MAP@K +(Mean Average Precision @ K) gives insight to how relevant the list of recommended +items are, whereas MAR@K (Mean Average Recall @ K) gives insight to how well the +recommender system is able to discover all the items the user has rated positively in the +test set. +In recommender systems, precision and recall are essentially the same as in machine +learning: +𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = # 𝑜𝑓 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑟𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛𝑠 +# 𝑜𝑓 𝑖𝑡𝑒𝑚𝑠 𝑟𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑒𝑑 + +𝑅𝑒𝑐𝑎𝑙𝑙 = # 𝑜𝑓 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑟𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛𝑠 +# 𝑜𝑓 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑖𝑡𝑒𝑚𝑠 + + o ever, these metrics don’t ta e ordering into account, and since the output of a +recommender system is usually an ordered list, the metrics at cut-off k are introduced, +MAP@K and MAR@K. + +25 + +𝑀𝐴𝑃@𝐾 = 1 +|𝑈| ∑ +1 +min (𝑚, 𝐾) +|𝑈| +𝑢=1 +∑ 𝑃𝑢(𝑘) ∙ 𝑟𝑒𝑙𝑢(𝑘) +𝐾 +𝑘=1 + +𝑀𝐴𝑅@𝐾 = 1 +|𝑈| ∑ 1 +𝑚 +|𝑈| +𝑢=1 +∑ 𝑟𝑢(𝑘) ∙ 𝑟𝑒𝑙𝑢(𝑘) +𝐾 +𝑘=1 + +Where 𝑈 is the set of users in the test set, 𝑚 is the number of relevant items for user 𝑢, +𝑃𝑢(𝑘) and 𝑟𝑢(𝑘), are the precision@k and recall@k, respectively, and 𝑟𝑒𝑙𝑢(𝑘) is a factor +equal to 1 if the 𝑘 th item is relevant, and 0 otherwise. + + +2. Coverage + +Coverage is the percentage of items on the training data that the recommender is able to +recommend on a test set. + +𝐶𝑜𝑣𝑒𝑟𝑎𝑔𝑒 = 𝐼 +𝑁 ∗ 100% + +Where 𝐼 is the number of unique items the model recommends in the test data and 𝑁 is +the total number of unique items in the training data. + + + +3. Personalization +Personalization is the dissimilarity between users lists of recommendations. A high score +indicates user lists are different between each other, while a low score indicates they are +very similar. Similarity between recommendation lists is calculated via the cosine +similarity between said lists and then by calculating the average of the upper triangle of +the cosine similarity matrix (avgCosim). The personalization is then given by: +𝑃𝑒𝑟𝑠𝑜𝑛𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 = 1 − 𝑎𝑣𝑔𝐶𝑜𝑠𝑖𝑚 + +4. Diversity +Diversity measures how different are the items being recommended to the user. +𝐷𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦 = 1 − 𝑖𝑙𝑠 + +26 + +Where 𝑖𝑙𝑠 corresponds to intra-list similarity, which is the average cosine similarity of all +items in a list of recommendations. This calculation uses features of the recommended +items (such as item metadata) to calculate the similarity. The feature matrix is indexed by +the item id and includes one-hot-encoded features. If a recommender system is +recommending lists of very similar items, the intra-list similarity will be high and +conversely, the diversity will be low. + +5. Novelty +Finally, novelty measures the capacity of recommender systems to propose novel and +unexpected items which a user is unlikely to know about already. It uses the self- +information of the recommended item, and it calculates the mean self-information per top- +N recommended list and averages them over all users. +𝑁𝑜𝑣𝑒𝑙𝑡𝑦 = 1 +|𝑈| ∑ ∑ +𝑙𝑜𝑔2 (𝑐𝑜𝑢𝑛𝑡(𝑖) +|𝑈| +) +|𝑁| +|𝑁| +𝑖=1 +|𝑈| +𝑢=1 + +Where 𝑈 is the user list, 𝑁 is the top n-list and 𝑐𝑜𝑢𝑛𝑡(𝑖) is the number of users that have +consumed the specific item. + +4.3 +CS with increasing data quantity +In this sub-section the previously presented datasets and the previously presented metrics are +employed to test and evaluate the RS in its various phases. For this to work, the datasets will be +gradually incremented, starting with very few users and no ratings, and ending with the full +datasets. This process is meant to mimic the natural evolution of a RS, from initial cold-start +conditions to thousands of users with thousands of reviews. In each phase different +recommenders are employed as was already mentioned in previous sections. + +4.3.1 CS in Phase 1 +As mentioned previously, phase 1 is characterized by little number of users and no ratings. At this +point, only content-based approaches are possible, and only if there is some input from the user +concerning his preferences, which the RS asks when the user first logs in. Otherwise, the RS +would be incapable of giving any recommendation short of a random context-filtered one. To +mimic this first stage, 98 initial users are added to the RS. Each user inputs their HL preference +vector related to Table 3, which the phase 1 content-based recommender uses to generate +recommendations. Unlike in Table 3, the HL preference vector takes either 0 or 1 values and thus +not conveying information on interest intensity. In the following tables, a sample of the 98 users +and their respective HL vectors are shown. + +27 + +Table 6 High-level preferences of the users. +userId +ViewPoints +Nature +Towns +Culture +Events +Leisure +Routes +Sports +1 +0 +0 +0 +0 +0 +1 +0 +0 +2 +1 +0 +1 +0 +0 +1 +1 +0 +3 +0 +0 +0 +0 +0 +1 +0 +0 +4 +0 +0 +0 +0 +0 +1 +1 +1 +5 +0 +0 +1 +0 +0 +0 +0 +0 +… +… +… +… +… +… +… +… +… +94 +0 +0 +0 +0 +0 +1 +0 +0 +95 +0 +0 +0 +0 +0 +1 +0 +0 +96 +1 +0 +1 +0 +0 +1 +1 +0 +97 +0 +0 +1 +0 +0 +0 +0 +0 +98 +1 +0 +1 +1 +0 +0 +1 +0 + +The recommendations given by the RS for each user are in the following table. We can apply all +previously presented metrics to these results, including MAP@K and MAR@K because we are +aware of some ratings given by the users, present in the User-Item ratings DF which we can use +for this purpose. + +Table 7 Sample of the recommendations given to the users by the content recommender. +userId +Recommendations +1 +[(6, 'Get a free pizza at Pizza Hut'), (7, 'Get a voucher for Sephora'), (8, 'Go +shopping in our new mall'), (27, 'go to the spa'), (28, 'visiting Disneyland')] +2 +[(6, 'Get a free pizza at Pizza Hut'), (7, 'Get a voucher for Sephora'), (8, 'Go +shopping in our new mall'), (14, 'Rest and relaxation at the spa'), (27, 'go to the +spa')] +3 +[(6, 'Get a free pizza at Pizza Hut'), (7, 'Get a voucher for Sephora'), (8, 'Go +shopping in our new mall'), (27, 'go to the spa'), (28, 'visiting Disneyland')] +4 +[(4, 'Get a discount for Comic-Con'), (6, 'Get a free pizza at Pizza Hut'), (7, 'Get a +voucher for Sephora'), (8, 'Go shopping in our new mall'), (14, 'Rest and relaxation +at the spa')] +5 +[(11, 'Medieval fair'), (1, 'A tavern that serves traditional food'), (13, 'One of the +main nightclubs in the city'), (2, 'Ancient history museum'), (0, 'A service that offers +you the opportunity to do bungee-jumping')] +… +… +94 +[(6, 'Get a free pizza at Pizza Hut'), (7, 'Get a voucher for Sephora'), (8, 'Go +shopping in our new mall'), (27, 'go to the spa'), (28, 'visiting Disneyland')] + +28 + +95 +[(6, 'Get a free pizza at Pizza Hut'), (7, 'Get a voucher for Sephora'), (8, 'Go +shopping in our new mall'), (27, 'go to the spa'), (28, 'visiting Disneyland')] +96 +[(6, 'Get a free pizza at Pizza Hut'), (7, 'Get a voucher for Sephora'), (8, 'Go +shopping in our new mall'), (14, 'Rest and relaxation at the spa'), (27, 'go to the +spa')] +97 +[(11, 'Medieval fair'), (1, 'A tavern that serves traditional food'), (13, 'One of the +main nightclubs in the city'), (2, 'Ancient history museum'), (0, 'A service that offers +you the opportunity to do bungee-jumping')] +98 +[(2, 'Ancient history museum'), (11, 'Medieval fair'), (13, 'One of the main +nightclubs in the city'), (1, 'A tavern that serves traditional food'), (14, 'Rest and +relaxation at the spa')] + + +Table 8 Values for the various metrics on the content model recommendations. +MAP@K +MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty +0.092 +0.092 +0.55 +0.51 +0.21 +0.76 +0.66 + +We can see that mean average precision and mean average recall have the same value, the +value at K is equal to 5, since the recommender recommends 5 items to each user. The two +diversity values pertain to high level and low-level preferences showing how diverse are the +recommendations in terms of recommending diverse items. It is expected for the high-level +diversity to be lower than the low-level diversity since the content recommender makes +recommendations based on high-level preferences of the users. Low-level preferences are linked +ontologically to high-level preferences, but they are greater in variety, hence the same higl-level +preference is linked to many low-level preferences, this justifies the larger value of Diversity LL +compared to Diversity HL. Coverage, personalization and both diversities return values from 0 to +1, where 1 represents maximum coverage, personalization and diversity. The value for novelty +can take any positive value, the greater the value the more unexpected recommendations are +given based on popularity. In this study, the metric for novelty may not be very useful due to the +relatively low cardinality of items and the fact that there are no less popular items per se, at least +not very noticeably. In any case, these metrics are more useful in when used to compare different +models. + + + + + +29 + +4.3.2 CS in Phase 2 +In phase 2 there are ratings in the system, although not enough users to feed the demographic- +based recommender. In this phase we can simulate an RS state where there are 98 users and +64 ratings. The hybrid recommender is a hybridization of the initial content-based recommender +with the new popularity-based recommender. The ratings are used to filter out items with average +rating below a given threshold. Once again, the same metrics are applied, and the results are +shown in the following table. + +Table 9 Values for the various metrics on the hybrid model recommendations. +MAP@K +MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty +0.219 +0.219 +0.17 +1.11e-16 +0.64 +0.91 +0.66 + +It is interesting to observe that the precision and recall have gone up, which makes sense because +the items are now being filtered according to rating and higher rating items are more prone to +having been liked by the users, at least the synthetic data was defined as such. The coverage +has gone down, which makes sense since less items are being recommended due to filtering. +Personalization has gone down since it now many users are being recommended the same items. +Diversity has gone up; this can be due to recommending some items outside of the natural +preference of the user due to ratings filtering. All in all, differences can be observed compared to +the content-recommender, these differences make sense and seem to go towards an expected +behavior by the recommender. + +4.3.3 CS in Phase 3 +In phase 3, enough users with ratings given have been introduced in the system to kickstart the +demographic-based recommender. This recommender works by defining user clusters based on +demographic features and then giving item recommendations based on the predictions of a kNN. +This phase 3 recommender works together with the hybrid recommender from phase 2. In the +following table, the metrics are applied, and the results shown. The number of users in this phase +total 198, with 191 ratings. + +Table 10 Values for the various metrics on the hybrid and demographic model recommendations. + +MAP@K MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty +Hybrid +0.178 +0.178 +0.34 +0.07 +0.64 +0.91 +0.66 +Demog +0.151 +0.151 +0.72 +0.57 +0.63 +0.90 +0.66 + + +30 + + +We can see these results in a bar chart where a min max scaler has been applied. This basically +shows which model wins in each category. + +Figure 9 Scaled metrics for both models. + +We can see that the hybrid model loses to the demographic model in coverage and +personalization and has higher values in the other metrics. However, we can see that results are +virtually equal in terms of Diversity and Novelty, and only on the Precision and Recall do we see +larger values for the hybrid model, which are not that much higher. On the other hand, the +demographic recommender has much larger personalization and coverage. Here we can see an +increment by the demographic model compared to the hybrid model. This makes sense because +the demographic model is more complex in how recommendations are given by finding similar +users in terms of demographic features and then recommending similar items to the user on a +more individual basis, whereas the hybrid model is again based on high level preferences. + +Table 11 Values for the various metrics on the hybrid phase 2 and hybrid phase 3 model recommendations. + +MAP@K MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty +Hybrid +P2 +0.219 +0.219 +0.17 +1.11e-16 +0.64 +0.91 +0.66 +Hybrid +P3 +0.178 +0.178 +0.34 +0.07 +0.64 +0.91 +0.66 + + +MAP@K +MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty31 + +It is also interesting to compare the metrics between the hybrid in phase 2 and phase 3. We can +see that most metrics remain similar with a slight decrease in precision and recall, which may be +just random, a slight increase in personalization, and a rather large increase in coverage. This +can be due to more items recommended and not filtered out due to poor ratings because of the +existence of more users and ratings on items. It is interesting to see a variation of the metrics of +the same recommender as the amount of data increases. + +4.3.4 CS in Phase 4 +Phase 4 starts when a given number of users and a given density of the user-item rating DF is +achieved. When this happens, the final recommender is initiated. This recommender is the +already mentioned FFM. In phase 4, the recommendations are, once again, the result of an +ensemble of recommenders, the same one in phase 3 with the addition of the new FFM. The +resulting metrics are once more applied to the recommendations and are shown in the following +table. In this phase we have 250 users and 191 ratings. + +Table 12 Values for the various metrics on the hybrid, demographic and collaborative model +recommendations. + +MAP@K +MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty +Hybrid +0.158 +0.158 +0.34 +0.06 +0.64 +0.91 +0.66 +Demog +0.137 +0.137 +0.68 +0.55 +0.66 +0.91 +0.66 +Collab +0.181 +0.181 +0.72 +0.54 +0.67 +0.91 +0.66 + +Comparing the recommenders, we can observe that the collaborative recommender, which was +added in this later stage has high levels of personalization and coverage and achieves the highest +values for precision and recall, compared to the other two models. The values for diversity are all +similar at this stage, and novelty again doesn’t provide useful information ith this number of total +items. In terms of precision and recall, coverage and personalization, the collaborative +recommender gives us expected results which is relatively high values in these metrics. We can +observe that each recommender brings different recommendations to the table with clear +improvements in some metrics as the recommender system matures. It would be interesting to +view this with a dataset comprising many more items and users. In the following figure we can +see the metrics in a scaled graph. + + +32 + + +Figure 10 Scaled metrics for all three models + +As said, we observe that the collaborative metrics are good in comparison to the other two, +however, the collaborative model is only useful when the recommender system has seen +sufficient data. The metrics for the other t o are not as high but they don’t suffer so much from +cold-start issues. We can see that between the demographic and the hybrid models there is a +trade-off in metrics. We had already seen this in the previous phase. + +Table 13 Values for the various metrics on the phase 1, phase 2 and phase 3 model recommendations of +hybrid and demographic models. + +MAP@K +MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty +Hybrid +P2 +0.219 +0.219 +0.17 +1.11e-16 +0.64 +0.91 +0.66 +Hybrid +P3 +0.178 +0.178 +0.34 +0.07 +0.64 +0.91 +0.66 +Hybrid +P4 +0.158 +0.158 +0.34 +0.06 +0.64 +0.91 +0.66 +Demog +P3 +0.151 +0.151 +0.72 +0.57 +0.63 +0.90 +0.66 +Demog +P4 +0.137 +0.137 +0.68 +0.55 +0.66 +0.91 +0.66 + + +MAP@K +MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty33 + +Here we can see a comparison between the metrics of the different models along each phase, +we can see a slight decrease of precision and recall in the evolving phases for hybrid and +demographic models, but this might have to do with insufficient ratings being added between +phase 3 and phase 4, which are important for the demographic recommender. With a further +increase in data, we can see further differences in the metrics. Feeding the recommender system +with 1000 users and 883 ratings, we attain the following results. + +Table 14 Values for the various metrics on the hybrid, demographic and collaborative model +recommendations, in the case of 250 users and 191 ratings as well as 1000 users and 883 ratings. + +MAP@K +MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty +Hybrid +0.158 +0.158 +0.34 +0.06 +0.64 +0.91 +0.66 +Demog +0.137 +0.137 +0.68 +0.55 +0.66 +0.91 +0.66 +Collab +0.181 +0.181 +0.72 +0.54 +0.67 +0.91 +0.66 +Hybrid +1000 +0.088 +0.088 +0.28 +0.19 +0.69 +0.89 +0.66 +Demog +1000 +0.128 +0.128 +0.97 +0.59 +0.48 +0.89 +0.66 +Collab +1000 +0.119 +0.119 +0.79 +0.61 +0.52 +0.89 +0.66 + + + +Figure 11 Scaled metrics for all three models. + +MAP@K +MAR@K +Coverage +Personalization +Diversity HL +Diversity LL +Novelty34 + +We can see that the metrics are qualitatively similar to the case before with less users and ratings. +Still the number of ratings is low, there is not a lot of rating density, which particularly penalizes +the collaborative model. Nonetheless, we can observe that the collaborative model is the one that +offers more personalization, which increased for all models with the increment in users and +ratings. Coverage also increased heavily for the demographic model while only increasing slightly +for the collaborative model. As for precision and recall, the demographic model maintains the +metric with only a slight decrease while the hybrid and collaborative model saw a rather significant +decrease. In regard to the collaborative model this might have to do with the low density in ratings. +All in all we see that the demographic and collaborative models clearly become more dominant +and useful as more data is added to the RS. The phases also make sense, by having the +collaborative model initiate after all others have been initiated, since the collaborative model is +very sensitive to rating density, while the demographic model is more robust in that sense. The +hybrid model by this phase has clearly been passed by the two other models in most metrics +which is exactly what would be expected. + +5 +Conclusion and future works +In this work an ontology-based context aware recommender system application for tourism was +presented where different recommenders are used at different stages of maturity of the +recommender system. The novel aspect is the evolution of the recommender system with different +types of recommenders entering the recommendation pool as the system’s maturity evolves. The +ontology extension of the recommender system allows items to be binned and recommended to +users based on user preference vectors with different degrees of detail that link to the item +ontology. These preference vectors will be ever changing based on user feedback, while other +recommenders based on demographic features and field-aware factorization machines join the +pool as data increases. +Along this work, the RS was presented and ultimately tested with synthetic data mimicking +different stages of maturity. One could observe that at each new phase the new recommenders +added value as observed from the comparison between the different adopted metrics, which were +MAP@K, MAR@K, Coverage, Personalization, Diversity HL, Diversity LL and finally Novelty. +These metrics are the state of the art for Recommender Systems because they attempt to go +beyond the usual metrics adopted in , hich don’t al ays have much meaning in RS. The +results obtained were expected where Collaborative and Demographic approaches essentially +brought more personalization and coverage to the table. However, the full extent of differences +between recommenders could not be captured mainly due to the relatively low cardinality of items +being offered, only 29. +Future works would entail a broader analysis with more items, and also context-aware data which +was not tested at this instance. Nonetheless, the context-aware would be essentially pre-filtering +which would not be of much interest regarding the results concerning the metrics. + +35 + +6 +Acknowledgements +The present paper was developed in the context of the PMP project – Partnership Management +Platform, code LISBOA-01-0247-FEDER-045411, co-financed by LISBOA 2020 and Portugal +2020 through the European Regional Development Fund. + +7 +References + +[1] +C. I. ee, . C. sia, . C. su, and J. Y. in, “ ntology-based tourism recommendation +system,” 2017 4th International Conference on Industrial Engineering and Applications, +ICIEA 2017, pp. 376–379, 2017, doi: 10.1109/IEA.2017.7939242. +[2] +J. Borràs, A. 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Rendle, “ actori ation machines,” in Proceedings - IEEE International Conference on +Data Mining, ICDM, 2010, pp. 995–1000. doi: 10.1109/ICDM.2010.127. + + diff --git a/99AyT4oBgHgl3EQf3fke/content/tmp_files/load_file.txt b/99AyT4oBgHgl3EQf3fke/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b144b19862dfb1db1e25c492f35af6f5f303338c --- /dev/null +++ b/99AyT4oBgHgl3EQf3fke/content/tmp_files/load_file.txt @@ -0,0 +1,2020 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf,len=2019 +page_content='1 Ontology-based Context Aware Recommender System Application for Tourism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Vitor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Camacho1, José Cruz2 1 PhD, vitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='camacho@syone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='com, R&D Data Science, Syone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2 MSc, jose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='cruz@syone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='com, R&D Data Science, Syone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Abstract In this work a novel recommender system (RS) for Tourism is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The RS is context aware as is now the rule in the state-of-the-art for recommender systems and works on top of a tourism ontology which is used to group the different items being offered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The presented RS mixes different types of recommenders creating an ensemble which changes on the basis of the RS’s maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Starting from simple content-based recommendations and iteratively adding popularity, demographic and collaborative filtering methods as rating density and user cardinality increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The result is a RS that mutates during its lifetime and uses a tourism ontology and natural language processing (NLP) to correctly bin the items to specific item categories and meta categories in the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This item classification facilitates the association between user preferences and items, as well as allowing to better classify and group the items being offered, which in turn is particularly useful for context-aware filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Keywords: recommender system, CARS, ontology, tourism, content-based, collaborative filtering, demographic-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 1 Introduction This work presents a novel recommender system (RS) approach, which builds on context awareness, domain ontology and different types of recommenders that enter the process at different stages of maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' From simple recommenders that are less prone to cold-start issues to more complex and powerful recommenders which struggle quite a bit with initial lack of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' At the final stage of maturity, when all the recommenders are already deployed in the recommender pool, the different recommenders analyze different aspects of the data, from demographic features to ratings, and provide an ensemble of recommendations to the users, based on different approaches and with varying degrees of personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The approach is novel 2 in how it uses several techniques, from domain ontology to bin the items using NLP to achieve concept similarity, and then from there applies content-based, demographic-based, popularity- based and collaborative filtering approaches to attain the recommended items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The collaborative filtering employed are field-aware factorization machines which are the state-of-the-art in matrix factorization, which can easily include context-awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The aim is to provide a powerful and adaptable recommender system framework which can adapt to any domain, given the respective domain ontology, and can overcome cold-start issues by using an approach with 4 stages of maturity, which are subsequently entered when given thresholds are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following the structure of the paper is presented, with an explanation of every section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the present section, the Introduction, an overview of the presented recommender system framework is provided as well as a literature review of the relevant works on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In section 2, the framework and all its components are presented, from adopted technologies to used algorithms and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' A presentation of the architecture is given as well as a mock-up of the designed UI to provide the link between user and recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In section 3, the technologies and techniques, mainly the ones central to the recommender system are better explained with some formulas being provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In section 4, the recommender system is tested with a synthetic dataset with varying stages of maturity, to show how the recommender system evolves as the data changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In section 5, conclusions are given as well as a brief discussion on future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Literature review Recommender systems (RS) have been the focus of research for many years now, both on the algorithm side and on the applied side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The study of RS started in the beginning of the 1990s but it was in the last 15 years that research and the number of publications on the topic surged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Concerning our application, tourism, RS have been the focus of studies since, at least, the start of the 2000s with many publications having been made since then [1]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' As for works that concern more an algorithmic approach without an explicit thematic application, several studies have been published on the different types of RS, from content-based approaches to collaborative filtering, as well as context-aware solutions, so called CARS [16]–[64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Going into more detail regarding the tourism themed recommenders, it is relevant to give particular attention to ontology-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' One of the more important examples concerning the present work is Moreno, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this work, an ontology-based approach (SigTur/E-destination) is developed to get recommendations for tourism in the region of Tarragona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The developed approach begins with the definition of a tourism domain ontology, which describes the tourist activities in a hierarchy, and bins the activities according to a given taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The ontology is thus used to explicitly classify the activities to recommend among a predefined set of distinctive main concepts, which are used by the intelligent recommender system in its reasoning processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The recommender than applies collaborative and content- based techniques to provide the recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Another relevant work is that of García- Crespo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [11], which proposes a semantic based expert system to provide 3 recommendations in the tourist domain (Sem-Fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The proposed system works based on the consumer’s experience about recommendations provided by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Sem-Fit uses the experience point of view in order to apply fuzzy logic techniques to relating customer and hotel characteristics, represented by means of domain ontologies and affect grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' An early and interesting work that applies Bayesian networks to attain personalized recommendations for tourist attractions by Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' and Bian, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [15] is also worth mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This work is from 2009 and uses ontologies to classify different types of tourist attractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It then uses a Bayesian network, to calculate the posterior probabilities of a given tourist’s preferred activities and the traveler category he fits into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Other works on recommender system tourism applications could also be mentioned but instead one can mention three surveys done on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' First, one from 2014, Borràs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [2] present a survey entitled “Intelligent tourism recommender systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this survey the various works in the state-of-the-art are analyzed and their different approaches concerning user interface, functionalities, recommendation techniques and use of AI techniques are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The second work that gives an overview on the topic is from Kzaz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [3] from 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this overview, the focus is essentially on recommender approaches and employed user and item data models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' A third survey on this topic is given to us by Renjith, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [60] in a work titled “An extensive study on the evolution of context-aware personalized travel recommender systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Herein, the authors start by defining the different recommender approaches that can be employed: content-based, collaborative, demographic-based, knowledge-based, hybrid, personalized and context-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The authors also go into detail on the different machine learning algorithms that are commonly employed, as well as the different employed metrics to evaluate the quality of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Finally, they present a table with many different works with the identification of whether or not they employ the previously mentioned techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' One of the aspects of the present work is that, as happens with some of the examples given above, it employs ontologies to organize and classify the items to be recommended in some way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Two works can also be mentioned concerning tourism domain ontologies, but in this case their formulation rather than their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These works are by Ruíz-Martinez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [65] and Barta, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [66] and they present different approaches to integrate and define tourism domain ontologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the latter work an approach is presented that shows how to cover the semantic space of tourism and be able to integrate different modularized ontologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the former, a strategy to automatically instantiate and populate a domain ontology by extracting semantic content from textual web documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This work deals essentially with natural language processing and named entity recognition, which are some of the techniques also employed in this paper in terms of ontology population or, in other words, the classification of the different items to recommend according to the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Many other works should also be referenced, this time not necessarily linked to the tourism theme, but instead due to their focus on the algorithmic aspect or rather the recommendation strategy regardless of its field of application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' One particular type of recommender system that is very much dominant in the literature in recent times is the context aware recommender system (CARS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The work by Kulkarni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [32] gives us a review on the state-of-the-art techniques employed in 4 context aware recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this work the authors list the most common algorithmic approaches from bio-inspired algorithms to other common and less common machine learning algorithms and then enumerate the works that employed each type of solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Another review study on context aware recommender systems is authored by Haruna, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this work, the authors particularly emphasize the manner in which the contextual filtration is applied, for which there are three variants, pre-filtering, post-filtering and context modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The difference between each approach has to do with how context filtering is applied together with the process of recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Hence, in pre-filtering the recommender filters the items prior to recommendation, while in post-filtering the opposite happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In context modelling there is a more complex integration of the context filtering and the recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The authors then go on to classify the different works in the literature according to this and other topics such as employed algorithms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' A third overview paper on the topic of CARS is the work by Raza, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this work, the authors focus on the type of algorithms, the dimensionality reduction techniques, user modelling techniques and finally the evaluation metrics and datasets employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Still focusing on CARS, a context-aware knowledge-based recommender system for movie showtimes called RecomMetz is presented in the work by Colombo-Mendoza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this work, the CARS developed has time awareness, crowd awareness and location awareness, as part of its context awareness composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It is interesting to verify that its location awareness employs an exponential distance decay that discards items that are far away from the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This sort of mechanism is also employed in the current work but with other goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' A last example on CARS is a genetic algorithm (GA) approach based on spatio-temporal aspects [68] by Linda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Here, the interesting aspect is the inclusion of a GA to optimize the temporal weights of each individual while employing collaborative filtering for the recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Lately, one of the most studied techniques for recommender systems have been Factorization Machines (FM) [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the present work, a field-aware version of this technique is employed, also known as an FFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This technique is a kind of collaborative filtering method that gained some notoriety for solving click-through prediction rates [64], among other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Several versions exist of these FMs in the literature, with ensembles with deep neural networks [45], for example, being one of such versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The value of FM is that they are more powerful than traditional matrix factorization techniques, being able to incorporate features and information such as implicit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' For these reasons, an FM, more specifically an FFM, is one of the recommenders employed in the proposed recommender system, constituting the collaborative filtering component of the proposed RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2 Description of the RS and field of application The proposed RS in this work is to be applied in the tourism industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' More specifically, the project entails the creation of a recommender system to be used by hotel companies to recommend to their guests their vast lists of partners in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It is very common that large hotel companies have hundreds of partners offering products and most hotel guests are unaware of most of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The partners usually offer a wide array of products, which need an ontology to be organized and better recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The proposed RS starts by having a Partner Management Platform (PMP) for the hotel’s partners where they can manually introduce the items they want to be recommended in the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The PMP, which is essentially an interface of the Item DB, feeds the Domain Ontology which exists in a graph DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The users are clients of the hotel that have checked- in, and they exist in the User DB, which houses not only demographic information but also user preferences which are collected and inferred by the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The RS interface is a web-app which is presented in a further section of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following sections more detail is provided concerning the various components of the RS, starting with the presentation of the RS architecture in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2 Architecture and frameworks of the recommender system The architecture of the RS can be essentially divided into 4 parts, the data repository, the context- aware subsystem, the recommender system per se and the user interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following figure the architecture is presented with each of its subcomponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' An overview of each of the subcomponents is given in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Figure 1 Architecture of the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Data repository The first element of the recommender system is its data repository, in the sense that this is where it starts, particularly with the Partner Management Platform (PMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It is through this PMP that we Location-aware Weather-aware Repetition-aware Userprofile manager Context-awareSubsystem Preference UserInterface manager ItemDB UserDB Domain Ontology Recommender pool Partner Management Recommender Platform Data Repository System6 have the introduction of the items, by the partners, to be recommended by the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the PMP, the partners introduce the items alongside necessary description and keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This information introduced in the PMP is organized into an Item DB and later inserted into the domain ontology, which is later explained in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Other than the PMP with its Item DB and the mentioned domain ontology, the data repository also has a User DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This DB has both the demographic information collected from the users that check-in to the hotel, but also the preference vectors that are inferred and managed by the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The RS uses these two components of the user info to make predictions and to build different recommendation models based on demographic, content, and collaborative filtering techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Domain ontology - Neo4j and automatic population of ontology As for the domain ontology, the initial approach was to adopt the ontology presented in SigTur [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In addition, Neo4j (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='neo4j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='com), which is a graph DB, was chosen to house the ontology and to facilitate the automatic ontological extension with the items from the PMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following figures, the original ontology is shown already inserted in a Neo4j graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Figure 2 Ontology inserted in Neo4j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Shopping Wine SCO Wine_E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='. Popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' SCO Music_F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' NightLife Sco SCO SCo sCO BookF Gastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='. 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Gastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Sport_ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Sco TownR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Relaxafi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='. $Co Arts_An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Sco ScO Towns SCO Sco SCO SCC NonAqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' sCO Sport_R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='. SCO Air_Spor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' SCo Culture SCo Tradition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='. Sco SCo - SCO Sco Sports Culture SCO Tradition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MotorSp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Sco Aquatic_ Nature Sco Climbing Sco Monume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' SCO Ethnogr OOS Saiting Culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' SCo Surfing Nature UnderW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' ViewPoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 8 SCO Archeol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' sco 8 $CO Protecte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Art_Mus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' LandSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Nature History.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' ichitect SCo SCO SCO Mountai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Coastal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Inland Rural_A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='7 Figure 3 Sample of the ontology (highlighted section in previous figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The advantage of using the Neo4j framework is that it facilitates the automation of ontological extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This ontological extension is achieved through the use of NLP techniques, such as named entity recognition and cosine similarity between semantic concepts, using the spaCy Python library integrated with Neo4j methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These processes start with the insertion of the items from the PMP or the Item DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These items are parsed and tokenized, using both the item descriptions and/or keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These parsed and tokenized items are then linked to the ontology by means of semantic similarity between its keywords and description with each of the ontological subclasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The similarity scores above a given threshold originate a link between the item and that specific ontological subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This process that ends with concept similarity and starts with parsing, removal of stopwords and tokenization is performed with methods in the spaCy library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The concept similarity is performed using spaCy’s vast pretrained word vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In addition, named entity recognition is also performed on the items, automatically linking a Wikipedia entry, if such entry exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In Figure 4, a representation of the ontology after being extended with some items, via the described process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' One can see the original nodes in orange, that belong to the ontology classes, some of which are now linked to grey nodes representing the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The green nodes represent the Wikipedia page object when such an object was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In Figure 5 a zoomed view of the highlighted zone in Figure 4 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' One can see two instances in which a Wikipedia page object was found from the Named Entity Recognition procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The items were linked to the ontology subclasses and one can observe that the links make sense in these cases, with driving an F1 racecar linked to “Motor Sports”, and golf lessons and discounts on clubs linked to “Golf”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='ie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='oi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='useums ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='istory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Culture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='A uatic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='usic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Sailing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='nder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Rural A ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Air Spor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='ature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Golf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='vents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='otorSp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='oo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Culture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='riving ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Shopping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Gastron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='eisure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Relaxati ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='eisure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='ine ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='onA u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='ountai ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='opular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Figure 4 Ontology extended with the addition of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 9 Figure 5 Sample of the extended ontology (highlighted section in previous figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The recommender system module then imports the extended ontology, both the classes and the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It will use the extended ontology to give content-based recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2 Context-aware subsystem module The context-aware subsystem module does item pre-filtering on the basis of three context submodules: location-aware, weather-aware and repetition-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the case of the location- aware submodule, the objective is to filter out the hotel partners that are not located close by to a specific instance of the hotel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Since the hotel company can have a wide array of partners that may, in many cases, be close to one specific hotel but not to other hotels in other locations, such as local or regional partners that only provide services to the hotels in the area, a first contextual filtering phase is to apply location pre-filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Then we go on to the weather-aware submodule, where the ontological sub-classes are associated with a given fuzzy definition of when they make sense to be recommended, for example the beach ontology class or the outdoor sports ontology class would tend to be penalized with bad weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Finally, a third module, which is very much novel, which is the repetition-aware module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Here, each ontological class would have a different elapsed time parameter that affects an inverse exponential penalization factor to mimic the repeatability of a given item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' For example, one would probably be more adept to repeat a restaurant than a museum in the same week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' So, different ontological classes have different factors that affect the inverse exponential function, that we may call the unwillingness to repeat function, which defines how soon a user may be willing to repeat a given item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='ie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='oi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='useums ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='istory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='ance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='onA u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='ountai ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='opular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3 Recommender system module The recommender system module is the main module as the name entails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This module is constituted by a user profile manager and a preference manager, besides the recommender pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Concerning the recommender pool and the models that compose it, that is addressed in depth in Section 3 of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Here it suffices to say that the recommender pool is the set of different recommender models that provide user recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The models create an ensemble, when more than one is active, that provides recommendations using different techniques and approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' As for the remainder of the recommender system module, the user profile and the preference manager, these two sub-modules manage the user related information, such as item ratings and other user feedback in the case of the former, while the latter manages the user preference vectors and propagates the user feedback on items to update the user preference vectors accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The way this is done will become clearer in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='4 User interface – web app The last component is the user interface, which in this case is a web app that connects to the recommender system module and other modules through a real-time and batch inference endpoints that connect to ML pipelines defined in Azure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 11 Figure 6 App mockup showing the four main screens: welcome, preference definition, home and user profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the previous figure one can observe the four different screens the user sees during his App experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The FILTER screen is only presented to the user on the first time he logs in and is, in essence, a series of check boxes where the user defines his preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These check boxes are used to give a first estimate on the user’s preferences concerning the ontology classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The user’s choices define his preference vectors which then are used to make content-based recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' As for the HOME screen, it shows the different recommendations made to the user by the RS, here the user can bookmark items, book items or mar an item as “uninteresting”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Finally, in the PROFILE screen, the user can observe his profile in terms preferences collected and inferred by the RS as well as demographic information, such as date of birth, nationality, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The different interactions the user can have with the App and the consequent interactions between the App and the RS and back to the user are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this figure one can see how these interactions cascade and what the user gets back from each action he undertakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" One can summarize the actions the user can take in the following: Logging in Preference input Viewing recommendations viser Adviser see syone syone YOU'RESTAYINGHERE Ldviser PAULAESTEVES Whatareyouinterested in?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" Hello HotelGoldenCrown MUSO Noture Paulo Esteves Vewpoinitsv HOTEL TMYPROFILE Concerte Le'sure Sports Walks Utorciaugue,faucibusatioculisid Nnrnec Pauio Esteves efficitursagittis diam." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Etiom eget nunc RestourantsV Foutes Cinema DOB: 10/03/1983 acus Adcress: Ruo Efficitur:sogittis dion,#3,1c Lisboc Finess Beatchv Top:5 Foryou Jeb: Sorior BIAnclyst PhasellusacportatellusVivamus EatProhik tempormattisultrces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content="Proinvitae Tellmemore conseouortortorguispnoretotortor A WHATWE'VELEARNEDABOUTYOU SKYDIVErush 50% Ipere Foucbusoticcuisidetficit LEISURE ROUTES EVENTS FooFightersy tiverpoolFo 22, gogittis Ciam." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='Etiam pgetnont locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Francesinho Guzado START TOWNS CULTURE NATURE Dive classes VEWPOINTS SPORTS 50% oucibuaticcuisi,eficitur 2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" pogittisdigmEtiomegetmunt I'MALSOINTO: WELCOME Search fortopics you/reinterestedin Shortintroduction 回 T Woles Andeboly FILTER Screen HOME ACTMITES HISTORY MYPROTRE ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" Hke Guizodo Collectthefirstlayer ofinformationfrom HOMEScreen HONE ACTMTES ISTORY MYPRORLE the user Present the activities andpositions the user PROFILEScreen Consult andeditthe user's information12 Item feedback Item booking Item rating Figure 7 User-App-RS interaction." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' User’s various possible actions and respective interactions between the App and the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3 Recommenders and stages in RS The recommender system module mentioned in the previous section is composed by three components: user profile manager, preference manager and recommender pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The two former ones have already been covered, and in this Section, the latter will be explained in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The recommender pool is composed by four recommenders of different types: content-based, popularity-based, demographic-based and collaborative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These four recommenders are modeled with specific algorithms or employ specific techniques and they come into play in different phases of maturity of the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" These phases of maturity concern amount of data, that is, number of users Yo RecSys User App 1 User's first log in 2 Asks preferences 3 Inputspreferences 4 Sendspreferences 5Returnsrecommendations 6Views recommendations 7 Gives feedback and/or makesabooking 8Sendsfeedback 9 Returns updated recommendations 10Ratesbooked item 11Sends itemrating (First rating in system) 12Hybrid Recommender initiated 13 Returns updated recommendations13 and rating density." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Only after certain pre-specified values of users and rating density have been reached are some of these methods activated, or in other words, are some of the phases reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following, the different phases and algorithms used are explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Phase 1 At the beginning, the RS is void of any ratings or users, and only items exist in the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' When a new user logs in for the first time, in order for the RS to make any meaningful recommendation, some information has to be provided in the form of user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This is, at this stage, the only way to overcome cold-start issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' he user’s preferences, which are associated to the predetermined ontology are given and used to give content-based recommendations to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The user then will provide explicit and implicit feedback, in the form of booking items, bookmarking items or explicitly indicating they don’t li e the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This feedback is then received by the RS who then uses the said feedbac to update the user’s preference vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This update originates new recommendations to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Preference vectors At the core of phase 1 are the user preference vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These preference vectors are ontology related and they are used to make content-based recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' There are three preference vectors per user: High-level preferences Low-level preferences Specific preferences The high-level preferences are the ones the user identifies in the beginning and are associated with the ontological super-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These classes are the most abstract classes and lower in number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' They are the first layer of ontological classes and are the ones that don’t have a parent class and only child classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Observing Figure 4, the Sports ontological class is an example of a high-level preference since there is no ontology class above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The low-level preferences are associated to the ontological classes that link directly to the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These ontological classes are more specific, less abstract and in larger number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Observing Figure 4 and Figure 5, Golf is an example of a low-level preference, because two items link to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Finally, the specific preferences relate directly to the items, and is a vector that results from the other two higher-level preference vectors and the user’s feedbac on the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The way these vectors interact is explained in the following: 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The user identifies the high-level preferences when he logs in for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These preferences are propagated by way of vector multiplication with the low-level ontological preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The low-level preferences are then propagated to the item level by way of vector multiplication as well, originating the specific preference vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The items are ranked, and a subset of the highest ranked items are recommended to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The user gives feedback on the recommendations by either bookmarking items, booking items or dismissing items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The feedback is propagated upwards to the higher-level preference vectors with different intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The low-level preference vector is strongly affected, while the high-level preference vector is less affected because it is higher upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This sort of “trickle-up” propagation of user feedback alters both high-level and low-level preference vectors with different magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' New item recommendations are calculated, this time using both the high-level and low- level preference vectors to predict whether an item should be recommended or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The prediction by each vector is weighed and aggregated originating an ensemble prediction using both high and low preference vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The items are ranked, and a subset of the highest ranked items are recommended to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Repeat step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2 Ontological content-based recommender The content-based recommender is essentially vector multiplication between preference vectors and content vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Content vectors are binary vectors which map one preference level to the items content or to another preference vector content, while preference vectors show the intensity levels of preference for each ontological category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In step 4, the high and low preference vectors multiply with their corresponding item content vector originating a content-based prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Both predictions are weighed and aggregated, and a subset of the highest ran ed items is recommended to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' After the user’s feedbac both preference vectors are updated according to the “tric le-up” propagation concept introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Then, new recommendations are calculated with the new preference vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2 Phase 2 If the user booked and used an item, he can then rate said item, which will kickstart the hybrid recommender composed by the initial content-based recommender and the new popularity-based appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This popularity-based recommender uses a so-called damped mean on every item so that little cardinality of ratings doesn’t give an exaggerated edge of an item over another, such as an item with a single 5-star rating having a 5-star average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 15 𝐷𝑎𝑚𝑝𝑒𝑑 𝑀𝑒𝑎𝑛𝑗 = ∑ 𝑟𝑗𝑖 + 𝑘 ∙ 𝑟̿𝐺 𝑛 𝑖=1 𝑛 + 𝑘 Where 𝑟𝑗𝑖 is item j’s rating i, 𝑘 is the damping coefficient, 𝑟̿𝐺 is the global mean rating or some other default value, and 𝑛 is the number of reviews of item j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Hybrid recommender (content-based + popularity-based) The start of the hybrid recommender marks the start of phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' At this point in the RS, there aren’t many users and there aren’t many ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' he lac in both mean that popularity-based, demographic-based or collaborative approaches are still of little use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' As more users join and more ratings are given, other recommenders can become increasingly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' As we reach a given threshold of user and rating numbers we can initiate the demographic-based recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The way in which the hybrid recommender uses both recommenders is by cascading ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' That is, the popularity recommender pre-filters the items according to a rating threshold and then the content-based recommender recommends items that were not eliminated by the popularity recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3 Phase 3 As more users are added to the RS, and as these users give feedback on recommended items, other types of recommenders can enter the recommender pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' A first set of threshold values for number of users and rating density is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' When these thresholds are reached, phase 3 is initiated with yet another recommender being added: the demographic-based recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Demographic-based recommender The demographic-based recommender is composed by two ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' One clustering algorithm and one classification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The clustering algorithm has the purpose of identifying clusters of similar users based on their demographic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' he user’s demographic features can be age, region/country, group composition, budget, academic degree, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These features can be a mix of numerical, ordinal and nominal features and so a clustering algorithm that can handle different data types is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' After the clustering has been performed, and the users are all organized in clusters, a classification algorithm is used to predict whether a user will enjoy each item based on the item feedback of other users in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' For clustering, the algorithm employed was K-Prototypes, which works similarly to K-Means but can deal with mixed data types, particularly ordinal and nominal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' To define the clustering model, a knee region identifier is employed to automatically identify the optimal (or close to 16 optimal) number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The clustering model is retrained from time to time when sufficient new users have been added since the last model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' For classification a k-Nearest Neighbor algorithm, or kNN, was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Here, the users from the same cluster are used to predict whether a given user will enjoy the items, based on those users’ feedbac .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' he uses a custom distance metric that ta es into account both Jaccard and Manhattan distance metrics for the ordinal and nominal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The kNN than weighs the opinion of the other users inversely proportional to their distance to the user to whom the predictions are being made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The predictions given by this algorithm are weighed and added to the predictions made by the hybrid recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='4 Phase 4 In phase 4, collaborative filtering is added to the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' As it happens with phase 3, the entry into phase 4 takes place when thresholds of user cardinality and rating density are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Once this happens the collaborative filtering model is fitted and starts giving recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The algorithm used for collaborative filtering is a Field-Aware Factorization Machine (FFM), which has already been introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following sub-section, the FFM application is explained in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Collaborative filtering with Field-Aware Factorization Machines (FFM) To use FFMs, a specific Python library (xLearn) is used and the data also has to be transformed into a specific format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' A sample of a dataset in said format is shown in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 1 Dataset in the FFM format where each column represents a feature, except for column 0 which represents the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 9 0 0 0:1:1 1:2:1 2:3:1 3:4:1 4:5:1 5:6:1 6:7:1 7:8:1 8:9:1 1 1 0:10:1 1:2:1 2:11:1 3:4:1 4:5:1 5:6:1 6:12:1 7:13:1 8:14:1 2 0 0:15:1 1:16:1 2:3:1 3:4:1 4:17:1 5:6:1 6:18:1 7:19:1 8:20:1 3 1 0:15:1 1:2:1 2:21:1 3:22:1 4:17:1 5:6:1 6:23:1 7:8:1 8:24:1 4 1 0:10:1 1:16:1 2:3:1 3:4:1 4:17:1 5:25:1 6:23:1 7:26:1 8:27:1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 686422 1 0:1:1 1:2:1 2:3:1 3:4:1 4:17:1 5:25:1 6:23:1 7:8:1 8:37:1 686423 1 0:34:1 1:2:1 2:21:1 3:4:1 4:5:1 5:25:1 6:35:1 7:8:1 8:36:1 686424 1 0:10:1 1:16:1 2:3:1 3:4:1 4:17:1 5:25:1 6:18:1 7:8:1 8:24:1 686425 1 0:34:1 1:16:1 2:21:1 3:22:1 4:17:1 5:25:1 6:50:1 7:13:1 8:49:1 686426 1 0:15:1 1:2:1 2:3:1 3:4:1 4:17:1 5:6:1 6:23:1 7:8:1 8:44:1 17 This format is more complex than that for the Standard FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This is due to the more complex information that is ingested by the FFM which uses information about the fields to define the latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' That is, while in FMs each feature (field) has one latent vector, in FFMs this single representation is broken down into multiple latent vectors, one to represent each other field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 𝑦̂(𝑥) ∶= 𝜔0 + ∑ 𝜔𝑖𝑥𝑖 + ∑ ∑ 〈𝕧𝑖, 𝕧𝑗〉𝑥𝑖𝑥𝑗 𝑛 𝑗=𝑖+1 𝑛 𝑖=1 𝑛 𝑖=1 In the equation that represents the FM, which is shown above, the feature interactions represented by 〈𝕧𝑖, 𝕧𝑗〉 would correspond to the following in our case scenario (user demographic features): 𝑣𝑚𝑎𝑙𝑒 ∙ 𝑣𝑏𝑙𝑢𝑒𝑐𝑜𝑙𝑙𝑎𝑟 + 𝑣𝑚𝑎𝑙𝑒 ∙ 𝑣𝑙𝑜𝑤𝑏𝑢𝑑𝑔𝑒𝑡 + 𝑣𝑚𝑎𝑙𝑒 ∙ 𝑣𝑛𝑜𝑟𝑡ℎ𝑒𝑢𝑟𝑜𝑝𝑒 + ⋯ That is, the male latent vector that multiplies with each other latent vector is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The idea behind FFM is that the weight of the male latent vector might not be the same when multiplying with the job latent vectors as they are with the budget latent vectors, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Thus, in the FFM, the latent vectors are field-aware, which results in the following: 𝑣𝑚𝑎𝑙𝑒,𝑗𝑜𝑏 ∙ 𝑣𝑏𝑙𝑢𝑒𝑐𝑜𝑙𝑙𝑎𝑟,𝑔𝑒𝑛𝑑𝑒𝑟 + 𝑣𝑚𝑎𝑙𝑒,𝑏𝑢𝑑𝑔𝑒𝑡 ∙ 𝑣𝑙𝑜𝑤𝑏𝑢𝑑𝑔𝑒𝑡,𝑔𝑒𝑛𝑑𝑒𝑟 + 𝑣𝑚𝑎𝑙𝑒,𝑟𝑒𝑔𝑖𝑜𝑛 ∙ 𝑣𝑛𝑜𝑟𝑡ℎ𝑒𝑢𝑟𝑜𝑝𝑒,𝑔𝑒𝑛𝑑𝑒𝑟 + ⋯ Besides demographic features, as is shown in this example, the latent-vectors can also easily incorporate item features as well as contextual features and can thus integrate context-awareness in a deeper sense than simple contextual pre-filtering or post-filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The FFM model represents the last phase addition to the recommender pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The predictions attained from it are weighed and then aggregated with the predictions given by the other two, the hybrid and the demographic recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The weighs given to each recommender may be set to change over time so that it accompanies the maturity and complexity of each of the recommenders in the pool, thus giving progressively larger weight to the FFM as more users and more ratings are added to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 18 Figure 8 Diagram of the various RS phases and interactions between RS and Data Repository (DB) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='0 4 Phase1 11 Phase2 Phase3 5 Phase4 8&9 Newuserlogsn Feedback Seneretes reApp Gets user from UserDB receivedfromAp toApp 12 Hybridt Rec ir Updates ontology Step4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='9811repealedh AftermanyStep A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='9&11 AnerSlep4 589811 GeneratestreAs Serenerates recs Phase3beqins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' all threemodeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Shokds for FFM reiraining en Rec initiated Recalculate clusters Demog ec cotnue Recaculalecstrs Clusters defined Collab FFMiniliated RetrainFFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Sends ue lo ecomnender Ses ies to GraoDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' GraphDB Sennsertsilemontology SsnsertsnewiteminOntoloy19 4 Recommender system - Case study (CS) with synthetic data One of the main challenges in designing the recommender system proposed in this work was the lack of data to perform any type of experiment or even just to aid and inspire in the definition of the algorithms to employ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The lack of data was absolute, both on the side of the items as on the side of the users and preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The main issue is the non-existence of a dataset with user demographic features and user preferences, since such a dataset would allow to overcome some of the cold-start issues as well as give some idea of the data schema to be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' As a result, and since no public datasets were found that could overcome this hinderance, the decision was made to generate a synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The generated dataset was done so by using many different techniques from gaussian copulas to fuzzy logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Further information on that work will be available in another paper by the author Camacho, VT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following sub-section, the synthetic data employed in this or ’s case study is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Besides the synthetic data, a set of metrics was chosen to get an idea about the quality of the results from the recommenders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Traditional ML metrics are not always adequate for RS, mainly because, by principle, the objective of an RS is not to emulate exactly the choices of a given user since, if that were the case, there ouldn’t be a need for an RS in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the metrics sub-section, the set of used metrics is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The remainder of this section is applying the recommenders introduced in the previous section and testing them with different amounts of data which will attempt to emulate the data present at the different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Synthetic data In the work mentioned above, a methodology for the generation of synthetic datasets for recommender systems is presented, thus allowing to overcome the obstacle of not having quality data in sufficient amount (or even at all) readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The difficulties that are associated with this task are essentially the definition of a dataset with multiple datatypes, such as numerical (continuous), ordinal and nominal, and with different levels of correlation among the data, as well as the definition of user-ratings based on well-defined latent user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' To overcome this, a methodology was devised where several different techniques are employed in sequence to create the datasets concerning user characteristics, item properties, item categories and latent user preferences associated to user and item features, and as a result, a user-item sparse ratings matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The output of the methodology is: 1) Item dataset with item names and categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2) User dataset with user characteristics (demographic features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3) User-item sparse ratings matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 20 4) Latent preferences and Multinomial Logit model to compare with the outputs of the Recommender System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 Data Schema From the output presented above, we can see 4 DataFrames with different information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These DataFrames each have their own schema and have features from different data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following, the created DataFrames are introduced: Demographic Features Preferences Item Features User Ratings Going into more detail regarding the user demographic features DataFrame: Demographic Features: o User ID o Age o Gender o Job o Academic Degree o Budget o Country/Region o Group Composition o Accommodation Concerning the type of feature, they can be divided essentially into three groups: numerical, categorical ordinal and categorical nominal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Concerning numerical and categorical ordinal features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" we have the following: Numerical o Age – numerical (can be transformed into age bins) Ordinal: o Age bins = ['18-30'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content="'31-40'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" '41-50'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" '51-60'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" '60+'] o Academic Degree = ['None'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'High School'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Some College'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'College Degree'] o Budget = ['Low'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Mid'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'High'] o Accommodation = ['Single'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Double'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Suite'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Villa'] As for categorical nominal features," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" the following were modelled: Gender = ['Male'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Female'] Job = ['Blue Collar'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'White Collar'] 21 Country/Region = ['South Europe'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'North Europe'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'East Europe'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'North America'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'South America'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Asia'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Africa'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Middle East'] Group Composition = ['1 Adult'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" '2 Adults'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" '2 Adults + Child'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Group of Friends'] 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2 Samples of the generated DataFrames The resulting DataFrames (DF) can be used to train and test RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the case of the present work, they are used to simulate the different phases of data availability, thus testing the recommenders employed in each of the four phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following, samples of the generated DFs are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The first sample shown is the User DF in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This DF is composed by the user demographic features and UserID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The demographic features are ordinal (Age, AcDeg, Budget, Accom) and nominal (Gender, Job, Region, GroupComp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The entire set of users created has cardinality of 100,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 2 User DF composed by the demographic features of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' UserID Age AcDeg Budget Accom Gender Job Region GroupComp 0 4 2 1 2 Female blue collar North Europe 2Adlt 1 5 4 2 3 Male white collar North Europe GrpFriends 2 3 3 2 2 Female blue collar North Europe 2Adlt+Child 3 4 4 2 2 Female white collar North Europe 2Adlt+Child 4 3 3 2 3 Female white collar South Europe 2Adlt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 99995 4 4 2 2 Female white collar North Europe 2Adlt+Child 99996 3 4 3 2 Male white collar Asia 2Adlt+Child 99997 1 1 1 1 Female blue collar South Europe 2Adlt 99998 1 3 1 2 Female blue collar South Europe 2Adlt+Child 99999 4 3 2 2 Male blue collar North America 2Adlt+Child The second DF is the User-Preference DF which contains the latent preferences and is presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These latent preferences are related to the ontology classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The latent preferences of each user were modeled through a multinomial logit model based on their demographic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This DF shows the relative interest of a given user in a given preference category versus any other preference category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The values between different users are not comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 3 User-Preference DF containing the latent preferences from the Multinomial Logit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' UserID Beach Relax Shop Nightlife Theme park Gastro Sports Culture Nature Events 22 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='408 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='026 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='020 0 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='003 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='002 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='003 The third DF sample presented is the Item DF in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Here a set of 29 items were included belonging to different categories which are the user latent preferences presented in the previous table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 4 Item DF with corresponding item category (ontology and latent preferences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" itemID Item Name Category 0 A service that offers you the opportunity to do bungee-jumping ['Leisure'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Sports'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Routes'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Nature'] 1 A tavern that serves traditional food ['Leisure'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Culture'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Towns'] 2 Ancient history museum ['Culture'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'ViewPoints'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Nature'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Routes'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Towns'] 3 Discount for Callaway clubs ['Sports'] 4 Get a discount for Comic-Con ['Sports'] 5 Get a free pint at the pub ['Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Leisure'] 6 Get a free pizza at Pizza Hut ['Leisure'] 7 Get a voucher for Sephora ['Leisure'] 8 Go shopping in our new mall ['Leisure'] 9 Golf lessons ['Sports'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Leisure'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'] 23 10 Great meals that are tasty ['Leisure'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'] 11 Medieval fair ['Culture'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Nature'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Towns'] 12 One day snorkeling with the fish ['Sports'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Leisure'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Nature'] 13 One of the main nightclubs in the city ['Culture'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Nature'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Leisure'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Routes'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Towns'] 14 Rest and relaxation at the spa ['Leisure'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Routes'] 15 Surfing lessons ['Sports'] 16 Take a trip in a hot-air balloon ['Sports'] 17 Try go-karts with your friends ['Sports'] 18 Try scubadiving ['Sports'] 19 Try spearfishing with a pro ['Sports'] 20 Watch a FC Porto match ['Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Sports'] 21 Watch a SL Benfica match ['Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Sports'] 22 Watch a Sporting CP match ['Sports'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'] 23 Watch a live concert of Mastodon ['Events'] 24 Watch a live football match ['Sports'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Events'] 25 Watch a motogp race ['Events'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Sports'] 26 drive a F1 racecar ['Sports'] 27 go to the spa ['Leisure'] 28 visiting Disneyland ['Leisure'] The last data sample is the result of an external product between the user preferences from the multinomial logit model and the item DF." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The result is the input of a Fuzzy Inference System, which along with other implicit information on user and items returns the User-Item ratings DF, a sample of which is shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 5 User-Item ratings DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 0 1 2 3 4 5 … 23 24 25 26 27 28 userId 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='41 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='61 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2 Metrics The metrics for a RS are not a trivial issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Many works tend to use common ML metrics, such as classification metrics like precision, recall, accuracy, or regression metrics such as RMSE or MAE when the goal is to perform a regression on 1-5 ratings, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' However, these metrics imply that the data available to us about user behavior is perfect, that is, users are aware of all the items they li e and the ones they haven’t tried aren’t as relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' If this were the case, no RS would be needed in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The drawback of using this type of metrics is that it can encourage the recommender to make obvious recommendations in some cases, by penalizing wrong recommendations too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In addition, these metrics do nothing to the tune of comparing recommenders based on how personalized its recommendations are, or how diversified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Other metrics have been developed for RS in recent years that try to address these issues, some of which are presented in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Mean Average Precision @ K and Mean Average Recall @ K As in more traditional machine learning, the dataset is split into training and test sets, and the test set is comprised of cases the learner did not train on and thus it is used to measure the model’s ability to generali e ith ne data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In recommender systems, the same is done, and the output of a recommender system is usually a list of K recommendations for each user in the test set, and to produce those recommendations the recommender only trained on the items that user enjoyed in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K (Mean Average Precision @ K) gives insight to how relevant the list of recommended items are, whereas MAR@K (Mean Average Recall @ K) gives insight to how well the recommender system is able to discover all the items the user has rated positively in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In recommender systems, precision and recall are essentially the same as in machine learning: 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = # 𝑜𝑓 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑟𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛𝑠 # 𝑜𝑓 𝑖𝑡𝑒𝑚𝑠 𝑟𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑒𝑑 𝑅𝑒𝑐𝑎𝑙𝑙 = # 𝑜𝑓 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑟𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛𝑠 # 𝑜𝑓 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑖𝑡𝑒𝑚𝑠 o ever, these metrics don’t ta e ordering into account, and since the output of a recommender system is usually an ordered list, the metrics at cut-off k are introduced, MAP@K and MAR@K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 25 𝑀𝐴𝑃@𝐾 = 1 |𝑈| ∑ 1 min (𝑚, 𝐾) |𝑈| 𝑢=1 ∑ 𝑃𝑢(𝑘) ∙ 𝑟𝑒𝑙𝑢(𝑘) 𝐾 𝑘=1 𝑀𝐴𝑅@𝐾 = 1 |𝑈| ∑ 1 𝑚 |𝑈| 𝑢=1 ∑ 𝑟𝑢(𝑘) ∙ 𝑟𝑒𝑙𝑢(𝑘) 𝐾 𝑘=1 Where 𝑈 is the set of users in the test set, 𝑚 is the number of relevant items for user 𝑢, 𝑃𝑢(𝑘) and 𝑟𝑢(𝑘), are the precision@k and recall@k, respectively, and 𝑟𝑒𝑙𝑢(𝑘) is a factor equal to 1 if the 𝑘 th item is relevant, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Coverage Coverage is the percentage of items on the training data that the recommender is able to recommend on a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 𝐶𝑜𝑣𝑒𝑟𝑎𝑔𝑒 = 𝐼 𝑁 ∗ 100% Where 𝐼 is the number of unique items the model recommends in the test data and 𝑁 is the total number of unique items in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Personalization Personalization is the dissimilarity between users lists of recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' A high score indicates user lists are different between each other, while a low score indicates they are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Similarity between recommendation lists is calculated via the cosine similarity between said lists and then by calculating the average of the upper triangle of the cosine similarity matrix (avgCosim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The personalization is then given by: 𝑃𝑒𝑟𝑠𝑜𝑛𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 = 1 − 𝑎𝑣𝑔𝐶𝑜𝑠𝑖𝑚 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Diversity Diversity measures how different are the items being recommended to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 𝐷𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦 = 1 − 𝑖𝑙𝑠 26 Where 𝑖𝑙𝑠 corresponds to intra-list similarity, which is the average cosine similarity of all items in a list of recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This calculation uses features of the recommended items (such as item metadata) to calculate the similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The feature matrix is indexed by the item id and includes one-hot-encoded features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' If a recommender system is recommending lists of very similar items, the intra-list similarity will be high and conversely, the diversity will be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Novelty Finally, novelty measures the capacity of recommender systems to propose novel and unexpected items which a user is unlikely to know about already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It uses the self- information of the recommended item, and it calculates the mean self-information per top- N recommended list and averages them over all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 𝑁𝑜𝑣𝑒𝑙𝑡𝑦 = 1 |𝑈| ∑ ∑ 𝑙𝑜𝑔2 (𝑐𝑜𝑢𝑛𝑡(𝑖) |𝑈| ) |𝑁| |𝑁| 𝑖=1 |𝑈| 𝑢=1 Where 𝑈 is the user list, 𝑁 is the top n-list and 𝑐𝑜𝑢𝑛𝑡(𝑖) is the number of users that have consumed the specific item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3 CS with increasing data quantity In this sub-section the previously presented datasets and the previously presented metrics are employed to test and evaluate the RS in its various phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' For this to work, the datasets will be gradually incremented, starting with very few users and no ratings, and ending with the full datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This process is meant to mimic the natural evolution of a RS, from initial cold-start conditions to thousands of users with thousands of reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In each phase different recommenders are employed as was already mentioned in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='1 CS in Phase 1 As mentioned previously, phase 1 is characterized by little number of users and no ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' At this point, only content-based approaches are possible, and only if there is some input from the user concerning his preferences, which the RS asks when the user first logs in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Otherwise, the RS would be incapable of giving any recommendation short of a random context-filtered one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' To mimic this first stage, 98 initial users are added to the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Each user inputs their HL preference vector related to Table 3, which the phase 1 content-based recommender uses to generate recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Unlike in Table 3, the HL preference vector takes either 0 or 1 values and thus not conveying information on interest intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following tables, a sample of the 98 users and their respective HL vectors are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 27 Table 6 High-level preferences of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' userId ViewPoints Nature Towns Culture Events Leisure Routes Sports 1 0 0 0 0 0 1 0 0 2 1 0 1 0 0 1 1 0 3 0 0 0 0 0 1 0 0 4 0 0 0 0 0 1 1 1 5 0 0 1 0 0 0 0 0 … … … … … … … … … 94 0 0 0 0 0 1 0 0 95 0 0 0 0 0 1 0 0 96 1 0 1 0 0 1 1 0 97 0 0 1 0 0 0 0 0 98 1 0 1 1 0 0 1 0 The recommendations given by the RS for each user are in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' We can apply all previously presented metrics to these results, including MAP@K and MAR@K because we are aware of some ratings given by the users, present in the User-Item ratings DF which we can use for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 7 Sample of the recommendations given to the users by the content recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' userId Recommendations 1 [(6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a free pizza at Pizza Hut')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a voucher for Sephora')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Go shopping in our new mall')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'go to the spa')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'visiting Disneyland')] 2 [(6," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a free pizza at Pizza Hut')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a voucher for Sephora')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Go shopping in our new mall')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Rest and relaxation at the spa')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'go to the spa')] 3 [(6," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a free pizza at Pizza Hut')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a voucher for Sephora')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Go shopping in our new mall')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'go to the spa')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'visiting Disneyland')] 4 [(4," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a discount for Comic-Con')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a free pizza at Pizza Hut')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a voucher for Sephora')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Go shopping in our new mall')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Rest and relaxation at the spa')] 5 [(11," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Medieval fair')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'A tavern that serves traditional food')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (13,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a voucher for Sephora')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Go shopping in our new mall')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'go to the spa')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'visiting Disneyland')] 28 95 [(6," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a free pizza at Pizza Hut')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a voucher for Sephora')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Go shopping in our new mall')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'go to the spa')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'visiting Disneyland')] 96 [(6," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a free pizza at Pizza Hut')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Get a voucher for Sephora')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Go shopping in our new mall')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Rest and relaxation at the spa')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'go to the spa')] 97 [(11," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Medieval fair')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'A tavern that serves traditional food')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'One of the main nightclubs in the city')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Ancient history museum')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'A service that offers you the opportunity to do bungee-jumping')] 98 [(2," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Ancient history museum')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Medieval fair')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'One of the main nightclubs in the city')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'A tavern that serves traditional food')," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' (14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=" 'Rest and relaxation at the spa')] Table 8 Values for the various metrics on the content model recommendations." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 We can see that mean average precision and mean average recall have the same value, the value at K is equal to 5, since the recommender recommends 5 items to each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The two diversity values pertain to high level and low-level preferences showing how diverse are the recommendations in terms of recommending diverse items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It is expected for the high-level diversity to be lower than the low-level diversity since the content recommender makes recommendations based on high-level preferences of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Low-level preferences are linked ontologically to high-level preferences, but they are greater in variety, hence the same higl-level preference is linked to many low-level preferences, this justifies the larger value of Diversity LL compared to Diversity HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Coverage, personalization and both diversities return values from 0 to 1, where 1 represents maximum coverage, personalization and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The value for novelty can take any positive value, the greater the value the more unexpected recommendations are given based on popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this study, the metric for novelty may not be very useful due to the relatively low cardinality of items and the fact that there are no less popular items per se, at least not very noticeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In any case, these metrics are more useful in when used to compare different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='2 CS in Phase 2 In phase 2 there are ratings in the system, although not enough users to feed the demographic- based recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this phase we can simulate an RS state where there are 98 users and 64 ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The hybrid recommender is a hybridization of the initial content-based recommender with the new popularity-based recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The ratings are used to filter out items with average rating below a given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Once again, the same metrics are applied, and the results are shown in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 9 Values for the various metrics on the hybrid model recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='11e-16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 It is interesting to observe that the precision and recall have gone up, which makes sense because the items are now being filtered according to rating and higher rating items are more prone to having been liked by the users, at least the synthetic data was defined as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The coverage has gone down, which makes sense since less items are being recommended due to filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Personalization has gone down since it now many users are being recommended the same items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Diversity has gone up;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' this can be due to recommending some items outside of the natural preference of the user due to ratings filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' All in all, differences can be observed compared to the content-recommender, these differences make sense and seem to go towards an expected behavior by the recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3 CS in Phase 3 In phase 3, enough users with ratings given have been introduced in the system to kickstart the demographic-based recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This recommender works by defining user clusters based on demographic features and then giving item recommendations based on the predictions of a kNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This phase 3 recommender works together with the hybrid recommender from phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following table, the metrics are applied, and the results shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The number of users in this phase total 198, with 191 ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 10 Values for the various metrics on the hybrid and demographic model recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty Hybrid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Demog 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 30 We can see these results in a bar chart where a min max scaler has been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This basically shows which model wins in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Figure 9 Scaled metrics for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' We can see that the hybrid model loses to the demographic model in coverage and personalization and has higher values in the other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' However, we can see that results are virtually equal in terms of Diversity and Novelty, and only on the Precision and Recall do we see larger values for the hybrid model, which are not that much higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' On the other hand, the demographic recommender has much larger personalization and coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Here we can see an increment by the demographic model compared to the hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This makes sense because the demographic model is more complex in how recommendations are given by finding similar users in terms of demographic features and then recommending similar items to the user on a more individual basis, whereas the hybrid model is again based on high level preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 11 Values for the various metrics on the hybrid phase 2 and hybrid phase 3 model recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty Hybrid P2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='11e-16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Hybrid P3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty31 It is also interesting to compare the metrics between the hybrid in phase 2 and phase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' We can see that most metrics remain similar with a slight decrease in precision and recall, which may be just random, a slight increase in personalization, and a rather large increase in coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This can be due to more items recommended and not filtered out due to poor ratings because of the existence of more users and ratings on items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It is interesting to see a variation of the metrics of the same recommender as the amount of data increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='4 CS in Phase 4 Phase 4 starts when a given number of users and a given density of the user-item rating DF is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' When this happens, the final recommender is initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' This recommender is the already mentioned FFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In phase 4, the recommendations are, once again, the result of an ensemble of recommenders, the same one in phase 3 with the addition of the new FFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The resulting metrics are once more applied to the recommendations and are shown in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In this phase we have 250 users and 191 ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 12 Values for the various metrics on the hybrid, demographic and collaborative model recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty Hybrid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Demog 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Collab 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Comparing the recommenders, we can observe that the collaborative recommender, which was added in this later stage has high levels of personalization and coverage and achieves the highest values for precision and recall, compared to the other two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The values for diversity are all similar at this stage, and novelty again doesn’t provide useful information ith this number of total items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In terms of precision and recall, coverage and personalization, the collaborative recommender gives us expected results which is relatively high values in these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' We can observe that each recommender brings different recommendations to the table with clear improvements in some metrics as the recommender system matures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' It would be interesting to view this with a dataset comprising many more items and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In the following figure we can see the metrics in a scaled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 32 Figure 10 Scaled metrics for all three models As said, we observe that the collaborative metrics are good in comparison to the other two, however, the collaborative model is only useful when the recommender system has seen sufficient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The metrics for the other t o are not as high but they don’t suffer so much from cold-start issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' We can see that between the demographic and the hybrid models there is a trade-off in metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' We had already seen this in the previous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 13 Values for the various metrics on the phase 1, phase 2 and phase 3 model recommendations of hybrid and demographic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty Hybrid P2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='11e-16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Hybrid P3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Hybrid P4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Demog P3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Demog P4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty33 Here we can see a comparison between the metrics of the different models along each phase, we can see a slight decrease of precision and recall in the evolving phases for hybrid and demographic models, but this might have to do with insufficient ratings being added between phase 3 and phase 4, which are important for the demographic recommender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' With a further increase in data, we can see further differences in the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Feeding the recommender system with 1000 users and 883 ratings, we attain the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Table 14 Values for the various metrics on the hybrid, demographic and collaborative model recommendations, in the case of 250 users and 191 ratings as well as 1000 users and 883 ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty Hybrid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Demog 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content='66 Collab 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' MAP@K MAR@K Coverage Personalization Diversity HL Diversity LL Novelty34 We can see that the metrics are qualitatively similar to the case before with less users and ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Still the number of ratings is low, there is not a lot of rating density, which particularly penalizes the collaborative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Nonetheless, we can observe that the collaborative model is the one that offers more personalization, which increased for all models with the increment in users and ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Coverage also increased heavily for the demographic model while only increasing slightly for the collaborative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' As for precision and recall, the demographic model maintains the metric with only a slight decrease while the hybrid and collaborative model saw a rather significant decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' In regard to the collaborative model this might have to do with the low density in ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' All in all we see that the demographic and collaborative models clearly become more dominant and useful as more data is added to the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The phases also make sense, by having the collaborative model initiate after all others have been initiated, since the collaborative model is very sensitive to rating density, while the demographic model is more robust in that sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The hybrid model by this phase has clearly been passed by the two other models in most metrics which is exactly what would be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 5 Conclusion and future works In this work an ontology-based context aware recommender system application for tourism was presented where different recommenders are used at different stages of maturity of the recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The novel aspect is the evolution of the recommender system with different types of recommenders entering the recommendation pool as the system’s maturity evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The ontology extension of the recommender system allows items to be binned and recommended to users based on user preference vectors with different degrees of detail that link to the item ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These preference vectors will be ever changing based on user feedback, while other recommenders based on demographic features and field-aware factorization machines join the pool as data increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Along this work, the RS was presented and ultimately tested with synthetic data mimicking different stages of maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' One could observe that at each new phase the new recommenders added value as observed from the comparison between the different adopted metrics, which were MAP@K, MAR@K, Coverage, Personalization, Diversity HL, Diversity LL and finally Novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' These metrics are the state of the art for Recommender Systems because they attempt to go beyond the usual metrics adopted in , hich don’t al ays have much meaning in RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' The results obtained were expected where Collaborative and Demographic approaches essentially brought more personalization and coverage to the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' However, the full extent of differences between recommenders could not be captured mainly due to the relatively low cardinality of items being offered, only 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Future works would entail a broader analysis with more items, and also context-aware data which was not tested at this instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' Nonetheless, the context-aware would be essentially pre-filtering which would not be of much interest regarding the results concerning the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 35 6 Acknowledgements The present paper was developed in the context of the PMP project – Partnership Management Platform, code LISBOA-01-0247-FEDER-045411, co-financed by LISBOA 2020 and Portugal 2020 through the European Regional Development Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' 7 References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQf3fke/content/2301.00768v1.pdf'} +page_content=' ee, .' metadata={'source': 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This includes iridescence, +parallax occlusion and interior mapping, (specular, regular, +diffuse, total-internal) reflections with many bounces, re- +fraction, subsurface scattering, transparency, and possibly +more. This method divides textures into a matrix of radiance +buckets, where each bucket represent some data at various +incidence angles. Data can show final pixel color, or deferred +rendering ambient occlusion, reflections, shadow map, etc. +Resolution of the final synthesized output is the radiance +bucket matrix size. Technique can be implemented with a +simple fragment shader. The computational footprint of this +technique is of simple diffuse-only graphics, but with vi- +sual fidelity of complex (off-line) ray-traced render at the +cost of storage memory footprint. Balance between com- +putational footprint and storage memory footprint can be +easily achieved with variable compression ratio of repetitive +radiance scene textures. +CCS Concepts: • Computing methodologies → Reflectance +modeling; Rasterization; Texturing; Ray tracing. +Keywords: 3D graphics, holography, light field, plenoptic, +radiance field, rasterization, ray tracing, reflectance field +© 2023 Jakub Maksymilian Fober +This work is licensed under Creative Commons BY-NC-ND 3.0 license. +https://creativecommons.org/licenses/by-nc-nd/3.0/ +For all other uses including commercial, contact the owner/author(s). +1 +Introduction +Radiance and reflectance field rendering techniques are a +class of algorithms used in computer graphics to generate im- +ages of three-dimensional scenes. These algorithms simulate +the way light interacts with surfaces in a virtual environment, +producing realistic and detailed images. +These techniques have been the subject of extensive re- +search in computer graphics and rendering, as they offer a +powerful and flexible way to generate high-quality images. +There is a wide range of applications for radiance and re- +flectance field algorithms, including film and video game +production, architectural visualization, and scientific visual- +ization. +In this paper, technique is presented to capture and render +complex precomputed light interactions, via radiance field +textures, embedded onto three-dimensional-object’s surface. +The presented technique utilizes a standard fragment pixel +shader and a two-dimensional texture lookup to render dy- +namic, view-independent, photo-realistic images at a fraction +of the computational cost associated with effects such as real- +time ray tracing, parallax mapping, and dynamic shadowing. +It is well-suited for real-time execution in video games, +virtual reality, and virtual production environments on mod- +ern hardware. It can take advantage of the direct storage +capability in ninth-generation gaming systems, providing +high-fidelity, high-performance images. +This technique can replace computationally heavy rendering- +pipeline chains, while preserving hardware-accelerated, highly- +optimized rasterization elements. It can also enable wider +implementation of real-time GPU ray-tracing, with ability +to combine bounce rays with precomputed radiance of the +environment. +1.1 +Previous work +Mainstream implementations of radiance field rendering +focus on volumetric data structures and spherical harmonics +for rendering images[Yu et al. 2021]. While volumetric data +can be sparse in order to exclude void regions[Yu et al. 2021], +the ultimate goal would logically be to perfectly match the +geometry of the represented object. And since the inside +volume of the object is of no interest (most of the time), only +half of the radiance sphere is considered practically useful. +Therefore, such fields could effectively be spread across the +surface of the object. +Some researchers embraced this approach, with neural +reflectance fields as texturing primitives[Baatz et al. 2022], +which rendered high-fidelity results. But while neural fields +produce fantastic results, they are computationally inten- +sive at rendering time[Yu et al. 2021] and therefore are not +suitable for real-time applications. +1.2 +Overview of the content +In this initial version of paper you will find theoretical ex- +planation and implementation of the subject, along with +equations and schematics. Some elements had been tested, +like mapping functions, some yet to be presented, as the +follow-up updates continue. +1.3 +Document naming convention +This document uses the following naming convention: +• Left-handed coordinate system. +arXiv:2301.01719v1 [cs.GR] 4 Jan 2023 + +Fober, J.M. +• Vectors presented natively in column. +• Row-major order matrix arranged, denoted “𝑀row col”. +• Matrix multiplication by “[column]𝑎 · [row]𝑏 = 𝑀𝑎 𝑏”. +• A single bar enclosure “|𝑢|” represents scalar absolute. +• A single bar enclosure “|�𝑣|” represents vector’s length. +• Vectors with an arithmetic sign, or without, are calcu- +lated component-wise and form another vector. +• Centered dot “·” represents the vector dot product. +• Square brackets with a comma “[𝑓 ,𝑐]” denote interval. +• Square brackets with blanks “[𝑥 𝑦]” denote vectors +and matrices. +• The power of “−1” implies the reciprocal of the value. +• QED symbol “□” marks the final result or output. +This naming convention simplifies the process of transform- +ing formulas into shader code. +2 +Methodology +Each pixel of the model’s texture contains discrete radiance +hemispherical map of size 𝑛 ×𝑛, called “bucket”. Buckets are +arranged in place of initial texture’s pixels, increasing overall +resolution to 𝑤 ·𝑛 ×ℎ ·𝑛 pixels, where 𝑤 and ℎ denote width +and height of the synthesized output texture, respectively. +Buckets are highly repetitive and change only slightly from +one to another. This is a great case for a simple compression. +To synthesize output texture for a given view position, +single sample per bucket is taken, giving normal resolution +texture output. +Model’s 𝑢, 𝑣 texture coordinates correspond to bucket ma- +trix position index, while incidence vector, correspond to +bucket’s internal 𝑢, 𝑣 position. Therefore radiance texture +sampling algorithm can be described as a four-dimensional +plenoptic function 𝐿(𝑢, 𝑣,𝜃,𝜙), where 𝑢, 𝑣 denote model’s +texture coordinates and 𝜃,𝜙 incidence angles. +Figure 1. Radiance texture sampling model, where the inci- +dence R3 vector (blue) is projected and squarified (orange) +to R2 texture coordinates (red and green), which map onto +hemispherical radiance bucket represented as a flat square. +Each radiance bucket should represent a hemisphere of +reflectivity. Equisolid azimuthal projection was chosen for +this task, for its properties, as it preserves area and resem- +bles spherical mirror reflection[Wikipedia contributors 2022]. +Resolution of the radiance bucket, in such projection, directly +corresponds to sin(𝜃/2) +√ +2, where 𝜃 is the incidence angle. +To efficiently spread information across square buckets, ad- +ditional disc-to-square mapping function was implemented, +providing uniform pixel count across both orthogonal direc- +tions and diagonal directions. +Equisolid azimuthal projection mapping can be easily im- +plemented in the vector domain without the use of anti- +trigonometric functions, as the orthographically projected +normalized sum of the incidence and normal vectors has +a length of sin(𝜃/2). This eliminates 𝜃,𝜙 angles from the +plenoptic function, resulting in new 𝐿′(𝑢, 𝑣,𝑥,𝑦,𝑧), where +𝑥,𝑦,𝑧 correspond to incidence unit-vector components in +orthogonal texture space. +2.1 +Mapping of incident vector to radiance bucket +For every visible pixel there is an incidence vector ˆ𝐼 ∈ R3. +This vector can be mapped and projected to R2 texture coor- +dinates using translation and R2×3-matrix transformation. +Following equation maps incidence vector to azimuthal +equisolid projection, with 𝑟 = 1, at Ω = 180°. + +�𝐴𝑥 +�𝐴𝑦 +√ +2 cos 𝜃/2 + += +√ +2 +������ + +ˆ𝐼𝑥 + ˆ𝑁𝑥 +ˆ𝐼𝑦 + ˆ𝑁𝑦 +ˆ𝐼𝑧 + ˆ𝑁𝑧 + +������ +(1a) +� �𝐴𝑥 +�𝐴𝑦 +� += +√ +2 +���ˆ𝐼𝑥 +ˆ𝐼𝑦 +ˆ𝐼𝑧 + 1 +��� +�ˆ𝐼𝑥 +ˆ𝐼𝑦 +� +, if ˆ𝑁𝑧 = 1 +(1b) +Inverse mapping: + +ˆ𝐴′ +𝑥 +ˆ𝐴′ +𝑦 +ˆ𝐴′ +𝑧 + += + +�𝐴𝑥 +√︁ +1/2 +�𝐴𝑦 +√︁ +1/2 +√︃ +1 − +�𝐴2𝑥 +2 − +�𝐴2𝑦 +2 + +(2a) + +ˆ𝐼𝑥 +ˆ𝐼𝑦 +ˆ𝐼𝑧 + += 2 +��� +� +ˆ𝐴′ · ˆ𝑁 + +ˆ𝐴′ +𝑥 +ˆ𝐴′ +𝑦 +ˆ𝐴′ +𝑧 + +− + +ˆ𝑁𝑥 +ˆ𝑁𝑦 +ˆ𝑁𝑧 + +��� +� ++ + +ˆ𝑁𝑥 +ˆ𝑁𝑦 +ˆ𝑁𝑧 + +(2b) += + +�𝐴𝑥 +√︃ +2 − �𝐴2𝑥 − �𝐴2𝑦 +�𝐴𝑦 +√︃ +2 − �𝐴2𝑥 − �𝐴2𝑦 +1 − �𝐴2 +𝑥 − �𝐴2 +𝑦 + +, if ˆ𝑁𝑧 = 1 +(2c) +where �𝐴 ∈ [−1, 1]2 is the azimuthal equisolid projection +coordinate. 𝜃 is the incidence angle. ˆ𝑁 ∈ R3 is the surface +normal vector. As the incidence ˆ𝐼 ∈ R3 is mapped to or from +orthogonal texture space, where ˆ𝑁𝑧 = 1, the transformation +can take form of equation 1b and 2c. + +Radiance Textures for Rasterizing Ray-Traced Data +Following equation transforms azimuthal projection vec- +tor, into square coordinates, for the radiance bucket sam- +pling.1 +� �𝐵𝑥 +�𝐵𝑦 +� += +�� � �𝐴𝑥 +�𝐴𝑦 +� �� +max �| �𝐴𝑥 |, | �𝐴𝑦|� +� �𝐴𝑥 +�𝐴𝑦 +� +if �𝐴𝑥 and �𝐴𝑦 ≠ 0 +(3) +where �𝐵 ∈ [−1, 1]2 is the bucket’s centered texture coordi- +nate and �𝐴 ∈ [−1, 1]2 is the azimuthal projection vector. +Note. It is important to prevent pixel blending between +edges of neighboring buckets. This can be done by clamping +bucket coordinates to �𝐵 ∈ [𝐵−1 +res − 1, 1 − 𝐵−1 +res]2 range. +Inverse transformation of bucked, centered coordinates +�𝐵 ∈ R2 to azimuthal projection coordinates ˆ𝐴 ∈ R2 can be +achieved with same, but inverted method. +� �𝐴𝑥 +�𝐴𝑦 +� += max �| �𝐵𝑥 |, | �𝐵𝑦|� +√︃ +�𝐵2𝑥 + �𝐵2𝑦 +� �𝐵𝑥 +�𝐵𝑦 +� +(4a) + +ˆ𝑅𝑥 +ˆ𝑅𝑦 +ˆ𝑅𝑧 + += + +− �𝐴𝑥 +√︃ +2 − �𝐴2𝑥 − �𝐴2𝑦 +− �𝐴𝑦 +√︃ +2 − �𝐴2𝑥 − �𝐴2𝑦 +1 − �𝐴2 +𝑥 − �𝐴2 +𝑦 + +(4b) +where ˆ𝑅 ∈ R3 denotes equisolid reflection vector. This vector +is used to sample ray-traced data onto radiance field texture. +It is a version of the vector mirrored along the normal, found +in equation 2c on the preceding page. +3 +Results +TBA +4 +Conclusion +I have theorized about possible implementation of radiance +field texturing using modern hardware shading capabilities, +and presented mathematical solution for executing such con- +cept. +Note. More conclusion are to be added, after the update to +the paper. +5 +Possible applications +Radiance field texture sampling can replace shading pipeline +or supplement it with enhanced effects. Some such effects +include: +Parallax interior mapping. This effect is used to mimic +interior of a room, as seen through a window, or it can simu- +late a portal to another place. +Proxy meshes with parallax mapping. Radiance tex- +ture with alpha mask can simulate more complex or furry +objects bound inside a proxy mesh. Similarly to neural radi- +ance fields texturing primitives[Baatz et al. 2022]. +1See figure 2 for visual reference. +(a) Picture of one cent American coin. +(b) One cent coin mapped to a rectangle, using equation 3. +Figure 2. A visual example of disc to square mapping using +the formulation found in equation 3. +Reflections. Many light bounces can be combined into a +single pixel of the radiance texture map. Dynamic objects +can then sample such radiance field to obtain environment +reflections. Also semi real-time ray-tracing can accumulate +dynamically generated reflections into such texture map, to +update and enhance environment one. +Shadowing. 1-bit radiance field texture map can repre- +sent shadowing of static objects. Here, incidence vector is +replaced with light direction vector for shadow occlusion +sampling. It can work with both parallel light sources and +point lights. With more than one sample per bucket, area +shadows are possible to produce. +Subsurface scattering. This computationally demand- +ing effect can be encoded in a radiance texture map, which +then replaces incidence vector, with the light direction vector +in relation to the view position for sampling. +References +H. Baatz, J. Granskog, M. Papas, F. Rousselle, and J. Novák. 2022. NeRF- +Tex: Neural Reflectance Field Textures. Computer Graphics Forum 41, 6 +(March 2022), 287–301. https://doi.org/10.1111/cgf.14449 + +W +L.188R 0 +2021 +DW +L1888809 +2021Fober, J.M. +Wikipedia contributors. 2022. Fisheye lens: Mapping function. Wikipedia, +The Free Encyclopedia. +https://en.wikipedia.org/w/index.php?title= +Fisheye_lens&oldid=1124809304#Mapping_function [Online]. +Alex Yu, Sara Fridovich-Keil, Matthew Tancik, Qinhong Chen, Benjamin +Recht, and Angjoo Kanazawa. 2021. Plenoxels: Radiance Fields without +Neural Networks. arXiv (Dec. 2021). https://doi.org/10.48550/ARXIV. +2112.05131 +Received January 2023 + diff --git a/B9AzT4oBgHgl3EQfwP5n/content/tmp_files/load_file.txt b/B9AzT4oBgHgl3EQfwP5n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..acf17d7ea42c889d6be3b4d390606aad8171c3b0 --- /dev/null +++ b/B9AzT4oBgHgl3EQfwP5n/content/tmp_files/load_file.txt @@ -0,0 +1,156 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf,len=155 +page_content='Radiance Textures for Rasterizing Ray-Traced Data Jakub Maksymilian Fober talk@maxfober.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='space Abstract Presenting real-time rendering of 3D surfaces using radiance textures for fast synthesis of complex incidence-variable ef- fects and environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This includes iridescence, parallax occlusion and interior mapping, (specular, regular, diffuse, total-internal) reflections with many bounces, re- fraction, subsurface scattering, transparency, and possibly more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This method divides textures into a matrix of radiance buckets, where each bucket represent some data at various incidence angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Data can show final pixel color, or deferred rendering ambient occlusion, reflections, shadow map, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Resolution of the final synthesized output is the radiance bucket matrix size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Technique can be implemented with a simple fragment shader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' The computational footprint of this technique is of simple diffuse-only graphics, but with vi- sual fidelity of complex (off-line) ray-traced render at the cost of storage memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Balance between com- putational footprint and storage memory footprint can be easily achieved with variable compression ratio of repetitive radiance scene textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' CCS Concepts: • Computing methodologies → Reflectance modeling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Rasterization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Texturing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Ray tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Keywords: 3D graphics, holography, light field, plenoptic, radiance field, rasterization, ray tracing, reflectance field © 2023 Jakub Maksymilian Fober This work is licensed under Creative Commons BY-NC-ND 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='org/licenses/by-nc-nd/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='0/ For all other uses including commercial, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 1 Introduction Radiance and reflectance field rendering techniques are a class of algorithms used in computer graphics to generate im- ages of three-dimensional scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' These algorithms simulate the way light interacts with surfaces in a virtual environment, producing realistic and detailed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' These techniques have been the subject of extensive re- search in computer graphics and rendering, as they offer a powerful and flexible way to generate high-quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' There is a wide range of applications for radiance and re- flectance field algorithms, including film and video game production, architectural visualization, and scientific visual- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' In this paper, technique is presented to capture and render complex precomputed light interactions, via radiance field textures, embedded onto three-dimensional-object’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' The presented technique utilizes a standard fragment pixel shader and a two-dimensional texture lookup to render dy- namic, view-independent, photo-realistic images at a fraction of the computational cost associated with effects such as real- time ray tracing, parallax mapping, and dynamic shadowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' It is well-suited for real-time execution in video games, virtual reality, and virtual production environments on mod- ern hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' It can take advantage of the direct storage capability in ninth-generation gaming systems, providing high-fidelity, high-performance images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This technique can replace computationally heavy rendering- pipeline chains, while preserving hardware-accelerated, highly- optimized rasterization elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' It can also enable wider implementation of real-time GPU ray-tracing, with ability to combine bounce rays with precomputed radiance of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='1 Previous work Mainstream implementations of radiance field rendering focus on volumetric data structures and spherical harmonics for rendering images[Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' While volumetric data can be sparse in order to exclude void regions[Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2021], the ultimate goal would logically be to perfectly match the geometry of the represented object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' And since the inside volume of the object is of no interest (most of the time), only half of the radiance sphere is considered practically useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Therefore, such fields could effectively be spread across the surface of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Some researchers embraced this approach, with neural reflectance fields as texturing primitives[Baatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2022], which rendered high-fidelity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' But while neural fields produce fantastic results, they are computationally inten- sive at rendering time[Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2021] and therefore are not suitable for real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='2 Overview of the content In this initial version of paper you will find theoretical ex- planation and implementation of the subject, along with equations and schematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Some elements had been tested, like mapping functions, some yet to be presented, as the follow-up updates continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='3 Document naming convention This document uses the following naming convention: Left-handed coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='01719v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='GR] 4 Jan 2023 Fober, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Vectors presented natively in column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Row-major order matrix arranged, denoted “𝑀row col”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Matrix multiplication by “[column]𝑎 · [row]𝑏 = 𝑀𝑎 𝑏”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' A single bar enclosure “|𝑢|” represents scalar absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' A single bar enclosure “|�𝑣|” represents vector’s length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Vectors with an arithmetic sign, or without, are calcu- lated component-wise and form another vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Centered dot “·” represents the vector dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Square brackets with a comma “[𝑓 ,𝑐]” denote interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Square brackets with blanks “[𝑥 𝑦]” denote vectors and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' The power of “−1” implies the reciprocal of the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' QED symbol “□” marks the final result or output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This naming convention simplifies the process of transform- ing formulas into shader code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2 Methodology Each pixel of the model’s texture contains discrete radiance hemispherical map of size 𝑛 ×𝑛, called “bucket”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Buckets are arranged in place of initial texture’s pixels, increasing overall resolution to 𝑤 ·𝑛 ×ℎ ·𝑛 pixels, where 𝑤 and ℎ denote width and height of the synthesized output texture, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Buckets are highly repetitive and change only slightly from one to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This is a great case for a simple compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' To synthesize output texture for a given view position, single sample per bucket is taken, giving normal resolution texture output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Model’s 𝑢, 𝑣 texture coordinates correspond to bucket ma- trix position index, while incidence vector, correspond to bucket’s internal 𝑢, 𝑣 position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Therefore radiance texture sampling algorithm can be described as a four-dimensional plenoptic function 𝐿(𝑢, 𝑣,𝜃,𝜙), where 𝑢, 𝑣 denote model’s texture coordinates and 𝜃,𝜙 incidence angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Radiance texture sampling model, where the inci- dence R3 vector (blue) is projected and squarified (orange) to R2 texture coordinates (red and green), which map onto hemispherical radiance bucket represented as a flat square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Each radiance bucket should represent a hemisphere of reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Equisolid azimuthal projection was chosen for this task, for its properties, as it preserves area and resem- bles spherical mirror reflection[Wikipedia contributors 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Resolution of the radiance bucket, in such projection, directly corresponds to sin(𝜃/2) √ 2, where 𝜃 is the incidence angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' To efficiently spread information across square buckets, ad- ditional disc-to-square mapping function was implemented, providing uniform pixel count across both orthogonal direc- tions and diagonal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Equisolid azimuthal projection mapping can be easily im- plemented in the vector domain without the use of anti- trigonometric functions, as the orthographically projected normalized sum of the incidence and normal vectors has a length of sin(𝜃/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This eliminates 𝜃,𝜙 angles from the plenoptic function, resulting in new 𝐿′(𝑢, 𝑣,𝑥,𝑦,𝑧), where 𝑥,𝑦,𝑧 correspond to incidence unit-vector components in orthogonal texture space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='1 Mapping of incident vector to radiance bucket For every visible pixel there is an incidence vector ˆ𝐼 ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This vector can be mapped and projected to R2 texture coor- dinates using translation and R2×3-matrix transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Following equation maps incidence vector to azimuthal equisolid projection, with 𝑟 = 1, at Ω = 180°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �𝐴𝑥 �𝐴𝑦 √ 2 cos 𝜃/2 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = √ 2 ������ \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˆ𝐼𝑥 + ˆ𝑁𝑥 ˆ𝐼𝑦 + ˆ𝑁𝑦 ˆ𝐼𝑧 + ˆ𝑁𝑧 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ������ (1a) � �𝐴𝑥 �𝐴𝑦 � = √ 2 ���ˆ𝐼𝑥 ˆ𝐼𝑦 ˆ𝐼𝑧 + 1 ��� �ˆ𝐼𝑥 ˆ𝐼𝑦 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' if ˆ𝑁𝑧 = 1 (1b) Inverse mapping: \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˆ𝐴′ 𝑥 ˆ𝐴′ 𝑦 ˆ𝐴′ 𝑧 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �𝐴𝑥 √︁ 1/2 �𝐴𝑦 √︁ 1/2 √︃ 1 − �𝐴2𝑥 2 − �𝐴2𝑦 2 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (2a) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˆ𝐼𝑥 ˆ𝐼𝑦 ˆ𝐼𝑧 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = 2 ��� � ˆ𝐴′ · ˆ𝑁 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˆ𝐴′ 𝑥 ˆ𝐴′ 𝑦 ˆ𝐴′ 𝑧 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb − \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˆ𝑁𝑥 ˆ𝑁𝑦 ˆ𝑁𝑧 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ��� � + \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˆ𝑁𝑥 ˆ𝑁𝑦 ˆ𝑁𝑧 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (2b) = \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �𝐴𝑥 √︃ 2 − �𝐴2𝑥 − �𝐴2𝑦 �𝐴𝑦 √︃ 2 − �𝐴2𝑥 − �𝐴2𝑦 1 − �𝐴2 𝑥 − �𝐴2 𝑦 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' if ˆ𝑁𝑧 = 1 (2c) where �𝐴 ∈ [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 1]2 is the azimuthal equisolid projection coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 𝜃 is the incidence angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' ˆ𝑁 ∈ R3 is the surface normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' As the incidence ˆ𝐼 ∈ R3 is mapped to or from orthogonal texture space, where ˆ𝑁𝑧 = 1, the transformation can take form of equation 1b and 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Radiance Textures for Rasterizing Ray-Traced Data Following equation transforms azimuthal projection vec- tor, into square coordinates, for the radiance bucket sam- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='1 � �𝐵𝑥 �𝐵𝑦 � = �� � �𝐴𝑥 �𝐴𝑦 � �� max �| �𝐴𝑥 |, | �𝐴𝑦|� � �𝐴𝑥 �𝐴𝑦 � if �𝐴𝑥 and �𝐴𝑦 ≠ 0 (3) where �𝐵 ∈ [−1, 1]2 is the bucket’s centered texture coordi- nate and �𝐴 ∈ [−1, 1]2 is the azimuthal projection vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' It is important to prevent pixel blending between edges of neighboring buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This can be done by clamping bucket coordinates to �𝐵 ∈ [𝐵−1 res − 1, 1 − 𝐵−1 res]2 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Inverse transformation of bucked, centered coordinates �𝐵 ∈ R2 to azimuthal projection coordinates ˆ𝐴 ∈ R2 can be achieved with same, but inverted method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' � �𝐴𝑥 �𝐴𝑦 � = max �| �𝐵𝑥 |, | �𝐵𝑦|� √︃ �𝐵2𝑥 + �𝐵2𝑦 � �𝐵𝑥 �𝐵𝑦 � (4a) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˆ𝑅𝑥 ˆ𝑅𝑦 ˆ𝑅𝑧 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 − �𝐴𝑥 √︃ 2 − �𝐴2𝑥 − �𝐴2𝑦 − �𝐴𝑦 √︃ 2 − �𝐴2𝑥 − �𝐴2𝑦 1 − �𝐴2 𝑥 − �𝐴2 𝑦 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (4b) where ˆ𝑅 ∈ R3 denotes equisolid reflection vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This vector is used to sample ray-traced data onto radiance field texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' It is a version of the vector mirrored along the normal, found in equation 2c on the preceding page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 3 Results TBA 4 Conclusion I have theorized about possible implementation of radiance field texturing using modern hardware shading capabilities, and presented mathematical solution for executing such con- cept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' More conclusion are to be added, after the update to the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 5 Possible applications Radiance field texture sampling can replace shading pipeline or supplement it with enhanced effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Some such effects include: Parallax interior mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This effect is used to mimic interior of a room, as seen through a window, or it can simu- late a portal to another place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Proxy meshes with parallax mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Radiance tex- ture with alpha mask can simulate more complex or furry objects bound inside a proxy mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Similarly to neural radi- ance fields texturing primitives[Baatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 1See figure 2 for visual reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' (a) Picture of one cent American coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' (b) One cent coin mapped to a rectangle, using equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' A visual example of disc to square mapping using the formulation found in equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Many light bounces can be combined into a single pixel of the radiance texture map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Dynamic objects can then sample such radiance field to obtain environment reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Also semi real-time ray-tracing can accumulate dynamically generated reflections into such texture map, to update and enhance environment one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Shadowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 1-bit radiance field texture map can repre- sent shadowing of static objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Here, incidence vector is replaced with light direction vector for shadow occlusion sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' It can work with both parallel light sources and point lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' With more than one sample per bucket, area shadows are possible to produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Subsurface scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' This computationally demand- ing effect can be encoded in a radiance texture map, which then replaces incidence vector, with the light direction vector in relation to the view position for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' References H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Baatz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Granskog, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Papas, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Rousselle, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Novák.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' NeRF- Tex: Neural 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='org/w/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='title= Fisheye_lens&oldid=1124809304#Mapping_function [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Alex Yu, Sara Fridovich-Keil, Matthew Tancik, Qinhong Chen, Benjamin Recht, and Angjoo Kanazawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' Plenoxels: Radiance Fields without Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' arXiv (Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content=' 2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} +page_content='05131 Received January 2023' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfwP5n/content/2301.01719v1.pdf'} diff --git a/B9FJT4oBgHgl3EQfACzo/content/tmp_files/2301.11418v1.pdf.txt b/B9FJT4oBgHgl3EQfACzo/content/tmp_files/2301.11418v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..30299f9d81a28f8ba42db13b0ad6a1dd4adf45d1 --- /dev/null +++ b/B9FJT4oBgHgl3EQfACzo/content/tmp_files/2301.11418v1.pdf.txt @@ -0,0 +1,980 @@ +Parkinson gait modelling from an anomaly deep +representation +Edgar Rangela, Fabio Martineza,∗ +a Biomedical Imaging, Vision and Learning Laboratory (BIVL2ab), Universidad Industrial +de Santander, 680002, Bucaramanga, Colombia +Abstract +Parkinson’s Disease is associated with gait movement disorders, such as pos- +tural instability, stiffness, and tremors. Today, some approaches implemented +learning representations to quantify kinematic patterns during locomotion, sup- +porting clinical procedures such as diagnosis and treatment planning. These +approaches assumes a large amount of stratified and labeled data to optimize +discriminative representations. Nonetheless, these considerations may restrict +the operability of approaches in real scenarios during clinical practice. +This +work introduces a self-supervised generative representation, under the pretext +of video reconstruction and anomaly detection framework. This architecture +is trained following a one-class weakly supervised learning to avoid inter-class +variance and approach the multiple relationships that represent locomotion. For +validation 14 PD patients and 23 control subjects were recorded, and trained +with the control population only, achieving an AUC of 86.9%, homoscedasticity +level of 80% and shapeness level of 70% in the classification task considering its +generalization. +Keywords: +Anomaly detection, Deep Learning, Weakly Supervised, Parkinson +Disease +1. Introduction +Parkinson’s Disease (PD) is the second most common neurodegenerative dis- +order, affecting more than 6.2 million people worldwide [1, 2]. According to the +World Health Organization, this number will increase by more than 12 million by +2030 [3]. PD is characterized by the progressive loss of dopamine, a neurotrans- +mitter involved in the execution of voluntary movements. For this reason, the +main diagnostic support is based on the observation and analysis of progressive +motor disorders, such as tremor, rigidity, slowness of movement (bradykinesia), +∗Corresponding author +Email addresses: edgar.rangel@correo.uis.edu.co (Edgar Rangel), +famarcar@saber.uis.edu.co (Fabio Martinez) +URL: https://bivl2ab.uis.edu.co/ (Fabio Martinez) +Preprint submitted to Pattern Recognition +January 30, 2023 +arXiv:2301.11418v1 [cs.CV] 26 Jan 2023 + +postural instability, among many other related symptoms [4]. Despite of impor- +tant advances to determine the sources of the disease and multiple symptoms, +today, there is not a definitive and universal biomarker to characterize, diagnose, +and follow the patient progression of PD patients. +Particularly, the gait is a multi-factorial and complex locomotion process +that involves several subsystems. The associated kinematics patterns are typ- +ically recovered over standard marker-based setups, that coarsely approximate +complex motion behaviors, resulting in restrictive, intrusive and, altering natu- +ral postural gestures for PD description. Alternative, markerless video strate- +gies together with discriminative learning approximations have emerged as key +solutions to support the PD characterization and classification from other dis- +eases [5–9]. These methodologies have been successful in controlled studies but +strongly require a stratified, balanced, and well-labeled dataset to avoid over- +fitting. Besides, these approaches are biased to the physicians’ experience to +determine the disease and limiting the quantification to general scale indexes +[10]. Even worst, these approaches solve classification tasks but remains limited +on further explanation about data representation to define the generalization +capability w.r.t the new data. +This work introduces a deep generative and anomaly architecture to learn a +hidden descriptor to represent locomotion patterns. Following a weakly super- +vised methodology, a 3D net is self-trained under a gait video reconstruction pre- +text. Then, the resultant embedding representation encodes complex dynamic +gait relationships, captured from control population, that allows to discrimi- +nate parkinson patients. The main contributions of this work are summarized +as follows: +• A new digital biomarker coded as an embedding vector with the capability +to represent hidden kinematic relationships of Parkinson disease. +• A 3D Convolutional GAN net dedicated to learn spatio-temporal pat- +terns of gait video-sequences. This architecture integrates an auto-encoder +net to learn video patterns in reconstruction tasks and a complementary +decoder that discriminates between reconstructed and original video se- +quences. +• A statistical test framework to validate the capability of the approach in +terms of generalization, coverage of data and discrimination capability for +any class with different groups between them, i.e. evaluate the general- +ization of Parkinsonian patients, at different stages of the disease, with +respect to a control population. +2. Current Work +Deep discriminative learning is nowadays the standard methodology in much +of the computer vision challenges, demonstrating remarkable results in very dif- +ferent domains. For instance, the Parkinson characterization is achieved from +2 + +sensor-based and vision-based approaches, following a supervised scheme to cap- +ture main observed relationships and to generate a particular prediction about +the condition of the patients [5]. These approaches in general are dedicated +to classify and discriminate between a control population and patients with the +Parkinson condition. The sensor-based approaches capture kinematics from mo- +tion signals, approximating to PD classification, but in many of the cases results +marker-invasive, alter natural gestures, and only have recognition capabilities +in advanced stages of the disease [11]. Contrary, the vision-based approaches +exploit postural and dynamic features, from video recordings, but the represen- +tations underlies on supervised schemes that requires a large amount of labeled +data to learn the inter and intra variability among classes [6–9]. Also, these +learning methodologies require that training data have well-balanced conditions +among classes, i.e., to have the same proportion of sample observations for each +of the considered class [12]. +Unsupervised, semi-supervised and weakly supervised approaches have emerged +as a key alternative to model biomedical problems, with significative variabil- +ity among observations but limited training samples. +However, to the best +of our knowledge, these learning methods have been poorly explored and ex- +ploited in Parkinson characterization, with some preliminary alternatives that +use principles of Minimum Distance Classifiers and K-means Clustering [5, 13– +17]. In such sense, the PD modelling from non-supervised perspective may be +addressed from reconstruction, prediction and generative tasks [18], that help +to determine sample distributions and determine future postural and kinematic +events. In fact, the PD pattern distribution results key to understand multi- +factorial nature of PD, being determinant to define variations such as laterality +affectation of disease, abnormality sources, but also to define patient prognosis, +emulating the development of a particular patient during the gait. +3. Proposed approach +This work introduces a digital PD biomarker that embedded gait motor pat- +terns, from anomaly video reconstruction task. Contrary to typical classification +modeling, we are dedicated to deal with one class learning, i.e., only to learn +control gait patterns, approaching the high variability on training samples, with- +out using explicit disease labels. Hence, we hypothesize that a digital biomarker +of the disease can be modeled as a mixture of distributions, composed of samples +that were labeled as outliers, from learned representation. In consequence, we +analyze the embedding, reconstruction, and discrimination space to later define +rules to separate Parkinson from control vectors, during test validation. The +general pipeline of the proposed approach is illustrated in Figure 1. +3.1. A volumetric autoencoder to recover gait embedding patterns +Here, we are interested on capture complex dynamic interactions during lo- +comotion, observed in videos as spatio-temporal textural interactions. From a +self-supervised strategy (video-reconstruction task), we implemented a 3D deep +3 + +Figure 1: Pipeline of the proposed model separated in volumetric auto-encoder to recover gait +patterns (a), Digital gait biomarker (b), Auxiliary task to discriminate reconstructions (c), +and statistical validation of learned classes distributions (d) +autoencoder that projects videos into low-dimensional vectors, learning the com- +plex gait dynamics into a latent space (see the architecture in Figure 1-a). For +doing so, 3D convolutional blocks were implemented, structured hierarchically, +with the main purpose to carry out a spatio-temporal reduction while increasing +feature descriptions. Formally, a gait sequence x ∈ Nf×h×w×c, where f denotes +the number of temporal frames, (h × w) are the spatial dimensions, and c is the +number of color channels in the video. This sequence is received as input in the +convolutional block which is convolved with a kernel κ of dimensions (kt, kh, +kw), where kt convolves on the temporal axis and kh, kw on the spatial axes. +At each level l of processing, we obtain a new volume xl ∈ Zf/2l×h/2l×w/2l×2lc +that represents a bank of spatio-temporal feature maps. Each of these volumet- +ric features are dedicated to stand out relevant gait patterns in a zG reduced +projection, that summarizes a multiscale gait motion representation. +The resultant embedding vector zG encodes principal dynamic non-linear +correlations, which are necessary to achieve a video reconstruction x′. In this +study, the validated datasets are recorded from a relative static background, so, +the major dependencies to achieve an effective reconstruction lies in temporal +and dynamic information expressed during the gait. Here, we adopt zG as a +digital gait biomarker that, among others, allows to study motion abnormalities +associated to the Parkinson disease. +To complete end-to-end learning, 3D transposed convolutional blocks were +implemented as decoder, positioned in a symmetrical configuration regarding the +encoder levels, and upsampling spatio-temporal dimensions to recover original +video-sequence. Formally, having the embedded feature vector zG ∈ Zn with +n coded features, we obtain x′l ∈ Z2lf×2lh×2lw×c/2l volumes from transpose +4 + +Generator +Conv 3D +Conv 3D +Conv 3D +ZG +Decoder +Encoder +2'G +Encoder +a +(a) +(b) +Discriminator +Statistical Validation +Xtest +control +-test +control +control? +Conv 3D +Encoder +ZD +Dense +Xtest +parkinson +(c) +(d)convolutional blocks until obtaining a video reconstruction x′ ∈ Nf×h×w×c. The +quality of reconstruction is key to guarantee the deep representation learning +in the autoencoder part of generator. To do this, an L1 loss is implemented +between x and x′ and its named contextual loss: Lcon = ∥x − x′∥1. +3.2. Auxiliary task to discriminate reconstructions +From a generative learning, the capability of the deep representations to code +locomotion patterns may be expressed in the quality of video reconstructions +x′. Hence, we hypothesize that embedding descriptors zG that properly repro- +duce videos x′ should encode sufficient kinematic information of trained class, +allowing to discriminate among locomotion populations, i.e. between control +and Parkinson samples. +To measure this reconstruction capability, an auxiliary task is here intro- +duced to receive tuples with original and reconstructed videos (x, x′), and out- +put a discriminatory decision y = {y, y′}, regarding video source. +In such +case, y corresponds to the label for real videos, while y′ as labels for embed- +dings from reconstructed sequences. For doing so, we implement an adversarial +L2 loss, expressed as: Ladv = ∥zD − z′ +D∥2. In such case, for large differences +between (zD, z′ +D) it will be a significant error that will be propagated to the +generator. It should be noted that such minimization rule optimizes only the +generator. Then discriminator is only minimized following a classical equally +weighted cross-entropy rule, as: Ldisc = log(y)+log(1−y′) +2 +. +The auxiliary task to monitor video reconstruction is implemented from a +discriminatory convolutional net that follows the same structure that encoder +in Figure 1-a, which halves the spatio-temporal dimension while increases the +features and finally dense layer determines its realness level (see in Figure 1- +c.). Interestingly, from such deep convolutional representation the input videos +are projected to an embedding vector zD ∈ Zm with m coded features, which +thereafter may be used as latent vectors descriptors that also encode motion +and realness information. To guarantee an optimal coding into low-dimensional +embeddings, the reconstructed video x′ is mapped to an additional encoder +projecting representation basis in a z′G embedding. In such sense, zG and z′G +must be similar, and lead to x and x′ to be equal which helps in generalization +of the generator, following an encoder L2 loss: Lenc = ∥zG − z′ +G∥2. +3.3. A Digital gait biomarker from anomaly embeddings +The video samples are high-dimensional motor observations that can be +projected into a low-dimensional embedding space, through the proposed model. +Formally, each video sample is an independent and random variable x(i) +ℓ +from the +class (i) that follows a distribution x(i) +ℓ +∈ Ψ(i)[µ(x(i)), σ(x(i))] with mean µ(x(i)), +and standard deviation σ(x(i)). We then considered the proposed model as an +operator that transform each sample F(x(i) +ℓ ) into a low dimensional space, while +preserves the original distribution, as: F(x(i) +ℓ ) ∈ Ψ(i)[F(µ(x(i))), F(σ(x(i)))]. +From this assumption we can measure statistical properties over low-dimensional +space and explore properties as the generalization of the modeling. +5 + +Figure 2: Field of action of standard metrics of the model, where the dataset used only cover +the intersection area but the model performance for new samples is not being evaluated +Hence, we can adopt a new digital kinematic descriptor by considering em- +bedding vector differences between (zG, z′G). For instance, large difference be- +tween zG, z′G may suggest a new motion class, regarding the original distribu- +tion of training. From such approximation, we can model a scheme of one-class +learning (in this case, anomaly learning) over the video distributions from the +low-embedding differences observations. This scheme learns data distribution +without any label constraint. Furthermore, if we train the architecture only with +videos of a control population (c), we can define a discriminatory problem from +the reconstruction, by inducing: ∥zG − z′G∥2 ≤ τ → c ∧ ∥zG − z′G∥2 > τ → p, +where p is a label imposed to a video with a significant error reconstruction and +projected to a Parkinson population. +3.4. Statistical validation setup +This new discriminatory descriptor can be validated following standard met- +rics into binary projection ˆy = {c, p}. For a particular threshold τ we can re- +cover metrics such as the accuracy, precision and recall. Also, ROC-AUC (the +Area Under the Curve) can estimate a performance by iterating over different +τ values. However, these metrics say us about the capability of the proposed +approach to discriminate classes but not about data distribution among classes +[19, 20]. To robustly characterize a Parkinson digital biomarker is then demand- +ing to explore more robust statistical alternatives that evidence the generaliza- +tion of the embedded descriptor and estimate the performance for new samples +(Figure 2 illustrates typical limitations of standard classification metrics for un- +seen data being positioned on unknown places). In fact, we hypothesize that +Parkinson and control distributions, observed from an embedding representa- +tion, should remain with equal properties from training and test samples. To +address such assumption, in this work is explored two statistical properties to +validate the shape and variance of motor population distributions: +6 + +Ctest +Ctest +parkinson +Conv 3D +Encoder3.4.1. Variance analysis from Homoscedasticity +Here, a equality among variance of data distributions is estimated through +homoscedasticity operators. Particularly, this analysis is carried out for two +independent groups ⟨k⟩, ⟨u⟩ with cardinality |x(i) +⟨k⟩|, |x(j) +⟨u⟩| of classes (i), (j). Here, +it was considered two dispersion metrics regarding the Levene mean (∆⟨g⟩ +ℓ += +|x⟨g⟩ +ℓ +− µ(x⟨g⟩)|), and the Brown-Forsythe median (∆⟨g⟩ +ℓ += |x⟨g⟩ +ℓ +− med(x⟨g⟩)|). +From such dispersion distances, the test statistic W between x(i) +⟨k⟩ and x(j) +⟨u⟩ can +be defined as: +W = N − |P| +|P| − 1 +� +g∈P [|x⟨g⟩|(µ(∆⟨g⟩) − µ(∆))2] +� +g∈P [� +ℓ∈x⟨g⟩ (∆⟨g⟩ +ℓ +− µ(∆⟨g⟩))2] +(1) +where P = {x(i) +⟨k⟩, x(j) +⟨u⟩, · · · } is the union set of every data group from all +classes, |P| is the cardinality of P, N is the sum of all |x⟨g⟩| cardinalities, µ(∆⟨g⟩) +correspond to the mean ⟨g⟩ of ∆⟨g⟩ +ℓ +values and µ(∆) is the overall mean of every +∆⟨g⟩ +ℓ +value in P. This estimation evaluates if the samples between two different +groups are equally in variance for the same class, leading us to the first step in +model generalization for any new sample related to trained data. Additionally, +the homoscedasticity property is useful when is needed to check if two groups +remains in the same distribution range, because two distribution can have the +same shape (frequency) but be placed at different domain range, indicating a +weakness for the model in new data domains. +From a statistical test perspective, the value W rejects the null hypothesis +of homocedasticity when W > fα,|P|−1,N−|P| where fα,|P|−1,N−|P| is the upper +critical value of Fischer distribution with |P|−1 and N −|P| degrees of freedom +at a significance level of α (generally 5%). This metric allows to estimate the +clustering level for the model and determine if new data samples from another +domain are contained in data distributions of control or Parkinson patients. +Then, the homoscedasticity value of x(i) +⟨k⟩ against x(j) +⟨u⟩ is defined as follow: +H(x(i) +⟨k⟩, x(j) +⟨u⟩) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +W(µ(x(i) +⟨k⟩, x(j) +⟨u⟩)) + W(med(x(i) +⟨k⟩, x(j) +⟨u⟩)) +2 +i = j ∧ k ̸= u +0 +i = j ∧ k = u +2 − (W(µ(x(i) +⟨k⟩, x(j) +⟨u⟩)) + W(med(x(i) +⟨k⟩, x(j) +⟨u⟩))) +2 +i ̸= j +(2) +3.4.2. Shapeness analysis from ChiSquare +Here, we quantify the “shapenes” focused in having equally distributions. +Following the ChiSquare test χ2 between x(i) +⟨k⟩ and x(j) +⟨u⟩ as: +7 + +χ2 = +� +ℓ +(x⟨k⟩ +ℓ +− x⟨u⟩ +ℓ +)2 +x⟨u⟩ +ℓ +(3) +From this rule, it should be considered that both groups must have the +same cardinality (|x⟨k⟩| = |x⟨u⟩|) and the respective data sorting determines +the direction of comparison (i.e. the direction goes from group ⟨k⟩ to have the +same distribution of ⟨u⟩). To address these issues we make that the lower group +will be repeated in its elements without adding new unknown data to preserve +its mean and standard deviation, and secondly, we evaluate both directions to +quantify the similarity when χ2(x(i) +⟨k⟩ → x(j) +⟨u⟩) and χ2(x(j) +⟨u⟩ → x(i) +⟨k⟩). +The value χ2 reject the null hypothesis of equal distributions when χ2 > +χ2 +α,|x⟨g⟩|−1 where χ2 +α,|x⟨g⟩|−1 is the upper critical value of Chi Square distribution +with |x⟨g⟩| − 1 degrees of freedom at a significance level of α. We define the +shapeness value as: +Sh(x(i) +⟨k⟩, x(j) +⟨u⟩) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +χ2(x(i) +⟨k⟩ → x(j) +⟨u⟩) + χ2(x(j) +⟨u⟩ → x(i) +⟨k⟩) +2 +i = j ∧ k ̸= u +0 +i = j ∧ k = u +2 − (χ2(x(i) +⟨k⟩ → x(j) +⟨u⟩) + χ2(x(j) +⟨u⟩ → x(i) +⟨k⟩)) +2 +i ̸= j +(4) +This test can be used directly as indicator of how relatively far are the +samples from each other. +Hence, a higher value of this metric means that +the samples will be clearly different and separated, but there is the possibility +that control patients’ distribution is near to parkinson’s while parkinson can be +clearly far. Finally, in algorithm 1 is showed the steps to calculate the proposed +homoscedasticity and shapeness level for the model. +4. Experimental setup +4.1. Datasets +In this study were recruited 37 patients from control (23 subjects with av- +erage age of 64.7 ± 13 ) and parkinson (14 subjects with an average age of +72.8 ± 6.8) populations. The patients were invited to walk (without any mark- +ers protocol), developing a natural locomotion gesture. Parkinson participants +were evaluated by a physiotherapist (with more than five years of experience) +and stratified according to the H&Y scale (level 1.0 = 2, level 1.5 = 1, level +2.5 = 5, and level 3.0 = 6 participants). These patients written an informed +consent and the total dataset count with the approval of the Ethics Committee +of Universidad Industrial de Santander. +For recording, during a natural walking in around 3 meters, the locomotion +was registered 8 times from a sagittal view, following a semi-controlled condi- +tions (a green background). In this study we use a conventional optical camera +8 + +Algorithm 1 Calculation of homoscedasticity and shapeness metric for any +quantity of data groups with any classes +Require: C = {c0, c1, · · · , cn} +▷ Classes in dataset +Require: Gci = +� +x(i) +⟨0⟩, x(i) +⟨1⟩, · · · , x(i) +⟨mi⟩ +� +∀ci ∈ C +▷ Partitions per classes +h ← 0 +s ← 0 +for any pair (ci, cj) in C do +for any pair (x(i) +⟨k⟩, x(j) +⟨u⟩) in �(Gci, Gcj) do +h ← h + H(x(i) +⟨k⟩, x(j) +⟨u⟩) +▷ H defined in eq. 2 +s ← s + Sh(x(i) +⟨k⟩, x(j) +⟨u⟩) +▷ Sh defined in eq. 4 +end for +end for +N ← �n +i |Gci| +d ← +�N +2 +� +▷ Combinatory of N in groups of 2 +h ← h +d +▷ Homocedasticity level metric +s ← s +d +▷ Shapeness level metric +Nikon D3500, that output sequences at 60 fps with a spatial resolution of 1080p. +The camera was localized to cover the whole participant silhouette. Every se- +quence was spatially resized to 64×64 pixels, and temporally cropped to 64 +frames. Besides, the videos were normalized and a subsequent subsampling was +carried out to ensure a complete gait cycle. To follow one learning class, the +proposed approach was trained only with control subjects. In such case, the set +of control patients was split in common train, validation and test partitions of +11, 3 and 9 randomly patients selected, respectively. For parkinson participants, +we take for validation and test partitions of 3 and 11 patients randomly selected +to complement validation and test control sets. Hence, we balanced data for +standard and statistical validation purposes. +4.1.1. External dataset validation +A main interest in this work is to measure the capability to generalize motion +patterns from anomaly deep representations. Also, we are interested in mea- +suring the capability of embedding descriptors to discriminate PD from other +classes, even for videos captured with external protocols. Hence, in this work +we only evaluate the proposed approach with a public dataset of walking videos +that include knee-osteoarthritis (50 subjects with an average age of 56.7 ± 12.7), +parkinson (16 subjects with an average age of 68.6 ± 8.3) and control (30 sub- +jects with an average age of 43.7 ± 9.3) patients [21]. The 96 participants were +recorded with a static green background, blurred faces and markers on their +bodies. Following the same methodology for owner data, each sequence was +spatially resized to 64×64 pixels, and temporally cropped to 64 frames, and +finally normalized and subsampled ensuring a complete gait cycle. +9 + +4.2. Model configuration +The introduced strategy has in the generator an autoencoder and encoder +net, while the discriminator has an encoder net. The encoders use three layers +that include 3D (4×4×4 and stride 2×2×2) convolutions, BatchNormalization +(momentum of 0.1 and epsilon of 1 × 10−5) and LeakyRelu (α = 0.2). +At +each progressive level, the input is reduced to half in spatial and temporal +dimensions while the features are increased twice. The decoder network follows +a symmetrical configuration against the encoder with same layers as encoder +(replacing 3D convolutions by 3D transpose convolutions). The overall structure +is summarized in table 1. +Table 1: Generator and Discriminator Networks structure summary +Module +Network +Levels +Input +Output +Generator +Encoder +5 +64×64×64×1 +1×1×1×n +Decoder +5 +1×1×1×n +64×64×64×1 +Discriminator +Encoder +5 +64×64×64×1 +1×1×1×1 +5. Evaluations and Results +The proposed strategy was exhaustively validated with respect to the ca- +pability to recognize parkinsonian inputs as abnormal class patterns in archi- +tectures trained only with control patterns and under challenging unbalanced +and scarce scenarios. Hence, in the first experiment, the proposed strategy was +trained only with control samples from owner dataset, following a video recon- +struction pretext task. Hence, encoder (∥zG − z′ +G∥2), contextual (∥x − x′∥1) +and adversarial (∥zD − z′ +D∥2) embedding errors were recovered as locomotor +descriptors of the observed sequences. For classification purposes, these errors +were binarized by imposing a threshold value, as: τzG = 1.768 for encoder, +τx = 0.147 for contextual, and τzD = 0.429 for adversarial errors. Table 2 sum- +marizes the achieved performance of three locomotor descriptors according to +standard classification metrics. In general, the proposed strategy reports a re- +markable capability to label parkinson patterns as abnormal samples, which are +excluded from trained representation. Interestingly, the contextual errors have +the highest value among the others to classify between control and parkinson +patients, reporting a remarkable 86.9% in AUC, with mistakes in only 64 video +clips (approximately 3 patients). +For robustness validation, we are also interested in the distribution out- +put of predictions, which may suggest the capability of generalization of the +model. For doing so, we also validate locomotion descriptors with respect to +10 + +Table 2: Model performance for encoder, contextual and adversarial losses using standard +metrics when the model trains with control patients. Acc, Pre, Rec, Spe, F1 are for accuracy, +precision, recall, specificity and f1 score respectively. +Loss +Acc +Pre +Rec +Spe +F1 +ROC-AUC +Encoder +53.8% +89.5% +20.4% +96.9% +33.2% +58.7% +Contextual +85.7% +96.6% +77.4% +96.4% +85.7% +86.9% +Adversarial +75.5% +94.3% +60% +95.4% +73.3% +77.7% +introduced homoscedasticity and shapeness validation. Table 3 summarizes the +results achieved by each locomotion embedding descriptor, contrasting with the +reported results from standard metrics. In such case, the validated metrics sug- +gest that contextual errors may be overfitted for the trained dataset and the +recording conditions, which may be restrictive for generalized architecture in +other datasets. Contrary, the encoder descriptor shows evident statistical ro- +bustness from variance and shapeness distributions. Furthermore, the encoder +losses evidence a clearly separation between the control and parkinson distribu- +tion in Figure 3, where even the proposed model can separate stages of Hoehn +& Yahr with the difference between 2.5 and 3.0 levels where the ChiSquare test +shows us that both distributions remains equals meaning that both stages are +difficult to model. +Table 3: Model performance for encoder, contextual and adversarial losses using the proposed +statistical metrics when the model trains with control patients. +Loss +Homocedasticity +Shapeness +Encoder +80% +70% +Contextual +50% +40% +Adversarial +50% +45% +To follow with one of the main interests in this work i.e, the generaliza- +tion capability, the proposed strategy was validated with an external public +dataset (without any extra training) that include parkinson (16 patients), knee- +osteoarthritis (50 patients) and control patients (30 patients) [21]. Table 4 sum- +marized the achieved results to discriminate among the three unseen classes, +evidencing a notable performance following encoder embedding representation. +It should be noted, that Encoder achieves the highest ROC-AUC, reporting an +average of 75%, being the more robust representation, as suggested by statistical +11 + +Figure 3: Data distribution given by the proposed model for control and parkinson samples +by Hoehn & Yahr levels. +homoscedasticity and shapeness validation. The contextual and the adversarial +losses have better accuracy, precision and recall, but the specificity suggests +that there is not any evidence of correctly classifying control subjects. In such +sense, the model label all samples as abnormal from trained representation. +In contrast, the encoder element in the network (Figure 1-a) capture relevant +gait patterns to distinguish between control, parkinson and knee-osteoarthritis +patients. +Table 4: Model performance for encoder, contextual and adversarial losses using the proposed +model without retraining and same thresholds as Table 2. Acc, Pre, Rec, Spe, F1 are for +accuracy, precision, recall, specificity and f1 score respectively. +Loss +Acc +Pre +Rec +Spe +F1 +ROC-AUC +Encoder +62.6% +97.9% +58.1% +91.9% +72.9% +75% +Contextual +86.7% +86.7% +100% +0% +92.9% +50% +Adversarial +87.8% +89.4% +97.4% +24.9% +93.3% +61.2% +Along the same line, the external dataset was also validated with respect +to homoscedasticity and shapeness metrics. Table 5 summarizes the achieved +results from the distribution representation of output probabilities. As expected, +the results enforce the fact that embeddings from the Encoder have much better +generalization against the other losses, allowing to discriminate among three +different unseen classes. Remarkably, the results suggest that control subjects +of the external dataset belong to the trained control set. This fact is relevant +because indicates that architecture is principally dedicated to coded locomotor +patterns without strict restrictions about captured conditions. To complement +such results, output probabilities from three classes are summarized in violin +plots, as illustrated in Figure 4 which shows the separation between the classes +of parkinson and knee-osteoarthritis, also, between levels of the diseases, being +remarkable the locomotor affectations produced by the patients diagnosed with +knee-Osteoarthritis. +12 + +25 +20 +p< 0.05 +p< 0.05 +15 +Encoder Errors +p<0.05 +p< 0.05 +10 +p<0.05 +p> 0.05 +Y +5 +0 +0 +-5 +-10 +Control +Stage 1.0 +Stage 1.5 +Stage 2.5 +Stage 3.0Table 5: Model performance for encoder, contextual and adversarial losses using the proposed +statistical metrics and model as Table 2. +Loss +Homocedasticity +Shapeness +Encoder +66.7% +66.7% +Contextual +83.4% +0% +Adversarial +16.7% +16.7% +Figure 4: Data distribution given by the proposed model for control, parkinson (PD) and +knee-osteoarthritis (KOA) samples by levels where EL is early, MD medium and SV severe. +Alternatively, in an additional experiment we train using only patients di- +agnosed with parkinson to force the architecture to extract these abnormal +locomotion patterns. In such cases, the videos from control subjects are associ- +ated with abnormal responses from trained architecture. Table 6 summarizes the +achieved results from standard and statistical distribution metrics. As expected, +from this configuration of the architecture is achieved a lower classification per- +formance because the high variability and complexity to code the disease. In +fact, parkinson patients may manifest totally different locomotion affectations +at the same stage. For such reason, the architecture has major challenges to +discriminate control subjects and therefore lower agreement with ground truth +labels. The statistical homoscedasticity and shapeness metrics confirm such is- +sue achieving scores lower than 50% and indicating that the model, from such +configuration, is not generalizable. In this configuration, it would be demanding +a larger amount of parkinson patients to deal with disease variability. +6. Discussion +This work presented a deep generative scheme, designed under the one-class- +learning methodology to model gait locomotion patterns in markerless video +sequences. The proposed architecture is trained under the reconstruction video +pretext task, being categorical to capture kinematic behaviors without the asso- +13 + +15.0 +p< 0.05 +p> 0.05 +T +12.5 +p<0.05 +p<0.05 +p< 0.05 +p<0.05 +11 +10.0 +Encoder Errors +7.5 +5.0 +2.5 +0.0 +-2.5 +-5.0 +Control +EL PD +MD PD +SV PD +EL KOA +MD KOA +SV KOATable 6: Model performance for encoder, contextual and adversarial losses using standard +metrics when the model trains with parkinson patients. Acc, Pre, Rec, Spe, Homo and Shape +are for accuracy, precision, recall, specificity, homocedasticity and shapeness respectively. +Loss +Acc +Pre +Rec +Spe +Homo +Shape +ROC-AUC +Encoder +62.5% +55.2% +88.9% +40.9% +45% +50% +64.9% +Contextual +71.5% +93.5% +73.7% +50% +50% +40% +61.9% +Adversarial +68.8% +64.1% +69.4% +68.2% +45% +40% +68.8% +ciation of expert diagnosis criteria. From an exhaustive experimental setup, the +proposed approach was trained with videos recorded from a control population, +while then parkinsonian patterns were associated with anomaly patterns from +the design of a discrimination metric that operates from embedding represen- +tations. From an owner dataset, the proposed approach achieves an ROC-AUC +of 86.9%, while for an external dataset without unseen training videos, the +proposed approach achieved an average ROC-AUC of 75%. +One of the main issues addressed in this work was to make efforts to train +generative architecture with a sufficient generalization capability to capture +kinematic patterns without a bias associated to the capture setups. To carefully +select such architectures, this study introduced homoscedasticity and shapeness +as complementary statistical rules to validate the models. From these metrics +was evidenced that encoder embeddings brings major capabilities to general- +ize models, against the contextual and adversarial losses, achieving in average +an 80% and 70% for homoscedasticity and shapeness, respectively. Once these +metrics defined the best architecture and embedding representation, we confirm +the selection by using the external dataset with different capture conditions and +even with the study of a new disease class into the population i.e., the Knee- +osteoarthritis. Remarkably, the proposed approach generates embeddings with +sufficient capabilities to discriminate among different unseen populations. +In the literature have been declared different efforts to develop computational +strategies to discriminate parkinson from control patterns, following markerless +and sensor-based observations [6–9, 22]. For instance, volumetric architectures +have been adjusted from discriminatory rules taking minimization rules associ- +ated with expert diagnosis annotations [6, 8]. These approaches have reported +remarkable results (average an 95% ROC-AUC with 22 patients). Also, Sun +et. al. proposed an architecture that takes frontal gait views and together with +volumetric convolution layers, discriminates the level of freeze in the gait for +parkinson patients with an accuracy of 79.3%. Likewise, Kour et. al. [22] de- +velops a sensor-based approach to correlate postural relationships with several +annotated disease groups (reports an accuracy = 92.4%, precision = 90.0% with +14 + +50 knee-ostheoarthritis, 16 parkinson and 30 control patients). +Nonetheless, +such schemes are restricted to a specific recording scenario and pose observa- +tional configurations. Besides, the minimization of these representations may be +biased by label annotations associated with expert diagnostics. Contrary, the +proposed approach adjusts the representation using only control video sequences +without any expert label intervention during the architecture tunning. In such +case, the architecture has major flexibility to code potential hidden relation- +ships associated with locomotor patterns. In fact, the proposed approach was +validated with raw video sequences, reported in [22], surpassing precision scores +without any additional training to observe such videos. Moreover, the proposed +approach uses video sequences instead of representation from key points, that +coarsely minimize dynamic complexity during locomotion. +Recovered generalization metrics scores (homocedasticity = 80%, shapeness += 70% ) suggest that some patients have different statistical distributions, an +expected result from variability in control population, as well as, the variability +associated to disease parkinson phenotyping. In such sense, it is demanding +a large set of training data to capture additional locomotion components, to- +gether with a sufficient variability spectrum. Nonetheless, the re-training of the +architecture should be supervised from output population distributions to avoid +overfitting regarding specific training scenarios. The output reconstruction may +also be extended as anomaly maps to evidence in the spatial domain the regions +with anomalies, which further may represent some association with the disease +to help experts in the correct identification of patient prediction. +7. Conclusions +This work presented a deep generative architecture with the capability of dis- +covering anomaly locomotion patterns, convolving entire video sequences into a +3D scheme. Interestingly, a parkinson disease population was projected to the +architecture, returning not only outlier rejection but coding a new locomotion +distribution with separable patterns with respect to the trained control popu- +lation. These results evidenced a potential use of this learning and architecture +scheme to recover potential digital biomarkers, coded into embedding represen- +tations. The proposed approach was validated with standard classification rules +but also with statistical measures to validate the capability of generalization. +Future works include the validation of proposals among different stages and +the use of federated scenarios with different experimental capture setups to test +performance on real scenarios. +8. Acknowledgements +The authors thank Ministry of science, technology and innovation of Colom- +bia (MINCIENCIAS) for supporting this research work by the project “Mecan- +ismos computacionales de aprendizaje profundo para soportar tareas de local- +izaci´on, segmentaci´on y pron´ostico de lesiones asociadas con accidentes cere- +brovasculares isqu´emicos.”, with code 91934. +15 + +References +[1] T. Vos, A. A. Abajobir, K. H. Abate, C. Abbafati, K. M. Abbas, F. Abd- +Allah, R. S. 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Arora, A vision-based clinical analysis for classifica- +tion of knee osteoarthritis, parkinson’s disease and normal gait with severity +based on k-nearest neighbour, Expert Systems 39 (6) (2022) e12955. +17 + diff --git a/B9FJT4oBgHgl3EQfACzo/content/tmp_files/load_file.txt b/B9FJT4oBgHgl3EQfACzo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9d9b0727eba8c9338446ccd3538be5e10116fdd --- /dev/null +++ b/B9FJT4oBgHgl3EQfACzo/content/tmp_files/load_file.txt @@ -0,0 +1,521 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf,len=520 +page_content='Parkinson gait modelling from an anomaly deep representation Edgar Rangela, Fabio Martineza,∗ a Biomedical Imaging, Vision and Learning Laboratory (BIVL2ab), Universidad Industrial de Santander, 680002, Bucaramanga, Colombia Abstract Parkinson’s Disease is associated with gait movement disorders, such as pos- tural instability, stiffness, and tremors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Today, some approaches implemented learning representations to quantify kinematic patterns during locomotion, sup- porting clinical procedures such as diagnosis and treatment planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' These approaches assumes a large amount of stratified and labeled data to optimize discriminative representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Nonetheless, these considerations may restrict the operability of approaches in real scenarios during clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This work introduces a self-supervised generative representation, under the pretext of video reconstruction and anomaly detection framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This architecture is trained following a one-class weakly supervised learning to avoid inter-class variance and approach the multiple relationships that represent locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For validation 14 PD patients and 23 control subjects were recorded, and trained with the control population only, achieving an AUC of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9%, homoscedasticity level of 80% and shapeness level of 70% in the classification task considering its generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Keywords: Anomaly detection, Deep Learning, Weakly Supervised, Parkinson Disease 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Introduction Parkinson’s Disease (PD) is the second most common neurodegenerative dis- order, affecting more than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2 million people worldwide [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' According to the World Health Organization, this number will increase by more than 12 million by 2030 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' PD is characterized by the progressive loss of dopamine, a neurotrans- mitter involved in the execution of voluntary movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For this reason, the main diagnostic support is based on the observation and analysis of progressive motor disorders, such as tremor, rigidity, slowness of movement (bradykinesia), ∗Corresponding author Email addresses: edgar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='rangel@correo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='co (Edgar Rangel), famarcar@saber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='co (Fabio Martinez) URL: https://bivl2ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='co/ (Fabio Martinez) Preprint submitted to Pattern Recognition January 30, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='11418v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='CV] 26 Jan 2023 postural instability, among many other related symptoms [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Despite of impor- tant advances to determine the sources of the disease and multiple symptoms, today, there is not a definitive and universal biomarker to characterize, diagnose, and follow the patient progression of PD patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Particularly, the gait is a multi-factorial and complex locomotion process that involves several subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The associated kinematics patterns are typ- ically recovered over standard marker-based setups, that coarsely approximate complex motion behaviors, resulting in restrictive, intrusive and, altering natu- ral postural gestures for PD description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Alternative, markerless video strate- gies together with discriminative learning approximations have emerged as key solutions to support the PD characterization and classification from other dis- eases [5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' These methodologies have been successful in controlled studies but strongly require a stratified, balanced, and well-labeled dataset to avoid over- fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Besides, these approaches are biased to the physicians’ experience to determine the disease and limiting the quantification to general scale indexes [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Even worst, these approaches solve classification tasks but remains limited on further explanation about data representation to define the generalization capability w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='t the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This work introduces a deep generative and anomaly architecture to learn a hidden descriptor to represent locomotion patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Following a weakly super- vised methodology, a 3D net is self-trained under a gait video reconstruction pre- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Then, the resultant embedding representation encodes complex dynamic gait relationships, captured from control population, that allows to discrimi- nate parkinson patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The main contributions of this work are summarized as follows: A new digital biomarker coded as an embedding vector with the capability to represent hidden kinematic relationships of Parkinson disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' A 3D Convolutional GAN net dedicated to learn spatio-temporal pat- terns of gait video-sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This architecture integrates an auto-encoder net to learn video patterns in reconstruction tasks and a complementary decoder that discriminates between reconstructed and original video se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' A statistical test framework to validate the capability of the approach in terms of generalization, coverage of data and discrimination capability for any class with different groups between them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' evaluate the general- ization of Parkinsonian patients, at different stages of the disease, with respect to a control population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Current Work Deep discriminative learning is nowadays the standard methodology in much of the computer vision challenges, demonstrating remarkable results in very dif- ferent domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For instance, the Parkinson characterization is achieved from 2 sensor-based and vision-based approaches, following a supervised scheme to cap- ture main observed relationships and to generate a particular prediction about the condition of the patients [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' These approaches in general are dedicated to classify and discriminate between a control population and patients with the Parkinson condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The sensor-based approaches capture kinematics from mo- tion signals, approximating to PD classification, but in many of the cases results marker-invasive, alter natural gestures, and only have recognition capabilities in advanced stages of the disease [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Contrary, the vision-based approaches exploit postural and dynamic features, from video recordings, but the represen- tations underlies on supervised schemes that requires a large amount of labeled data to learn the inter and intra variability among classes [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Also, these learning methodologies require that training data have well-balanced conditions among classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=', to have the same proportion of sample observations for each of the considered class [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Unsupervised, semi-supervised and weakly supervised approaches have emerged as a key alternative to model biomedical problems, with significative variabil- ity among observations but limited training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' However, to the best of our knowledge, these learning methods have been poorly explored and ex- ploited in Parkinson characterization, with some preliminary alternatives that use principles of Minimum Distance Classifiers and K-means Clustering [5, 13– 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such sense, the PD modelling from non-supervised perspective may be addressed from reconstruction, prediction and generative tasks [18], that help to determine sample distributions and determine future postural and kinematic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In fact, the PD pattern distribution results key to understand multi- factorial nature of PD, being determinant to define variations such as laterality affectation of disease, abnormality sources, but also to define patient prognosis, emulating the development of a particular patient during the gait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Proposed approach This work introduces a digital PD biomarker that embedded gait motor pat- terns, from anomaly video reconstruction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Contrary to typical classification modeling, we are dedicated to deal with one class learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=', only to learn control gait patterns, approaching the high variability on training samples, with- out using explicit disease labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Hence, we hypothesize that a digital biomarker of the disease can be modeled as a mixture of distributions, composed of samples that were labeled as outliers, from learned representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In consequence, we analyze the embedding, reconstruction, and discrimination space to later define rules to separate Parkinson from control vectors, during test validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The general pipeline of the proposed approach is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' A volumetric autoencoder to recover gait embedding patterns Here, we are interested on capture complex dynamic interactions during lo- comotion, observed in videos as spatio-temporal textural interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' From a self-supervised strategy (video-reconstruction task),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' we implemented a 3D deep 3 Figure 1: Pipeline of the proposed model separated in volumetric auto-encoder to recover gait patterns (a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Digital gait biomarker (b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Auxiliary task to discriminate reconstructions (c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' and statistical validation of learned classes distributions (d) autoencoder that projects videos into low-dimensional vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' learning the com- plex gait dynamics into a latent space (see the architecture in Figure 1-a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For doing so, 3D convolutional blocks were implemented, structured hierarchically, with the main purpose to carry out a spatio-temporal reduction while increasing feature descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Formally, a gait sequence x ∈ Nf×h×w×c, where f denotes the number of temporal frames, (h × w) are the spatial dimensions, and c is the number of color channels in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This sequence is received as input in the convolutional block which is convolved with a kernel κ of dimensions (kt, kh, kw), where kt convolves on the temporal axis and kh, kw on the spatial axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' At each level l of processing, we obtain a new volume xl ∈ Zf/2l×h/2l×w/2l×2lc that represents a bank of spatio-temporal feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Each of these volumet- ric features are dedicated to stand out relevant gait patterns in a zG reduced projection, that summarizes a multiscale gait motion representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The resultant embedding vector zG encodes principal dynamic non-linear correlations, which are necessary to achieve a video reconstruction x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In this study, the validated datasets are recorded from a relative static background, so, the major dependencies to achieve an effective reconstruction lies in temporal and dynamic information expressed during the gait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Here, we adopt zG as a digital gait biomarker that, among others, allows to study motion abnormalities associated to the Parkinson disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To complete end-to-end learning, 3D transposed convolutional blocks were implemented as decoder, positioned in a symmetrical configuration regarding the encoder levels, and upsampling spatio-temporal dimensions to recover original video-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=" Formally, having the embedded feature vector zG ∈ Zn with n coded features, we obtain x′l ∈ Z2lf×2lh×2lw×c/2l volumes from transpose 4 Generator Conv 3D Conv 3D Conv 3D ZG Decoder Encoder 2'G Encoder a (a) (b) Discriminator Statistical Validation Xtest control test control control?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Conv 3D Encoder ZD Dense Xtest parkinson (c) (d)convolutional blocks until obtaining a video reconstruction x′ ∈ Nf×h×w×c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The quality of reconstruction is key to guarantee the deep representation learning in the autoencoder part of generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To do this, an L1 loss is implemented between x and x′ and its named contextual loss: Lcon = ∥x − x′∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Auxiliary task to discriminate reconstructions From a generative learning, the capability of the deep representations to code locomotion patterns may be expressed in the quality of video reconstructions x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Hence, we hypothesize that embedding descriptors zG that properly repro- duce videos x′ should encode sufficient kinematic information of trained class, allowing to discriminate among locomotion populations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' between control and Parkinson samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To measure this reconstruction capability, an auxiliary task is here intro- duced to receive tuples with original and reconstructed videos (x, x′), and out- put a discriminatory decision y = {y, y′}, regarding video source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such case, y corresponds to the label for real videos, while y′ as labels for embed- dings from reconstructed sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For doing so, we implement an adversarial L2 loss, expressed as: Ladv = ∥zD − z′ D∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such case, for large differences between (zD, z′ D) it will be a significant error that will be propagated to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' It should be noted that such minimization rule optimizes only the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Then discriminator is only minimized following a classical equally weighted cross-entropy rule, as: Ldisc = log(y)+log(1−y′) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The auxiliary task to monitor video reconstruction is implemented from a discriminatory convolutional net that follows the same structure that encoder in Figure 1-a, which halves the spatio-temporal dimension while increases the features and finally dense layer determines its realness level (see in Figure 1- c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Interestingly, from such deep convolutional representation the input videos are projected to an embedding vector zD ∈ Zm with m coded features, which thereafter may be used as latent vectors descriptors that also encode motion and realness information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To guarantee an optimal coding into low-dimensional embeddings, the reconstructed video x′ is mapped to an additional encoder projecting representation basis in a z′G embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such sense, zG and z′G must be similar, and lead to x and x′ to be equal which helps in generalization of the generator, following an encoder L2 loss: Lenc = ∥zG − z′ G∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' A Digital gait biomarker from anomaly embeddings The video samples are high-dimensional motor observations that can be projected into a low-dimensional embedding space, through the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Formally, each video sample is an independent and random variable x(i) ℓ from the class (i) that follows a distribution x(i) ℓ ∈ Ψ(i)[µ(x(i)), σ(x(i))] with mean µ(x(i)), and standard deviation σ(x(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' We then considered the proposed model as an operator that transform each sample F(x(i) ℓ ) into a low dimensional space, while preserves the original distribution, as: F(x(i) ℓ ) ∈ Ψ(i)[F(µ(x(i))), F(σ(x(i)))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' From this assumption we can measure statistical properties over low-dimensional space and explore properties as the generalization of the modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 5 Figure 2: Field of action of standard metrics of the model, where the dataset used only cover the intersection area but the model performance for new samples is not being evaluated Hence, we can adopt a new digital kinematic descriptor by considering em- bedding vector differences between (zG, z′G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For instance, large difference be- tween zG, z′G may suggest a new motion class, regarding the original distribu- tion of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' From such approximation, we can model a scheme of one-class learning (in this case, anomaly learning) over the video distributions from the low-embedding differences observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This scheme learns data distribution without any label constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Furthermore, if we train the architecture only with videos of a control population (c), we can define a discriminatory problem from the reconstruction, by inducing: ∥zG − z′G∥2 ≤ τ → c ∧ ∥zG − z′G∥2 > τ → p, where p is a label imposed to a video with a significant error reconstruction and projected to a Parkinson population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Statistical validation setup This new discriminatory descriptor can be validated following standard met- rics into binary projection ˆy = {c, p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For a particular threshold τ we can re- cover metrics such as the accuracy, precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Also, ROC-AUC (the Area Under the Curve) can estimate a performance by iterating over different τ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' However, these metrics say us about the capability of the proposed approach to discriminate classes but not about data distribution among classes [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To robustly characterize a Parkinson digital biomarker is then demand- ing to explore more robust statistical alternatives that evidence the generaliza- tion of the embedded descriptor and estimate the performance for new samples (Figure 2 illustrates typical limitations of standard classification metrics for un- seen data being positioned on unknown places).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In fact, we hypothesize that Parkinson and control distributions, observed from an embedding representa- tion, should remain with equal properties from training and test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To address such assumption, in this work is explored two statistical properties to validate the shape and variance of motor population distributions: 6 Ctest Ctest parkinson Conv 3D Encoder3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Variance analysis from Homoscedasticity Here, a equality among variance of data distributions is estimated through homoscedasticity operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Particularly, this analysis is carried out for two independent groups ⟨k⟩, ⟨u⟩ with cardinality |x(i) ⟨k⟩|, |x(j) ⟨u⟩| of classes (i), (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Here, it was considered two dispersion metrics regarding the Levene mean (∆⟨g⟩ ℓ = |x⟨g⟩ ℓ − µ(x⟨g⟩)|), and the Brown-Forsythe median (∆⟨g⟩ ℓ = |x⟨g⟩ ℓ − med(x⟨g⟩)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' From such dispersion distances, the test statistic W between x(i) ⟨k⟩ and x(j) ⟨u⟩ can be defined as: W = N − |P| |P| − 1 � g∈P [|x⟨g⟩|(µ(∆⟨g⟩) − µ(∆))2] � g∈P [� ℓ∈x⟨g⟩ (∆⟨g⟩ ℓ − µ(∆⟨g⟩))2] (1) where P = {x(i) ⟨k⟩, x(j) ⟨u⟩, · · · } is the union set of every data group from all classes, |P| is the cardinality of P, N is the sum of all |x⟨g⟩| cardinalities, µ(∆⟨g⟩) correspond to the mean ⟨g⟩ of ∆⟨g⟩ ℓ values and µ(∆) is the overall mean of every ∆⟨g⟩ ℓ value in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This estimation evaluates if the samples between two different groups are equally in variance for the same class, leading us to the first step in model generalization for any new sample related to trained data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Additionally, the homoscedasticity property is useful when is needed to check if two groups remains in the same distribution range, because two distribution can have the same shape (frequency) but be placed at different domain range, indicating a weakness for the model in new data domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' From a statistical test perspective, the value W rejects the null hypothesis of homocedasticity when W > fα,|P|−1,N−|P| where fα,|P|−1,N−|P| is the upper critical value of Fischer distribution with |P|−1 and N −|P| degrees of freedom at a significance level of α (generally 5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This metric allows to estimate the clustering level for the model and determine if new data samples from another domain are contained in data distributions of control or Parkinson patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Then, the homoscedasticity value of x(i) ⟨k⟩ against x(j) ⟨u⟩ is defined as follow: H(x(i) ⟨k⟩, x(j) ⟨u⟩) = � � � � � � � � � � � � � � � W(µ(x(i) ⟨k⟩, x(j) ⟨u⟩)) + W(med(x(i) ⟨k⟩, x(j) ⟨u⟩)) 2 i = j ∧ k ̸= u 0 i = j ∧ k = u 2 − (W(µ(x(i) ⟨k⟩, x(j) ⟨u⟩)) + W(med(x(i) ⟨k⟩, x(j) ⟨u⟩))) 2 i ̸= j (2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Shapeness analysis from ChiSquare Here, we quantify the “shapenes” focused in having equally distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Following the ChiSquare test χ2 between x(i) ⟨k⟩ and x(j) ⟨u⟩ as: 7 χ2 = � ℓ (x⟨k⟩ ℓ − x⟨u⟩ ℓ )2 x⟨u⟩ ℓ (3) From this rule, it should be considered that both groups must have the same cardinality (|x⟨k⟩| = |x⟨u⟩|) and the respective data sorting determines the direction of comparison (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' the direction goes from group ⟨k⟩ to have the same distribution of ⟨u⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To address these issues we make that the lower group will be repeated in its elements without adding new unknown data to preserve its mean and standard deviation, and secondly, we evaluate both directions to quantify the similarity when χ2(x(i) ⟨k⟩ → x(j) ⟨u⟩) and χ2(x(j) ⟨u⟩ → x(i) ⟨k⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The value χ2 reject the null hypothesis of equal distributions when χ2 > χ2 α,|x⟨g⟩|−1 where χ2 α,|x⟨g⟩|−1 is the upper critical value of Chi Square distribution with |x⟨g⟩| − 1 degrees of freedom at a significance level of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' We define the shapeness value as: Sh(x(i) ⟨k⟩, x(j) ⟨u⟩) = � � � � � � � � � � � � � � � χ2(x(i) ⟨k⟩ → x(j) ⟨u⟩) + χ2(x(j) ⟨u⟩ → x(i) ⟨k⟩) 2 i = j ∧ k ̸= u 0 i = j ∧ k = u 2 − (χ2(x(i) ⟨k⟩ → x(j) ⟨u⟩) + χ2(x(j) ⟨u⟩ → x(i) ⟨k⟩)) 2 i ̸= j (4) This test can be used directly as indicator of how relatively far are the samples from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Hence, a higher value of this metric means that the samples will be clearly different and separated, but there is the possibility that control patients’ distribution is near to parkinson’s while parkinson can be clearly far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Finally, in algorithm 1 is showed the steps to calculate the proposed homoscedasticity and shapeness level for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Experimental setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Datasets In this study were recruited 37 patients from control (23 subjects with av- erage age of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7 ± 13 ) and parkinson (14 subjects with an average age of 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='8 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='8) populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The patients were invited to walk (without any mark- ers protocol), developing a natural locomotion gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Parkinson participants were evaluated by a physiotherapist (with more than five years of experience) and stratified according to the H&Y scale (level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 = 2, level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 = 1, level 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 = 5, and level 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 = 6 participants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' These patients written an informed consent and the total dataset count with the approval of the Ethics Committee of Universidad Industrial de Santander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For recording, during a natural walking in around 3 meters, the locomotion was registered 8 times from a sagittal view, following a semi-controlled condi- tions (a green background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In this study we use a conventional optical camera 8 Algorithm 1 Calculation of homoscedasticity and shapeness metric for any quantity of data groups with any classes Require: C = {c0, c1, · · · , cn} ▷ Classes in dataset Require: Gci = � x(i) ⟨0⟩, x(i) ⟨1⟩, · · · , x(i) ⟨mi⟩ � ∀ci ∈ C ▷ Partitions per classes h ← 0 s ← 0 for any pair (ci, cj) in C do for any pair (x(i) ⟨k⟩, x(j) ⟨u⟩) in �(Gci, Gcj) do h ← h + H(x(i) ⟨k⟩, x(j) ⟨u⟩) ▷ H defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 2 s ← s + Sh(x(i) ⟨k⟩, x(j) ⟨u⟩) ▷ Sh defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 4 end for end for N ← �n i |Gci| d ← �N 2 � ▷ Combinatory of N in groups of 2 h ← h d ▷ Homocedasticity level metric s ← s d ▷ Shapeness level metric Nikon D3500, that output sequences at 60 fps with a spatial resolution of 1080p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The camera was localized to cover the whole participant silhouette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Every se- quence was spatially resized to 64×64 pixels, and temporally cropped to 64 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Besides, the videos were normalized and a subsequent subsampling was carried out to ensure a complete gait cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To follow one learning class, the proposed approach was trained only with control subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such case, the set of control patients was split in common train, validation and test partitions of 11, 3 and 9 randomly patients selected, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For parkinson participants, we take for validation and test partitions of 3 and 11 patients randomly selected to complement validation and test control sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Hence, we balanced data for standard and statistical validation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' External dataset validation A main interest in this work is to measure the capability to generalize motion patterns from anomaly deep representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Also, we are interested in mea- suring the capability of embedding descriptors to discriminate PD from other classes, even for videos captured with external protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Hence, in this work we only evaluate the proposed approach with a public dataset of walking videos that include knee-osteoarthritis (50 subjects with an average age of 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7), parkinson (16 subjects with an average age of 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='6 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='3) and control (30 sub- jects with an average age of 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='3) patients [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The 96 participants were recorded with a static green background, blurred faces and markers on their bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Following the same methodology for owner data, each sequence was spatially resized to 64×64 pixels, and temporally cropped to 64 frames, and finally normalized and subsampled ensuring a complete gait cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Model configuration The introduced strategy has in the generator an autoencoder and encoder net, while the discriminator has an encoder net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The encoders use three layers that include 3D (4×4×4 and stride 2×2×2) convolutions, BatchNormalization (momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='1 and epsilon of 1 × 10−5) and LeakyRelu (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' At each progressive level, the input is reduced to half in spatial and temporal dimensions while the features are increased twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The decoder network follows a symmetrical configuration against the encoder with same layers as encoder (replacing 3D convolutions by 3D transpose convolutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The overall structure is summarized in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Table 1: Generator and Discriminator Networks structure summary Module Network Levels Input Output Generator Encoder 5 64×64×64×1 1×1×1×n Decoder 5 1×1×1×n 64×64×64×1 Discriminator Encoder 5 64×64×64×1 1×1×1×1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Evaluations and Results The proposed strategy was exhaustively validated with respect to the ca- pability to recognize parkinsonian inputs as abnormal class patterns in archi- tectures trained only with control patterns and under challenging unbalanced and scarce scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Hence, in the first experiment, the proposed strategy was trained only with control samples from owner dataset, following a video recon- struction pretext task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Hence, encoder (∥zG − z′ G∥2), contextual (∥x − x′∥1) and adversarial (∥zD − z′ D∥2) embedding errors were recovered as locomotor descriptors of the observed sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For classification purposes, these errors were binarized by imposing a threshold value, as: τzG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='768 for encoder, τx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='147 for contextual, and τzD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='429 for adversarial errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Table 2 sum- marizes the achieved performance of three locomotor descriptors according to standard classification metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In general, the proposed strategy reports a re- markable capability to label parkinson patterns as abnormal samples, which are excluded from trained representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Interestingly, the contextual errors have the highest value among the others to classify between control and parkinson patients, reporting a remarkable 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% in AUC, with mistakes in only 64 video clips (approximately 3 patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For robustness validation, we are also interested in the distribution out- put of predictions, which may suggest the capability of generalization of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For doing so, we also validate locomotion descriptors with respect to 10 Table 2: Model performance for encoder, contextual and adversarial losses using standard metrics when the model trains with control patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Acc, Pre, Rec, Spe, F1 are for accuracy, precision, recall, specificity and f1 score respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Loss Acc Pre Rec Spe F1 ROC-AUC Encoder 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='8% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% Contextual 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='6% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% Adversarial 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='3% 60% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='3% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% introduced homoscedasticity and shapeness validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Table 3 summarizes the results achieved by each locomotion embedding descriptor, contrasting with the reported results from standard metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such case, the validated metrics sug- gest that contextual errors may be overfitted for the trained dataset and the recording conditions, which may be restrictive for generalized architecture in other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Contrary, the encoder descriptor shows evident statistical ro- bustness from variance and shapeness distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Furthermore, the encoder losses evidence a clearly separation between the control and parkinson distribu- tion in Figure 3, where even the proposed model can separate stages of Hoehn & Yahr with the difference between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 levels where the ChiSquare test shows us that both distributions remains equals meaning that both stages are difficult to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Table 3: Model performance for encoder, contextual and adversarial losses using the proposed statistical metrics when the model trains with control patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Loss Homocedasticity Shapeness Encoder 80% 70% Contextual 50% 40% Adversarial 50% 45% To follow with one of the main interests in this work i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='e, the generaliza- tion capability, the proposed strategy was validated with an external public dataset (without any extra training) that include parkinson (16 patients), knee- osteoarthritis (50 patients) and control patients (30 patients) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Table 4 sum- marized the achieved results to discriminate among the three unseen classes, evidencing a notable performance following encoder embedding representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' It should be noted, that Encoder achieves the highest ROC-AUC, reporting an average of 75%, being the more robust representation, as suggested by statistical 11 Figure 3: Data distribution given by the proposed model for control and parkinson samples by Hoehn & Yahr levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' homoscedasticity and shapeness validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The contextual and the adversarial losses have better accuracy, precision and recall, but the specificity suggests that there is not any evidence of correctly classifying control subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such sense, the model label all samples as abnormal from trained representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In contrast, the encoder element in the network (Figure 1-a) capture relevant gait patterns to distinguish between control, parkinson and knee-osteoarthritis patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Table 4: Model performance for encoder, contextual and adversarial losses using the proposed model without retraining and same thresholds as Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Acc, Pre, Rec, Spe, F1 are for accuracy, precision, recall, specificity and f1 score respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Loss Acc Pre Rec Spe F1 ROC-AUC Encoder 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='6% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='1% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% 75% Contextual 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% 100% 0% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% 50% Adversarial 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='8% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='3% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2% Along the same line, the external dataset was also validated with respect to homoscedasticity and shapeness metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Table 5 summarizes the achieved results from the distribution representation of output probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' As expected, the results enforce the fact that embeddings from the Encoder have much better generalization against the other losses, allowing to discriminate among three different unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Remarkably, the results suggest that control subjects of the external dataset belong to the trained control set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' This fact is relevant because indicates that architecture is principally dedicated to coded locomotor patterns without strict restrictions about captured conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To complement such results, output probabilities from three classes are summarized in violin plots, as illustrated in Figure 4 which shows the separation between the classes of parkinson and knee-osteoarthritis, also, between levels of the diseases, being remarkable the locomotor affectations produced by the patients diagnosed with knee-Osteoarthritis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 12 25 20 p< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 p< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 15 Encoder Errors p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 p< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 10 p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 p> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 Y 5 0 0 5 10 Control Stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 Stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 Stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 Stage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0Table 5: Model performance for encoder, contextual and adversarial losses using the proposed statistical metrics and model as Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Loss Homocedasticity Shapeness Encoder 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% Contextual 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4% 0% Adversarial 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% Figure 4: Data distribution given by the proposed model for control, parkinson (PD) and knee-osteoarthritis (KOA) samples by levels where EL is early, MD medium and SV severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Alternatively, in an additional experiment we train using only patients di- agnosed with parkinson to force the architecture to extract these abnormal locomotion patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such cases, the videos from control subjects are associ- ated with abnormal responses from trained architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Table 6 summarizes the achieved results from standard and statistical distribution metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' As expected, from this configuration of the architecture is achieved a lower classification per- formance because the high variability and complexity to code the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In fact, parkinson patients may manifest totally different locomotion affectations at the same stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For such reason, the architecture has major challenges to discriminate control subjects and therefore lower agreement with ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The statistical homoscedasticity and shapeness metrics confirm such is- sue achieving scores lower than 50% and indicating that the model, from such configuration, is not generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In this configuration, it would be demanding a larger amount of parkinson patients to deal with disease variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Discussion This work presented a deep generative scheme, designed under the one-class- learning methodology to model gait locomotion patterns in markerless video sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The proposed architecture is trained under the reconstruction video pretext task, being categorical to capture kinematic behaviors without the asso- 13 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 p< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 p> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 T 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 p< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='05 11 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 Encoder Errors 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0 Control EL PD MD PD SV PD EL KOA MD KOA SV KOATable 6: Model performance for encoder, contextual and adversarial losses using standard metrics when the model trains with parkinson patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Acc, Pre, Rec, Spe, Homo and Shape are for accuracy, precision, recall, specificity, homocedasticity and shapeness respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Loss Acc Pre Rec Spe Homo Shape ROC-AUC Encoder 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% 45% 50% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% Contextual 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='5% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='7% 50% 50% 40% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9% Adversarial 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='8% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='1% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='2% 45% 40% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='8% ciation of expert diagnosis criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' From an exhaustive experimental setup, the proposed approach was trained with videos recorded from a control population, while then parkinsonian patterns were associated with anomaly patterns from the design of a discrimination metric that operates from embedding represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' From an owner dataset, the proposed approach achieves an ROC-AUC of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='9%, while for an external dataset without unseen training videos, the proposed approach achieved an average ROC-AUC of 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' One of the main issues addressed in this work was to make efforts to train generative architecture with a sufficient generalization capability to capture kinematic patterns without a bias associated to the capture setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' To carefully select such architectures, this study introduced homoscedasticity and shapeness as complementary statistical rules to validate the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' From these metrics was evidenced that encoder embeddings brings major capabilities to general- ize models, against the contextual and adversarial losses, achieving in average an 80% and 70% for homoscedasticity and shapeness, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Once these metrics defined the best architecture and embedding representation, we confirm the selection by using the external dataset with different capture conditions and even with the study of a new disease class into the population i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=', the Knee- osteoarthritis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Remarkably, the proposed approach generates embeddings with sufficient capabilities to discriminate among different unseen populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In the literature have been declared different efforts to develop computational strategies to discriminate parkinson from control patterns, following markerless and sensor-based observations [6–9, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' For instance, volumetric architectures have been adjusted from discriminatory rules taking minimization rules associ- ated with expert diagnosis annotations [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' These approaches have reported remarkable results (average an 95% ROC-AUC with 22 patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Also, Sun et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' proposed an architecture that takes frontal gait views and together with volumetric convolution layers, discriminates the level of freeze in the gait for parkinson patients with an accuracy of 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Likewise, Kour et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' [22] de- velops a sensor-based approach to correlate postural relationships with several annotated disease groups (reports an accuracy = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='4%, precision = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content='0% with 14 50 knee-ostheoarthritis, 16 parkinson and 30 control patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Nonetheless, such schemes are restricted to a specific recording scenario and pose observa- tional configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Besides, the minimization of these representations may be biased by label annotations associated with expert diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Contrary, the proposed approach adjusts the representation using only control video sequences without any expert label intervention during the architecture tunning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such case, the architecture has major flexibility to code potential hidden relation- ships associated with locomotor patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In fact, the proposed approach was validated with raw video sequences, reported in [22], surpassing precision scores without any additional training to observe such videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Moreover, the proposed approach uses video sequences instead of representation from key points, that coarsely minimize dynamic complexity during locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Recovered generalization metrics scores (homocedasticity = 80%, shapeness = 70% ) suggest that some patients have different statistical distributions, an expected result from variability in control population, as well as, the variability associated to disease parkinson phenotyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' In such sense, it is demanding a large set of training data to capture additional locomotion components, to- gether with a sufficient variability spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Nonetheless, the re-training of the architecture should be supervised from output population distributions to avoid overfitting regarding specific training scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The output reconstruction may also be extended as anomaly maps to evidence in the spatial domain the regions with anomalies, which further may represent some association with the disease to help experts in the correct identification of patient prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Conclusions This work presented a deep generative architecture with the capability of dis- covering anomaly locomotion patterns, convolving entire video sequences into a 3D scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Interestingly, a parkinson disease population was projected to the architecture, returning not only outlier rejection but coding a new locomotion distribution with separable patterns with respect to the trained control popu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' These results evidenced a potential use of this learning and architecture scheme to recover potential digital biomarkers, coded into embedding represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' The proposed approach was validated with standard classification rules but also with statistical measures to validate the capability of generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Future works include the validation of proposals among different stages and the use of federated scenarios with different experimental capture setups to test performance on real scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FJT4oBgHgl3EQfACzo/content/2301.11418v1.pdf'} +page_content=' Acknowledgements The authors thank Ministry of science, technology and innovation of Colom- bia (MINCIENCIAS) for supporting this research work by the project “Mecan- ismos computacionales de aprendizaje profundo para soportar tareas de local- izaci´on, segmentaci´on y pron´ostico de lesiones asociadas con accidentes cere- brovasculares isqu´emicos.”, with code 91934.' metadata={'source': 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effects in the chiral antiferro- +magnetic antiperovskite Mn3NiN +E. Triana-Ramíreza,b, W. Ibarra-Hernandezc +and A. C. Garcia-Castroa,∗ +Magnetic antiperovskites, holding chiral noncollinear antiferromagnetic ordering, have shown remark- +able properties that cover from negative thermal expansion to anomalous Hall effect. Nevertheless, +details on the electronic structure related to the oxidation states and the octahedral center’s site +effect are still scarce. Here, we show a theoretical study, based on first-principles calculations in the +framework of the density-functional theory, DFT, on the electronic details associated with the nitro- +gen site effect into the structural, electronic, magnetic, and topological degrees of freedom. Thus, we +show that the nitrogen vacancy increases the values of the anomalous Hall conductivity and retains +the chiral Γ4g antiferromagnetic ordering. Moreover, we reveal, based on the Bader charges and +the electronic structure analysis, the negative and positive oxidation states in the Ni and Mn sites, +respectively. The latter is in agreement with the expected Aα+ +3 +Bβ−Xδ− oxidation states to satisfy +the charge neutrality in the antiperovskites, but rare for transition metals. Finally, we extrapolate +our findings on the oxidation states to several Mn3BN compounds showing that the antiperovskite +structure is an ideal platform to encounter negative oxidation states in metals sitting at the corner +B-site. +1 +Introduction: +In the modern quest for novel and exciting phenomena in new +materials, antiperovskite, or inversed-perovskite with the formu- +lae A3BX, has stood out as an astonishing type of materials show- +ing large anomalous Hall conductivity1–4, superconductivity5–8, +good performance for new batteries9,10, tunable hybrid-improper +ferroelectricity11, tangible magnetocaloric effects12, and large +spin-lattice coupling13–17. All the latter are just a few examples +that can be mentioned among the vast functionalities offered by +these materials. Interestingly, in antiperovskites18,19 the electro- +static balance and the oxidation site occupation are apparently +reversed with respect to the known perovskites, ABX3 20. Most +of the mentioned phenomena in the antiperovskites owe most of +their properties to the reversed occupation of their anionic and +cationic sites forming the reversed XA6 octahedra. This subtle +change in the coordination and atomic occupations gives rise to, +for example, the triangular geometric coordination of the mag- +netic sites that in turn, induces a strong magnetic frustration con- +verging into chiral noncollinear antiferromagnetic orderings21. +Another example is the topologically related properties in the +a School of Physics, Universidad Industrial de Santander, Carrera 27 Calle 09, 680002, +Bucaramanga, Colombia +b Centro de Investigación y Estudios Avanzados del IPN - CINVESTAV-Querétaro, MX- +76230, Querétaro, México. +c Facultad de Ingeniería, Benemérita Universidad Autónoma de Puebla, Apartado Postal +J-39, Puebla, Pue. 72570, México. +*E-mail: acgarcia@uis.edu.co +Sr3SnO (Sr3PbO) with bands crossing at the Fermi level between +the Sr:4d and Sn:5p (Pb:6p) electronic states22,23. Here, the Sn +and Pb atomic sites hold formal negative oxidation states con- +firmed by Mössbauer22 and X-ray photoemission spectroscopy, +XPS23. Interestingly, the negative oxidation states in metals have +attracted considerable attention due to the possible new physics +that may unveil24,25. Despite these reports, few metallic species +are known in the literature. Some examples of negative oxida- +tion states in metals have been demonstrated in compounds such +as CsAu26 and Na-Au binary compounds27,28. In the latter com- +pounds, gold’s oxidation state is Au1− achieving a full 6s25d10 +electronic occupation in the outer shell. Other examples are the +Pt’s negative oxidation at Ba-Pt systems29,30 and dimethylfor- +mamide’s surface31,32 as well as the negative oxidation state in +Zn at octahedrally coordinated Zn2M4 (M = Li and Na)33. More- +over, multiple molecular compounds have shown metals in their +structures with formal negative oxidation states24,25. As such, an- +tiperovskites appear to be potential candidates to explore possi- +ble negative oxidation states in metals, and their induced proper- +ties, due to the expected stoichiometric relationship Aα+ +3 +Bβ−Xδ− +in contrast to Aα+Bβ+Xδ− +3 +to hold the charge neutrality. Then, +when going from the perovskite to the antiperovskite, the B-site +switches from Bβ+ to Bβ−. This is the case of the SrSnO3 and +Sr3SnO where the Sn oxidation state goes from 4+ to 4−22 to +maintain the charge neutrality in both compounds, while the Sr +and O sites keep the 2+ and 2− oxidation, respectively. These +findings are also in agreement with other gold-based antiper- +ovskites oxides Cs3AuO and Rb3AuO34. Interestingly as observed +Journal Name, [year], [vol.], +1–7 | 1 +arXiv:2301.04242v1 [cond-mat.mtrl-sci] 10 Jan 2023 + +ARTICLETYPEReceivedDate +AcceptedDate +D01:00.0000/xxxxxxxxxxin perovskites, the anionic vacancies, such as oxygen deficiency +in perovskite oxides,35,36 could alter the electronic structure by +inducing an electronic reconstruction that directly affects the pre- +sented oxidation states of the atomic species37. +These vacan- +cies can be present despite the advances in growth techniques +nitrogen vacancies are expected to be formed, as in the Mn3PtNx +case38. Therefore, exploring the effect of the nitrogen deficiency +in the Mn3BN antiperovskites, and particularly in the Mn3NiN +prototype, is essential in order to unveil its influence on the struc- +tural and electronic degrees of freedom that may affect the mag- +netic response and anomalous Hall conductivity. Additionally, the +N-site is strongly correlated with the oxidation states at the Mn- +and Ni-sites in the Mn3NiN antiperovskite. In this paper, we study +from first-principles calculations, the electronic structure of the +chiral antiferromagnetic antiperovskite Mn3NiN. Thus, we per- +formed a detailed study of this antiperovskite’s electronic struc- +ture and explored the formal charges of the atomic species cou- +pled with the understanding of the N-site effect in the physics +beneath the electronic structure of this compound. This antiper- +ovskite stands as a prototype in this family of materials and we +pay special attention to the influence of the nitrogen vacancy in +the topological features, such as the anomalous Hall conductivity. +In Section 2 we present all the computational details and the the- +oretical approaches used for the development of this work. This +section is followed by the presentation of the obtained results and +the consequent analysis, in Section 3. Finally, we draw our con- +clusions, in Section 4, and highlight some perspectives associated +with our findings around the oxidation states and the nitrogen’s +effect in the Mn3NiN antiperovskite. Furthermore, we extrapolate +these analyzes and include our results in several Mn3BN antiper- +ovskites. +2 +Theoretical and computational details: +We performed first-principles calculations within the density- +functional theory (DFT)39,40 approach by using the VASP code +(version 5.4.4)41,42. The projected-augmented waves, PAW43,44 +scheme was used to represent the valence and core electrons. +The electronic configurations considered in the pseudo-potentials +as valence electrons are Mn: (3p63d54s2, version 02Aug2007), +Ni: +(3p63d84s2, version 06Sep2000), and N: (2s22p5, ver- +sion 08Apr2002). +The exchange-correlation, Exc, was repre- +sented within the generalized gradient approximation, GGA in +the PBEsol parametrization45, and the Exc of the d-electrons was +corrected through the DFT+U approximation within the Liecht- +enstein formalism46. We used a Coulomb on-site value of U = +2.0 eV parameter. The latter optimized to reproduce the exper- +imentally observed lattice parameter. Also, a metaGGA formal- +ism47, within the SCAN implementation48, and the hybrid based- +functional, HSE0649 were adopted to correlate with the Hubbard +correction within the PBEsol+U calculations. The periodic solu- +tion of the crystal was represented by using Bloch states with a +Monkhorst-Pack50 k-point mesh of 13×13×13 and 600 eV en- +ergy cut-off to give forces convergence of less than 0.001 eV·Å−1 +and an error in the energy less than 0.5 meV. The spin-orbit cou- +pling (SOC) was included to consider noncollinear magnetic con- +figurations51. The phonon calculations were performed within +the finite-differences methodology52,53 and analyzed through the +PHONOPY interface54,55. The latter calculations were performed +in the 2×2×2 supercell to properly map the lattice dynamics at +the zone-boundary. For these calculations in the supercell, the +k-mesh was then set to 6×6×6 and the noncollinear magnetic or- +derings were also considered. To evaluate the anomalous Hall +conductivity, and the changes in the Berry curvature, we have +used the Wannier functions methodology for which the wannier- +ization was performed with the WANNIER90 code56,57 and post- +processed with the WANNIERBERRI package58. Here, s, p, and +d orbitals were considered in the Mn and Ni cases, while s and +p were considered at the N site. +Additionally, a 3.0 eV win- +dow was used around the Fermi level for the wannierization. +Bader charges were evaluated by the methodology developed by +G. Henkelman et al.59. Finally, the atomic structure figures were +elaborated with the VESTA code60. +3 +Results and discussion: +In what follows, we start by describing the electronic properties +related to the N-site effect in the Mn3NiN antiperovskite. In Fig. +1 are shown the Mn3Ni and Mn3NiN cubic Pm¯3m (SG. 221) an- +tiperovskites as well as the symmetry allowed noncollinear chiral +antiferromagnetic Γ4g and Γ5g orderings. Thus, in the nitrogen +deficiency antiperovskite, the Γ4g ordering is more stable over the +Γ5g explained in terms of the MAE energy ∆E = EΓ4g − EΓ5g = +−0.58 meV·f.u.−1. Thus, as in the Mn3NiN case, the magnetic +ground state in the Mn3Ni is the chiral Γ4g antiferromagnetic or- +dering that allows the anomalous Hall effect61, as will be dis- +cussed further. Therefore, it is worth recalling that all the cal- +culations contained in this work were performed considering the +spin-orbit coupling and the noncollinear antiferromagnetic states +for Mn3NiN, as well as for Mn3Ni antiperovskites. After fully re- +laxing the Mn3NiN and Mn3Ni the obtained lattice parameters +are a = 3.889 Å and a = 3.707 Å respectively. It can be noted +that, in the Mn3NiN case, the lattice parameter is in good agree- +ment with the experimentally reported value of a = 3.886 Å62. In +the Mn3Ni, although there is no experimentally reported param- +eter, as the exchange-correlation correction was also considered +in the Mn:3d orbitals, it can be expected a close value to the one +reported here by us. When comparing the lattice parameter, it +can be observed that the inclusion of the N-site in the octahedral +center induces a tangible lattice expansion, equivalent to 0.182 +Å. However, the symmetry space groups remain the same being +Pm¯3m (SG. 221) without considering the magnetic ground state +and R¯3m′ (MSG. 166.101) once the chiral noncollinear antifer- +romagnetic ground state is accounted into the symmetry opera- +tions. Then, only changes in the volume and electronic structure +were found. Moreover, both Mn3NiN and Mn3Ni antiperovskite +are fully dynamically stable in the cubic configuration under the +noncollinear Γ4g antiferromagnetic ordering, see Fig. 1. As it +can be observed, the high-frequency modes are absent in Mn3Ni +in which case, these modes are nitrogen driven. For instance, +the antiperovskite Mn3BN structure can be viewed as magnetic +Mn-based kagome lattices with B-sites embedded into them and +separated by nitrogen layers. +In Fig. 2 we present the entire electronic characterization for +2 | +1–7 +Journal Name, [year], [vol.], + +FIG.1. Structure….?? +Γ4g +Γ5g +Mn3Ni +Mn3NiN +y +x +z +y +x +z +Mn3Ni +Mn3NiN +a) +b) +c) +Fig. 1 (Color online) (a) Mn3Ni and Mn3NiN Pm¯3m structures where the N-site octahedral center is shown in the latter and absent in the former. In +(b) are shown the chiral antiferromagnetic noncollinear Γ4g and Γ5g orderings. Here, the magnetic moments per Mn atom are shown as grey arrows +by notation. In (c) are presented the full phonon-dispersion curves obtained for the Mn3NiN, as well as the nitrogen-deficient Mn3Ni antiperovskites. +The latter were computed with U = 2.0 eV. In both cases, we consider the Γ4g chiral antiferromagnetic ordering ground state. +the Mn3NiN and Mn3Ni antiperovskites. Here, the full orbitally- +projected band electronic structure is presented, in Fig. 2(a), as +well as the local density of states, in Fig. 2(b), and the computed +anomalous Hall conductivity for the σxy, and σ111 terms, in Fig. +2(c). At first glance, we can observe from Fig. 2(a) that there is +a substantial reduction of the available states close to the Fermi +energy, here located at EF = 0.0 eV by notation, once the N-site +is introduced in the antiperovskite. It can be appreciated that the +major contribution at and above the Fermi level is associated with +the Mn:3d orbitals in both cases. As for the Ni states, those ap- +pear to be located well below E = −0.5 eV and are quite localized +around −1.5 eV as in an insulator case, see Fig. 2(b). Never- +theless, a small contribution from the Ni states can be observed +above the Fermi level. The latter is expected because the antiper- +ovskite structure can be understood, as commented before, as +(111) Mn-based kagome planes with Ni sites embedded and sep- +arated by the N-sites. Importantly, as the N-site is located at the +octahedral center, the Mn3NiN and the Mn3Ni hold the same crys- +tallographic and magnetic symmetry. Thus, only modifications in +the electronic structure are observed, but the AHC tensor is kept +fixed. In this case, the anomalous Hall conductivity component, +σxy, has been computed by the relationship: +σxy = −2πe2 +h +occ +∑ +n +� +BZ +d3k +(2π)3 fn(k)Ωn,xy(k), +(1) +where Ωxy(k)=∑occ +n +fn(k)Ωn,xy(k) is the summation of all the oc- +cupied n-bands and fn(k) represents the Fermi distribution. More- +over, the symmetry-allowed AHC components within the Γ4g or- +dering in the R¯3m′ magnetic symmetry group are: +σR¯3m′ = +� +� +� +0 +σxy +−σxy +−σxy +0 +σxy +σxy +−σxy +0 +� +� +� +(2) +The charge at the Ni-site is expected to be the same and only +changes in the allowed electronic states in the proximity to the +Fermi level might influence the anomalous Hall conductivity. Ad- +ditionally, as the AHC is strongly dependent on the spin-orbit cou- +pling strength63, the absence or presence of the nitrogen octahe- +dral center site has a negligible effect on σxy. In Fig. 2(c) we +show the computed anomalous Hall conductivity for the σxy in +both compounds, as well as the σ111 component in the magnetic +kagome lattice at the (111) lattice plane. The σ111 component +is computed as σ111 ≡ +1 +√ +3 +� +σxy +σyz +σzx +� +and corresponds to the +conductivity on the (111) kagome lattice. We then found that +in absence of the N-site σxy = 139 S·cm−1 (σ111 = 241 S·cm−1) +whereas in the Mn3NiN is σxy = 78 S·cm−1 (σ111 = 135 S·cm−1), +both at the EF level,. Our findings show a considerable increase +of the σxy in the nitrogen-deficient antiperovskite that can be cor- +related to the increase of the available electronic states close to +Fermi. The latter enhancement of the σxy component is in agree- +ment with the electronic band structure, also shown and dis- +cussed before. Therefore, the N-site is directly influencing the +fn(k) function into Eq. 1 modifying the σxy value but keeping the +symmetry operations. +In regards to the electronic structure, formally, the oxidation +states according to the IUPAC64–66, quantifies the oxidation de- +gree of an atom defined based-on the electron counting of such +atomic species after the bonding is reached. Therefore, the ox- +idation number can be obtained by following a set of rules, as +exposed by A. Walsh et al.67,68 that, as mentioned before, can +be ascribed to the electron counting. Then, aiming to estimate +the potential oxidation states, hold by each atomic component in +the antiperovskite, we proceded by obtaining the charges around +each site. As in the Mn3NiN and Mn3Ni antiperovskites, the elec- +tronic structure is metallic, the Born effective charges, Z∗, are +Journal Name, [year], [vol.], +1–7 | 3 + +FIG.2. Electronic Structure….?? +a) +b) +c) +Mn3NiN +Mn3NiN +Mn3NiN +Mn3Ni +Mn3Ni +Mn3Ni +DOS (states/eV) +DOS (states/eV) +BZ-path +BZ-path +Fig. 2 (Color online) (a) Atomically projected band structure, and (b) atomically projected density of states, DOS. Here, the Mn, Ni, and N states +are denoted in violet, blue, and green colors respectively. Additionally, in (b) the total DOS is denoted in grey color. (c) Anomalous Hall conductivity, +σxy and σ111 components, computed at the Γ4g orderings in the Mn3NiN and Mn3Ni. +Table 1 Bader charges, in e− units, computed for the Mn, Ni, and N +sites in the Mn3NiN and Mn3Ni considering the chiral antiferromagnetic +Γ4g ordering. The latter values were extracted under several exchange- +correlation representations. Additionally, we present the magnetic mo- +ment, per Mn atom, in each case. +XCPBEsol +ZMn +ZNi +ZN +m (µB·Mn−1) +Mn3NiN ++0.907 +−0.723 +−2.006 +3.560 +Mn3Ni ++0.259 +−0.704 +— +3.583 +XCSCAN +ZMn +ZNi +ZN +m (µB·Mn−1) +Mn3NiN ++0.806 +−0.745 +−1.957 +3.418 +Mn3Ni ++0.262 +−0.788 +— +3.470 +XCHSE06 +ZMn +ZNi +ZN +m (µB·Mn−1) +Mn3NiN ++0.973 +−0.790 +−2.130 +3.854 +Mn3Ni ++0.324 +−0.718 +— +3.883 +not accessible due to the ill-defined polarization in metals*, and +therefore, the Bader charges offer an alternative route to esti- +mate the charges in the atomic species. In Table 1 are condensed +the results related to the Bader charges computed in the Mn3NiN +and Mn3Ni antiperovskites. These values were obtained for the +PBEsol+U, SCAN, and HSE06 exchange-correlation approaches. +We can observe that, independently of the exchange-correlation +considerations, following the Jacob’s ladder from the GGA+U +to the hybrid-functional approach69, the computed charges are +close to +0.9e−, −0.7e−, and −2.0e− for the Mn, Ni, and N sites, +respectively, in the Mn3NiN case. The previous charges are in +contrast with the computed charges of +0.3e− and −0.7e− for +Mn and Ni in the Mn3Ni. These results are suggesting a good rep- +resentation of the charges with the PBEsol+U approach. Thus, +the PBEsol+U exchange-correlation is used for further analysis. +As expected, the Mn-sites hold, in both antiperovskite cases, a +positive charge associated with a Mnα+ oxidation state. Mean- +* The Z∗ tensor is defined as Z∗ +αβ,κ = +∂Pβ +∂τα,κ |E =0 where α and β are the cartesian +coordinates, Pβ is the polarization and τ are the atomic displacements of the κ atom. +while, the corner Ni-site shows a negative charge leading to a +Niβ− oxidation state. In the nitrogen case, as expected the Bader +charge is negative and it is related to the Nδ−, (δ = 3) oxida- +tion state expected in this anionic site. Interestingly, the Mn’s +Bader charge is +0.259e− in the Mn3Ni whereas is +0.907e− in +the Mn3NiN. This can be explained due to the charge localized in +the nitrogen site when incorporated in the antiperovskite and that +it is transferred from the manganese sites. Moreover, this result +is in agreement with larger electronic states, close to the Fermi +level, available in the Mn3Ni in comparison to Mn3NiN, and also +explaining the AHC results. Aiming to compare with other in- +sulating antiperovskites, such as Ca3SnO and Ca3BiN, we have +computed the Bader charges and found that ZCa = +1.308e−, +ZSn = −2.364e−, and ZO = −1.558e− for the Ca3SnO oxide, +and ZCa = +1.333e−, ZBi = −1.955e−, ZN = −2.043e−, for the +Ca3BiN nitride antiperovskite case. As for Born effective charges, +Z∗ accessible in these compounds, we observed that the diag- +onal terms are Z∗ +Ca = +2.388e−, Z∗ +Sn = −3.023e−, and Z∗ +O = +−3.381e− in the Ca3SnO oxide, and Z∗ +Ca = +2.380e−, Z∗ +Bi = +−2.899e−, Z∗ +N = −4.397e− for the Ca3BiN nitride. The devia- +tion of the Born effective charges, with respect to the nominal +values (ZCa = +2e−, ZSn = −4e−, ZBi = −3e−, ZO = −2e− and +ZN = −3e−,), can be explained in terms of the large polarizability +of the Sn–O and Bi–N bondings widely observed and reported in +ferroelectric perovskite oxides70,71. Despite the charge underes- +timation, shown by the Bader analysis, and overestimation, ob- +tained with the Born effective charges, the latter results are in +fair agreement with the expected oxidation states of A2+ +3 B4−O2− +and A2+ +3 B3−N3− compounds, respectively. As such, these findings +are consistent with the experimentally measured, by Mössbauer +spectroscopy and X-ray photoemission spectroscopy, XPS, oxida- +tion states of the atomic constituents in the Sr3SnO and Sr3PbO +antiperovskites22,23. In such compounds, the oxidation state was +associated with Sn4− and Pb4− states based on the experimental +results. Additionally, these results on the oxidation states are also +in agreement with the calculations in other antiperovskite insu- +lators such as Ba3SiO and Ba3SiO/Ba3GeO ferroelectric superlat- +tices in which, the Z∗ values are +2.396e−, −4.720e−, −4.594e−, +and −2.801e− for the Ba, Si, Ge, and O sites, respectively11,72. +4 | +1–7 +Journal Name, [year], [vol.], + +MMS'O +-1'2-S'0 +-1'2-J'O +-0'20.0 +0'2T'O5'O-1'O0'0E-E (eV)0'2 +T'O1'2 +s'OTable 2 Computed Bader charges, in e− units, for the Mn, B-sites, and +N sites in the Mn3BN within the chiral antiferromagnetic Γ4g ordering. +Here, we also present the magnetic moment, per Mn atom in each case as +well as the electronic configuration for each B-site with the outer valence +electrons in neutral state. In the Mn and Ni cases, the outer electrons +are [Ar]4s23d5 and [He]2s22p3, respectively. +Mn3BN +ZMn +ZB +ZN +B-site +Mn3NiN ++0.907 +−0.723 +−2.006 +Ni:[Ar]4s23d8 +Mn3PdN ++0.884 +−1.023 +−1.629 +Pd:[K]5s04d10 +Mn3PtN ++0.982 +−1.351 +−1.596 +Pt:[Xe]6s15d9 +Mn3IrN ++0.946 +−1.273 +−1.566 +Ir:[Xe]6s25d7 +Mn3NiN ++0.907 +−0.723 +−2.006 +Ni:[Ar]4s23d8 +Mn3CuN ++0.754 +−0.601 +−1.661 +Cu:[Ar]4s13d10 +Mn3ZnN ++0.682 +−0.391 +−1.656 +Zn:[Ar]4s23d10 +Mn3GaN ++0.593 +−0.136 +−1.643 +Ga:[Ar]4s23d104p1 +Mn3SnN ++0.661 +−0.355 +−1.630 +Sn:[Kr]5s24d105p2 +To contrast the obtained oxidation states in the antiperovskite +Mn3NiN, we defined a hypothetical perovskite compound as +NiMnN3 by inverting the Mn and N sites. Thus, the Mn occupies +the octahedral center whereas the N sites form the octahedra, +i.e. MnN6. Here, the Ni sites remain in the cell’s corner site. In +such a compound, we have fully relaxed the structural and elec- +tronic degrees of freedom and found a metallic behavior with a +tangible magnetic response in which, the Mn holds a m = 2.501 +µB·Mn−1 and the Ni is m = −1.080 µB·Ni−1. +After extracting +the Bader charges we obtained values of ZMn = +1.981e−, ZNi += +0.888e−, ZN = −0.957e−. As expected, the A+B+X− +3 is kept +as Niα+Mnβ+Nδ− +3 . It is worth mentioning that the Bader charges +seem to underestimate the computed charge in the atomic sites, +as observed, for example in Ca3BiN possibly due to the partition- +ing methodology and exchange-correlation approach73. Never- +theless, it can be concluded that as perovskite and antiperovskite +structures are considered, the Ni-site oxidation state is reversed +from positive, in the former, to negative in the latter. +Moving forward, we applied our analysis to several members of +the Mn3BN family in order to extrapolate our findings. In Table +2 we present our calculations of the Bader charges across several +reported Mn-based antiperovskites. In all the cases, the chiral +antiferromagnetic Γ4g ordering was considered as the magnetic +ground state in the calculations. As expected, the N-sites remain +negative with values between −2.0e− to −1.6e− whereas the Mn- +sites vary from +0.9e− to +0.6e−. Thus, we followed the trend +vertically along the periodic table from the 3d in Ni to 5d in Pt +and horizontally from Ni to Ga. +We found an increase of the +charge at the B-site, from Ni to Pt, suggesting an increase of the +negative oxidation state, for example, β ∼ 1− in Ni to β ∼ 2− +in Pt. In the Mn3IrN case, the Bader charge is close to the value +observed for the Pt site. Interestingly, the received charge could +be possibly located at the open s- and d-orbitals. On the contrary, +the charge decreases from Ni to Ga possibly due to the tendency, +in this case, to keep the outer electronic shells closed. As for the +magnetic moment value, per Mn site, remains in values between +3.5 µB·Mn−1 to 3.8 µB·Mn−1. In contrast, we observed that the +charge at the B-sites decreases when going from Ni to Ga. This +can be explained in terms of the closing electronic shell limiting +the space for acquired charge and therefore, diminishing the pos- +sible negative oxidation state. +Although we are aware of the underestimated charge obtained +through the Bader charges approach, and fluctuations in the val- +ues of the charges could be expected due to the partitioning +method employed68,73, our findings consistently suggest negative +oxidation states in metals when located at the antiperovskite’s +corner B-site. Finally, it is worth noticing that as the spin-orbit +coupling, SOC, increases with the oxidation state74, the B-site’s +oxidation is paramount to understanding its contribution to the +anomalous Hall conductivity. +Thus, in the case of the Mn3BN +compounds, the SOC possibly decreases due to the negative ox- +idation state in the B-site when compared with perovskite com- +pounds. +4 +Conclusions: +In this paper, we have studied the electronic structure of the +Mn3NiN and Mn3Ni magnetically chiral noncollinear antiferro- +magnetic antiperovskites by utilizing first-principles calculations. +We found that the N-site expands the cell when it is located at the +center of the octahedral. Nonetheless, due to the center position, +the symmetry operations and expected properties are conserved. +We observed a tangible increase of the available electronic states +close to the Fermi level that favors the conductivity, as in the case +of the anomalous Hall effect in which, σxy = 139 S·cm−1 (σ111 += 241 S·cm−1) in absence of the N-site in contrast to σxy = 78 +S·cm−1 (σ111 = 135 S·cm−1) in the Mn3NiN counterpart. Our +findings suggest that the nitrogen inclusion in the Mn3NiN sys- +tem enhances a positive oxidation state, possibly ∼1+, in the Mn +whereas, and more interestingly, the Ni sites hold a negative, po- +tentially ∼1−, oxidation state. This behavior is observed although +the overall electronic structure remains metallic. Finally, our find- +ings also suggest that several transition metals may exhibit neg- +ative oxidation states when located at the B-site in the Mn3BN +antiperovskite. We thus hope that our result will motivate further +studies in antiperovskite structures that might be ideal candidates +to further investigate the negative oxidation states in metals. +Author contributions +All of the authors were involved in the preparation and develop- +ment of the manuscript. Moreover, all of the authors read and +approved the final manuscript. +Conflict of interest +The authors declare no personnel or financial conflict of interests +with any person or organization. +Acknowledgments: +The calculations presented in this paper were carried out using +the Grid UIS-2 experimental testbed, being developed under the +Universidad Industrial de Santander (SC3-UIS) High Performance +and Scientific Computing Centre, development action with sup- +Journal Name, [year], [vol.], +1–7 | 5 + +port from UIS Vicerrectoría de Investigación y Extension (VIE- +UIS) and several UIS research groups as well as other funding +resources. Additionally, we acknowledge the computational sup- +port extended to us by Laboratorio de Supercomputo del Sureste +de México (LNS), Benemérita Universidad Autónoma de Puebla, +BUAP, for performing heavy theoretical calculations. A. C. Garcia- +Castro acknowledge the grant No. 2677 entitled “Quiralidad y Or- +denamiento Magnético en Sistemas Cristalinos: Estudio Teórico +desde Primeros Principios” supported by the VIE – UIS. +Notes and references +1 G. Gurung, D. F. Shao, T. R. Paudel and E. Y. Tsymbal, Physical +Review Materials, 2019, 3, 1–7. +2 Z. Q. Liu, H. Chen, J. M. Wang, J. H. Liu, K. Wang, Z. X. +Feng, H. Yan, X. R. Wang, C. B. Jiang, J. M. Coey and A. H. +Macdonald, Nature Electronics, 2018, 1, 172–177. +3 H. +Tsai, +T. +Higo, +K. +Kondou, +T. +Nomoto, +A. +Sakai, +A. Kobayashi, T. Nakano, K. Yakushiji, R. Arita, S. Miwa, +Y. Otani and S. Nakatsuji, Nature, 2020, 1–6. +4 D. +Torres-Amaris, +A. +Bautista-Hernandez, +R. +González- +Hernández, A. H. Romero and A. C. 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Whangbo, Journal of Computa- +tional Chemistry, 2008, 29, 2187–2209. +Journal Name, [year], [vol.], +1–7 | 7 + diff --git a/G9E2T4oBgHgl3EQf-wmW/content/tmp_files/load_file.txt b/G9E2T4oBgHgl3EQf-wmW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b36989adbb9e1666bfca701f00be7a616e1df8a --- /dev/null +++ b/G9E2T4oBgHgl3EQf-wmW/content/tmp_files/load_file.txt @@ -0,0 +1,930 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf,len=929 +page_content='Anionic nickel and nitrogen effects in the chiral antiferro- magnetic antiperovskite Mn3NiN E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Triana-Ramíreza,b, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Ibarra-Hernandezc and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Garcia-Castroa,∗ Magnetic antiperovskites, holding chiral noncollinear antiferromagnetic ordering, have shown remark- able properties that cover from negative thermal expansion to anomalous Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Nevertheless, details on the electronic structure related to the oxidation states and the octahedral center’s site effect are still scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Here, we show a theoretical study, based on first-principles calculations in the framework of the density-functional theory, DFT, on the electronic details associated with the nitro- gen site effect into the structural, electronic, magnetic, and topological degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, we show that the nitrogen vacancy increases the values of the anomalous Hall conductivity and retains the chiral Γ4g antiferromagnetic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Moreover, we reveal, based on the Bader charges and the electronic structure analysis, the negative and positive oxidation states in the Ni and Mn sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The latter is in agreement with the expected Aα+ 3 Bβ−Xδ− oxidation states to satisfy the charge neutrality in the antiperovskites, but rare for transition metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Finally, we extrapolate our findings on the oxidation states to several Mn3BN compounds showing that the antiperovskite structure is an ideal platform to encounter negative oxidation states in metals sitting at the corner B-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 1 Introduction: In the modern quest for novel and exciting phenomena in new materials, antiperovskite, or inversed-perovskite with the formu- lae A3BX, has stood out as an astonishing type of materials show- ing large anomalous Hall conductivity1–4, superconductivity5–8, good performance for new batteries9,10, tunable hybrid-improper ferroelectricity11, tangible magnetocaloric effects12, and large spin-lattice coupling13–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' All the latter are just a few examples that can be mentioned among the vast functionalities offered by these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Interestingly, in antiperovskites18,19 the electro- static balance and the oxidation site occupation are apparently reversed with respect to the known perovskites, ABX3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Most of the mentioned phenomena in the antiperovskites owe most of their properties to the reversed occupation of their anionic and cationic sites forming the reversed XA6 octahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' This subtle change in the coordination and atomic occupations gives rise to, for example, the triangular geometric coordination of the mag- netic sites that in turn, induces a strong magnetic frustration con- verging into chiral noncollinear antiferromagnetic orderings21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Another example is the topologically related properties in the a School of Physics, Universidad Industrial de Santander, Carrera 27 Calle 09, 680002, Bucaramanga, Colombia b Centro de Investigación y Estudios Avanzados del IPN - CINVESTAV-Querétaro, MX- 76230, Querétaro, México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' c Facultad de Ingeniería, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-39, Puebla, Pue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 72570, México.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' E-mail: acgarcia@uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='co Sr3SnO (Sr3PbO) with bands crossing at the Fermi level between the Sr:4d and Sn:5p (Pb:6p) electronic states22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Here, the Sn and Pb atomic sites hold formal negative oxidation states con- firmed by Mössbauer22 and X-ray photoemission spectroscopy, XPS23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Interestingly, the negative oxidation states in metals have attracted considerable attention due to the possible new physics that may unveil24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Despite these reports, few metallic species are known in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Some examples of negative oxida- tion states in metals have been demonstrated in compounds such as CsAu26 and Na-Au binary compounds27,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In the latter com- pounds, gold’s oxidation state is Au1− achieving a full 6s25d10 electronic occupation in the outer shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Other examples are the Pt’s negative oxidation at Ba-Pt systems29,30 and dimethylfor- mamide’s surface31,32 as well as the negative oxidation state in Zn at octahedrally coordinated Zn2M4 (M = Li and Na)33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' More- over, multiple molecular compounds have shown metals in their structures with formal negative oxidation states24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As such, an- tiperovskites appear to be potential candidates to explore possi- ble negative oxidation states in metals, and their induced proper- ties, due to the expected stoichiometric relationship Aα+ 3 Bβ−Xδ− in contrast to Aα+Bβ+Xδ− 3 to hold the charge neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Then, when going from the perovskite to the antiperovskite, the B-site switches from Bβ+ to Bβ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' This is the case of the SrSnO3 and Sr3SnO where the Sn oxidation state goes from 4+ to 4−22 to maintain the charge neutrality in both compounds, while the Sr and O sites keep the 2+ and 2− oxidation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' These findings are also in agreement with other gold-based antiper- ovskites oxides Cs3AuO and Rb3AuO34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Interestingly as observed Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' ], 1–7 | 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='04242v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='mtrl-sci] 10 Jan 2023 ARTICLETYPEReceivedDate AcceptedDate D01:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='0000/xxxxxxxxxxin perovskites, the anionic vacancies, such as oxygen deficiency in perovskite oxides,35,36 could alter the electronic structure by inducing an electronic reconstruction that directly affects the pre- sented oxidation states of the atomic species37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' These vacan- cies can be present despite the advances in growth techniques nitrogen vacancies are expected to be formed, as in the Mn3PtNx case38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Therefore, exploring the effect of the nitrogen deficiency in the Mn3BN antiperovskites, and particularly in the Mn3NiN prototype, is essential in order to unveil its influence on the struc- tural and electronic degrees of freedom that may affect the mag- netic response and anomalous Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Additionally, the N-site is strongly correlated with the oxidation states at the Mn- and Ni-sites in the Mn3NiN antiperovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In this paper, we study from first-principles calculations, the electronic structure of the chiral antiferromagnetic antiperovskite Mn3NiN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, we per- formed a detailed study of this antiperovskite’s electronic struc- ture and explored the formal charges of the atomic species cou- pled with the understanding of the N-site effect in the physics beneath the electronic structure of this compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' This antiper- ovskite stands as a prototype in this family of materials and we pay special attention to the influence of the nitrogen vacancy in the topological features, such as the anomalous Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In Section 2 we present all the computational details and the the- oretical approaches used for the development of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' This section is followed by the presentation of the obtained results and the consequent analysis, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Finally, we draw our con- clusions, in Section 4, and highlight some perspectives associated with our findings around the oxidation states and the nitrogen’s effect in the Mn3NiN antiperovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Furthermore, we extrapolate these analyzes and include our results in several Mn3BN antiper- ovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2 Theoretical and computational details: We performed first-principles calculations within the density- functional theory (DFT)39,40 approach by using the VASP code (version 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='4)41,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The projected-augmented waves, PAW43,44 scheme was used to represent the valence and core electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The electronic configurations considered in the pseudo-potentials as valence electrons are Mn: (3p63d54s2, version 02Aug2007), Ni: (3p63d84s2, version 06Sep2000), and N: (2s22p5, ver- sion 08Apr2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The exchange-correlation, Exc, was repre- sented within the generalized gradient approximation, GGA in the PBEsol parametrization45, and the Exc of the d-electrons was corrected through the DFT+U approximation within the Liecht- enstein formalism46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' We used a Coulomb on-site value of U = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='0 eV parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The latter optimized to reproduce the exper- imentally observed lattice parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Also, a metaGGA formal- ism47, within the SCAN implementation48, and the hybrid based- functional, HSE0649 were adopted to correlate with the Hubbard correction within the PBEsol+U calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The periodic solu- tion of the crystal was represented by using Bloch states with a Monkhorst-Pack50 k-point mesh of 13×13×13 and 600 eV en- ergy cut-off to give forces convergence of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='001 eV·Å−1 and an error in the energy less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='5 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The spin-orbit cou- pling (SOC) was included to consider noncollinear magnetic con- figurations51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The phonon calculations were performed within the finite-differences methodology52,53 and analyzed through the PHONOPY interface54,55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The latter calculations were performed in the 2×2×2 supercell to properly map the lattice dynamics at the zone-boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' For these calculations in the supercell, the k-mesh was then set to 6×6×6 and the noncollinear magnetic or- derings were also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' To evaluate the anomalous Hall conductivity, and the changes in the Berry curvature, we have used the Wannier functions methodology for which the wannier- ization was performed with the WANNIER90 code56,57 and post- processed with the WANNIERBERRI package58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Here, s, p, and d orbitals were considered in the Mn and Ni cases, while s and p were considered at the N site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Additionally, a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='0 eV win- dow was used around the Fermi level for the wannierization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Bader charges were evaluated by the methodology developed by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Henkelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Finally, the atomic structure figures were elaborated with the VESTA code60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 3 Results and discussion: In what follows, we start by describing the electronic properties related to the N-site effect in the Mn3NiN antiperovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 1 are shown the Mn3Ni and Mn3NiN cubic Pm¯3m (SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 221) an- tiperovskites as well as the symmetry allowed noncollinear chiral antiferromagnetic Γ4g and Γ5g orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, in the nitrogen deficiency antiperovskite, the Γ4g ordering is more stable over the Γ5g explained in terms of the MAE energy ∆E = EΓ4g − EΓ5g = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='58 meV·f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, as in the Mn3NiN case, the magnetic ground state in the Mn3Ni is the chiral Γ4g antiferromagnetic or- dering that allows the anomalous Hall effect61, as will be dis- cussed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Therefore, it is worth recalling that all the cal- culations contained in this work were performed considering the spin-orbit coupling and the noncollinear antiferromagnetic states for Mn3NiN, as well as for Mn3Ni antiperovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' After fully re- laxing the Mn3NiN and Mn3Ni the obtained lattice parameters are a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='889 Å and a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='707 Å respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' It can be noted that, in the Mn3NiN case, the lattice parameter is in good agree- ment with the experimentally reported value of a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='886 Å62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In the Mn3Ni, although there is no experimentally reported param- eter, as the exchange-correlation correction was also considered in the Mn:3d orbitals, it can be expected a close value to the one reported here by us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' When comparing the lattice parameter, it can be observed that the inclusion of the N-site in the octahedral center induces a tangible lattice expansion, equivalent to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='182 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' However, the symmetry space groups remain the same being Pm¯3m (SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 221) without considering the magnetic ground state and R¯3m′ (MSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='101) once the chiral noncollinear antifer- romagnetic ground state is accounted into the symmetry opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Then, only changes in the volume and electronic structure were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Moreover, both Mn3NiN and Mn3Ni antiperovskite are fully dynamically stable in the cubic configuration under the noncollinear Γ4g antiferromagnetic ordering, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As it can be observed, the high-frequency modes are absent in Mn3Ni in which case, these modes are nitrogen driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' For instance, the antiperovskite Mn3BN structure can be viewed as magnetic Mn-based kagome lattices with B-sites embedded into them and separated by nitrogen layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2 we present the entire electronic characterization for 2 | 1–7 Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' ], FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Structure….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Γ4g Γ5g Mn3Ni Mn3NiN y x z y x z Mn3Ni Mn3NiN a) b) c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 1 (Color online) (a) Mn3Ni and Mn3NiN Pm¯3m structures where the N-site octahedral center is shown in the latter and absent in the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In (b) are shown the chiral antiferromagnetic noncollinear Γ4g and Γ5g orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Here, the magnetic moments per Mn atom are shown as grey arrows by notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In (c) are presented the full phonon-dispersion curves obtained for the Mn3NiN, as well as the nitrogen-deficient Mn3Ni antiperovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The latter were computed with U = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In both cases, we consider the Γ4g chiral antiferromagnetic ordering ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' the Mn3NiN and Mn3Ni antiperovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Here, the full orbitally- projected band electronic structure is presented, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2(a), as well as the local density of states, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2(b), and the computed anomalous Hall conductivity for the σxy, and σ111 terms, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' At first glance, we can observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2(a) that there is a substantial reduction of the available states close to the Fermi energy, here located at EF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='0 eV by notation, once the N-site is introduced in the antiperovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' It can be appreciated that the major contribution at and above the Fermi level is associated with the Mn:3d orbitals in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As for the Ni states, those ap- pear to be located well below E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='5 eV and are quite localized around −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='5 eV as in an insulator case, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Never- theless, a small contribution from the Ni states can be observed above the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The latter is expected because the antiper- ovskite structure can be understood, as commented before, as (111) Mn-based kagome planes with Ni sites embedded and sep- arated by the N-sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Importantly, as the N-site is located at the octahedral center, the Mn3NiN and the Mn3Ni hold the same crys- tallographic and magnetic symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, only modifications in the electronic structure are observed, but the AHC tensor is kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In this case, the anomalous Hall conductivity component, σxy, has been computed by the relationship: σxy = −2πe2 h occ ∑ n � BZ d3k (2π)3 fn(k)Ωn,xy(k), (1) where Ωxy(k)=∑occ n fn(k)Ωn,xy(k) is the summation of all the oc- cupied n-bands and fn(k) represents the Fermi distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' More- over, the symmetry-allowed AHC components within the Γ4g or- dering in the R¯3m′ magnetic symmetry group are: σR¯3m′ = � � � 0 σxy −σxy −σxy 0 σxy σxy −σxy 0 � � � (2) The charge at the Ni-site is expected to be the same and only changes in the allowed electronic states in the proximity to the Fermi level might influence the anomalous Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Ad- ditionally, as the AHC is strongly dependent on the spin-orbit cou- pling strength63, the absence or presence of the nitrogen octahe- dral center site has a negligible effect on σxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2(c) we show the computed anomalous Hall conductivity for the σxy in both compounds, as well as the σ111 component in the magnetic kagome lattice at the (111) lattice plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The σ111 component is computed as σ111 ≡ 1 √ 3 � σxy +σyz +σzx � and corresponds to the conductivity on the (111) kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' We then found that in absence of the N-site σxy = 139 S·cm−1 (σ111 = 241 S·cm−1) whereas in the Mn3NiN is σxy = 78 S·cm−1 (σ111 = 135 S·cm−1), both at the EF level,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Our findings show a considerable increase of the σxy in the nitrogen-deficient antiperovskite that can be cor- related to the increase of the available electronic states close to Fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The latter enhancement of the σxy component is in agree- ment with the electronic band structure, also shown and dis- cussed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Therefore, the N-site is directly influencing the fn(k) function into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 1 modifying the σxy value but keeping the symmetry operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In regards to the electronic structure, formally, the oxidation states according to the IUPAC64–66, quantifies the oxidation de- gree of an atom defined based-on the electron counting of such atomic species after the bonding is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Therefore, the ox- idation number can be obtained by following a set of rules, as exposed by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='67,68 that, as mentioned before, can be ascribed to the electron counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Then, aiming to estimate the potential oxidation states, hold by each atomic component in the antiperovskite, we proceded by obtaining the charges around each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As in the Mn3NiN and Mn3Ni antiperovskites, the elec- tronic structure is metallic, the Born effective charges, Z∗, are Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' ], 1–7 | 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Electronic Structure….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' a) b) c) Mn3NiN Mn3NiN Mn3NiN Mn3Ni Mn3Ni Mn3Ni DOS (states/eV) DOS (states/eV) BZ-path BZ-path Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2 (Color online) (a) Atomically projected band structure, and (b) atomically projected density of states, DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Here, the Mn, Ni, and N states are denoted in violet, blue, and green colors respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Additionally, in (b) the total DOS is denoted in grey color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' (c) Anomalous Hall conductivity, σxy and σ111 components, computed at the Γ4g orderings in the Mn3NiN and Mn3Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Table 1 Bader charges, in e− units, computed for the Mn, Ni, and N sites in the Mn3NiN and Mn3Ni considering the chiral antiferromagnetic Γ4g ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The latter values were extracted under several exchange- correlation representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Additionally, we present the magnetic mo- ment, per Mn atom, in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' XCPBEsol ZMn ZNi ZN m (µB·Mn−1) Mn3NiN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='907 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='723 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='006 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='560 Mn3Ni +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='259 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='704 — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='583 XCSCAN ZMn ZNi ZN m (µB·Mn−1) Mn3NiN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='806 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='745 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='957 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='418 Mn3Ni +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='262 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='788 — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='470 XCHSE06 ZMn ZNi ZN m (µB·Mn−1) Mn3NiN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='973 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='790 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='130 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='854 Mn3Ni +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='324 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='718 — 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='883 not accessible due to the ill-defined polarization in metals*, and therefore, the Bader charges offer an alternative route to esti- mate the charges in the atomic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In Table 1 are condensed the results related to the Bader charges computed in the Mn3NiN and Mn3Ni antiperovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' These values were obtained for the PBEsol+U, SCAN, and HSE06 exchange-correlation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' We can observe that, independently of the exchange-correlation considerations, following the Jacob’s ladder from the GGA+U to the hybrid-functional approach69, the computed charges are close to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='9e−, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='7e−, and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='0e− for the Mn, Ni, and N sites, respectively, in the Mn3NiN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The previous charges are in contrast with the computed charges of +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='3e− and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='7e− for Mn and Ni in the Mn3Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' These results are suggesting a good rep- resentation of the charges with the PBEsol+U approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, the PBEsol+U exchange-correlation is used for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As expected, the Mn-sites hold, in both antiperovskite cases, a positive charge associated with a Mnα+ oxidation state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Mean- The Z∗ tensor is defined as Z∗ αβ,κ = ∂Pβ ∂τα,κ |E =0 where α and β are the cartesian coordinates, Pβ is the polarization and τ are the atomic displacements of the κ atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' while, the corner Ni-site shows a negative charge leading to a Niβ− oxidation state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In the nitrogen case, as expected the Bader charge is negative and it is related to the Nδ−, (δ = 3) oxida- tion state expected in this anionic site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Interestingly, the Mn’s Bader charge is +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='259e− in the Mn3Ni whereas is +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='907e− in the Mn3NiN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' This can be explained due to the charge localized in the nitrogen site when incorporated in the antiperovskite and that it is transferred from the manganese sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Moreover, this result is in agreement with larger electronic states, close to the Fermi level, available in the Mn3Ni in comparison to Mn3NiN, and also explaining the AHC results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Aiming to compare with other in- sulating antiperovskites, such as Ca3SnO and Ca3BiN, we have computed the Bader charges and found that ZCa = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='308e−, ZSn = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='364e−, and ZO = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='558e− for the Ca3SnO oxide, and ZCa = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='333e−, ZBi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='955e−, ZN = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='043e−, for the Ca3BiN nitride antiperovskite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As for Born effective charges, Z∗ accessible in these compounds, we observed that the diag- onal terms are Z∗ Ca = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='388e−, Z∗ Sn = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='023e−, and Z∗ O = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='381e− in the Ca3SnO oxide, and Z∗ Ca = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='380e−, Z∗ Bi = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='899e−, Z∗ N = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='397e− for the Ca3BiN nitride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' The devia- tion of the Born effective charges, with respect to the nominal values (ZCa = +2e−, ZSn = −4e−, ZBi = −3e−, ZO = −2e− and ZN = −3e−,), can be explained in terms of the large polarizability of the Sn–O and Bi–N bondings widely observed and reported in ferroelectric perovskite oxides70,71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Despite the charge underes- timation, shown by the Bader analysis, and overestimation, ob- tained with the Born effective charges, the latter results are in fair agreement with the expected oxidation states of A2+ 3 B4−O2− and A2+ 3 B3−N3− compounds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As such, these findings are consistent with the experimentally measured, by Mössbauer spectroscopy and X-ray photoemission spectroscopy, XPS, oxida- tion states of the atomic constituents in the Sr3SnO and Sr3PbO antiperovskites22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In such compounds, the oxidation state was associated with Sn4− and Pb4− states based on the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Additionally, these results on the oxidation states are also in agreement with the calculations in other antiperovskite insu- lators such as Ba3SiO and Ba3SiO/Ba3GeO ferroelectric superlat- tices in which, the Z∗ values are +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='396e−, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='720e−, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='594e−, and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='801e− for the Ba, Si, Ge, and O sites, respectively11,72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 4 | 1–7 Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=" ], MMS'O 1'2-S'0 1'2-J'O 0'20." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content="0 0'2T'O5'O-1'O0'0E-E (eV)0'2 T'O1'2 s'OTable 2 Computed Bader charges, in e− units, for the Mn, B-sites, and N sites in the Mn3BN within the chiral antiferromagnetic Γ4g ordering." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Here, we also present the magnetic moment, per Mn atom in each case as well as the electronic configuration for each B-site with the outer valence electrons in neutral state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In the Mn and Ni cases, the outer electrons are [Ar]4s23d5 and [He]2s22p3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Mn3BN ZMn ZB ZN B-site Mn3NiN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='907 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='723 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='006 Ni:[Ar]4s23d8 Mn3PdN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='884 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='023 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='629 Pd:[K]5s04d10 Mn3PtN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='982 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='351 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='596 Pt:[Xe]6s15d9 Mn3IrN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='946 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='273 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='566 Ir:[Xe]6s25d7 Mn3NiN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='907 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='723 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='006 Ni:[Ar]4s23d8 Mn3CuN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='754 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='601 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='661 Cu:[Ar]4s13d10 Mn3ZnN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='682 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='391 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='656 Zn:[Ar]4s23d10 Mn3GaN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='593 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='136 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='643 Ga:[Ar]4s23d104p1 Mn3SnN +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='661 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='355 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='630 Sn:[Kr]5s24d105p2 To contrast the obtained oxidation states in the antiperovskite Mn3NiN, we defined a hypothetical perovskite compound as NiMnN3 by inverting the Mn and N sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, the Mn occupies the octahedral center whereas the N sites form the octahedra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' MnN6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Here, the Ni sites remain in the cell’s corner site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In such a compound, we have fully relaxed the structural and elec- tronic degrees of freedom and found a metallic behavior with a tangible magnetic response in which, the Mn holds a m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='501 µB·Mn−1 and the Ni is m = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='080 µB·Ni−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' After extracting the Bader charges we obtained values of ZMn = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='981e−, ZNi = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='888e−, ZN = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='957e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As expected, the A+B+X− 3 is kept as Niα+Mnβ+Nδ− 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' It is worth mentioning that the Bader charges seem to underestimate the computed charge in the atomic sites, as observed, for example in Ca3BiN possibly due to the partition- ing methodology and exchange-correlation approach73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Never- theless, it can be concluded that as perovskite and antiperovskite structures are considered, the Ni-site oxidation state is reversed from positive, in the former, to negative in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Moving forward, we applied our analysis to several members of the Mn3BN family in order to extrapolate our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In Table 2 we present our calculations of the Bader charges across several reported Mn-based antiperovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In all the cases, the chiral antiferromagnetic Γ4g ordering was considered as the magnetic ground state in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As expected, the N-sites remain negative with values between −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='0e− to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='6e− whereas the Mn- sites vary from +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='9e− to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='6e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, we followed the trend vertically along the periodic table from the 3d in Ni to 5d in Pt and horizontally from Ni to Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' We found an increase of the charge at the B-site, from Ni to Pt, suggesting an increase of the negative oxidation state, for example, β ∼ 1− in Ni to β ∼ 2− in Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In the Mn3IrN case, the Bader charge is close to the value observed for the Pt site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Interestingly, the received charge could be possibly located at the open s- and d-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' On the contrary, the charge decreases from Ni to Ga possibly due to the tendency, in this case, to keep the outer electronic shells closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' As for the magnetic moment value, per Mn site, remains in values between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='5 µB·Mn−1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content='8 µB·Mn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' In contrast, we observed that the charge at the B-sites decreases when going from Ni to Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' This can be explained in terms of the closing electronic shell limiting the space for acquired charge and therefore, diminishing the pos- sible negative oxidation state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Although we are aware of the underestimated charge obtained through the Bader charges approach, and fluctuations in the val- ues of the charges could be expected due to the partitioning method employed68,73, our findings consistently suggest negative oxidation states in metals when located at the antiperovskite’s corner B-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Finally, it is worth noticing that as the spin-orbit coupling, SOC, increases with the oxidation state74, the B-site’s oxidation is paramount to understanding its contribution to the anomalous Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Thus, in the case of the Mn3BN compounds, the SOC possibly decreases due to the negative ox- idation state in the B-site when compared with perovskite com- pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 4 Conclusions: In this paper, we have studied the electronic structure of the Mn3NiN and Mn3Ni magnetically chiral noncollinear antiferro- magnetic antiperovskites by utilizing first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' We found that the N-site expands the cell when it is located at the center of the octahedral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Nonetheless, due to the center position, the symmetry operations and expected properties are conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' We observed a tangible increase of the available electronic states close to the Fermi level that favors the conductivity, as in the case of the anomalous Hall effect in which, σxy = 139 S·cm−1 (σ111 = 241 S·cm−1) in absence of the N-site in contrast to σxy = 78 S·cm−1 (σ111 = 135 S·cm−1) in the Mn3NiN counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Our findings suggest that the nitrogen inclusion in the Mn3NiN sys- tem enhances a positive oxidation state, possibly ∼1+, in the Mn whereas, and more interestingly, the Ni sites hold a negative, po- tentially ∼1−, oxidation state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' This behavior is observed although the overall electronic structure remains metallic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Finally, our find- ings also suggest that several transition metals may exhibit neg- ative oxidation states when located at the B-site in the Mn3BN antiperovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' We thus hope that our result will motivate further studies in antiperovskite structures that might be ideal candidates to further investigate the negative oxidation states in metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Author contributions All of the authors were involved in the preparation and develop- ment of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Moreover, all of the authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Conflict of interest The authors declare no personnel or financial conflict of interests with any person or organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Acknowledgments: The calculations presented in this paper were carried out using the Grid UIS-2 experimental testbed, being developed under the Universidad Industrial de Santander (SC3-UIS) High Performance and Scientific Computing Centre, development action with sup- Journal Name, [year], [vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' ], 1–7 | 5 port from UIS Vicerrectoría de Investigación y Extension (VIE- UIS) and several UIS research groups as well as other funding resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Additionally, we acknowledge the computational sup- port extended to us by Laboratorio de Supercomputo del Sureste de México (LNS), Benemérita Universidad Autónoma de Puebla, BUAP, for performing heavy theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Garcia- Castro acknowledge the grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 2677 entitled “Quiralidad y Or- denamiento Magnético en Sistemas Cristalinos: Estudio Teórico desde Primeros Principios” supported by the VIE – UIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Notes and references 1 G.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=', 1965, 140, A1133– A1138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' 41 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Kresse and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Furthmüller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E2T4oBgHgl3EQf-wmW/content/2301.04242v1.pdf'} 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sha256:57b2c531e3f67d440f69737436c22929922b46e3c45c70939334776c8ca6b4d8 +size 93071 diff --git a/J9FOT4oBgHgl3EQfyjQh/content/tmp_files/2301.12928v1.pdf.txt b/J9FOT4oBgHgl3EQfyjQh/content/tmp_files/2301.12928v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b91eb6b0bf70ab8b39d0c84639fef6311b65e568 --- /dev/null +++ b/J9FOT4oBgHgl3EQfyjQh/content/tmp_files/2301.12928v1.pdf.txt @@ -0,0 +1,1249 @@ +arXiv:2301.12928v1 [math.RA] 27 Dec 2022 +Mock-Lie bialgebras and mock-Lie analogue of the classical +Yang-Baxter equation +K. Benali1 *, T. Chtioui1 †, A. Hajjaji1 ‡, and S. Mabrouk2 § +1. University of Sfax, Faculty of Sciences, BP 1171, 3038 Sfax, Tunisia. +2. University of Gafsa, Faculty of Sciences, 2112 Gafsa, Tunisia. +Abstract +The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent +to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie +bialgebra leads to the introduction the mock-Lie Yang-Baxter equation on a mock-Lie algebra which +is an analogue of the classical Yang-Baxter equation on a Lie algebra. Note that a skew-symmetric +solution of mock-Lie Yang-Baxter equation gives a mock-Lie bialgebra. Finally, the notation of +O-operators are studied to construct skew-symmetric solution of mock-Lie Yang-Baxter equation. +Key words: mock-Lie algebra, mock-Lie bialgebra, Matched pair, Manin triple, mock-Lie Yang-Baxter +equation, O-operators. +M. S. C (2020):16W10, 16T10, 16T15, 16T25, 17B38. +Contents +1 +Introduction +1 +2 +Preliminaries +3 +3 +Matched pairs, Manin triples and mock-Lie bialgebras +5 +4 +Coboundary mock-Lie bialgebras and the mock-Lie Yang-Baxter equation +9 +5 +O-operators of mock-Lie algebras and mock-Lie Yang-Baxter equation +14 +1 +Introduction +A while ago, a new class of algebras emerged in the literature the so called mock-Lie algebras. These +are commutative algebras satisfying the Jacobi identity. They were appeared for the first time in [16] and +since then a lot of works are done on this subject, note for example [23, 35]. These algebras live a dual +life: as member of a very particular class of Jordan algebras and as strange cousins of Lie algebras. +The theory of Lie bialgebra and Poisson Lie groups dates back to the early 80s. +*E-mail: karimabenali172@yahoo.fr +†E-mail: chtioui.taoufik@yahoo.fr +‡E-mail: atefhajjaji100@gmail.com +§E-mail: mabrouksami00@yahoo.fr (Corresponding author) +1 + +Poisson Lie groups are Lie groups equipped with an additional structure, a Poisson bracket satisfy- +ing a compatibility condition with the group multiplication. The infinitesimal object associated with a +poisson Lie group is the tangent vector space at the origin of the group, which is in a naturel way a Lie +algebra g, see for instance [14, 30].The Poisson structure on the group induces on the Lie algebra an +additional structure which is nothing but a Lie algebra structure on the dual vector space g∗ satisfying a +compatibility condition with the Lie bracket on g itself. Such a Lie algebra together with its additional +structure is called a Lie bialgebra. So a bialgebra structure on a given algebra is obtained by a corre- +sponding set of comultiplication together with the set of compatibility conditions between multiplication +and comultiplication [7]. For example take a finite dimensional vector space V with a given algebraic +structure, this can be acheived by equipping the dual space V ∗ with the same algebraic structure and a set +of compatibility conditions between the structures on V and those on V ∗. Among the well-known bial- +gebra structures, we have the associative bialgebra and infinitesimal bialgebra introduced in [1,25]. Note +that these two structures have the same associative multiplications on V and V ∗. They are distinguished +only by the compatibility conditions, with the comultiplication acting as an homomorphism (respectively +a derivation) on the multiplication for the associative bialgebra (respectively the infinitesimal bialgebra). +In general, it is quite common to have multiple bialgebra structures that differ only by their compati- +bility conditions. A good compatibility condition is prescribed on one hand by a strong motivation and +potential applications, and on the other hand by a rich structure theory and effective constructions. See +also [4,11,15,17–21,27,28,32–34] for more details. +One reason for the usefulness of the Lie bialgebra is that it has a coboundary theory, which leads to +the construction of Lie bialgebras from solutions of the classical Yang-Baxter equations. The origin of +the Yang-Baxter-equations is purely physics. They were first introduced by Baxter, McGuire, and Yang +in [12, 13, 31]. Later on, this equation attracts the attention of scientists and becomes one of the most +basic equation in mathematical physics [6,8]. Namely it plays a crucial role for introducing the theory of +quantum groups. This exceptional importance can be seen in many other domaines like: quantum groups, +knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative +descent theory, quantum computing, non-commutative geometry, etc. The various forms of the Yang- +Baxter-equation and some of their uses in physics are summarized in [29]. Many scientists have found +solutions for the Yang-Baxter equation, however the full classification of its solutions remains an open +problem. In the theory of Lie bialgebras, it is essential to consider the coboundary case, which is related +to the theory of the classical Yang-Baxter equation [3,5,7,24]. We aim to have an anlogue in th mock-Lie +case. +This paper is organized as follows: In Section 2 we recall some basic definitions and constructions +about mock-Lie algebras. Section 3 deals with matched pairs, manin triple and mock-Lie bialgebras. +In Section 4, we introduce and develop the notion of coboundary mock-Lie bialgebras and the mock- +Lie Yang-Baxter equation. In Section 5, we give the O-operators of mock-Lie algebras and construct a +solution mock-Lie Yang-Baxter equation. +Unless otherwise specified, all the vector spaces and algebras are finite dimensional over a field K of +characteristic zero. +Notations. Let V and W be two vector spaces: +1. Denote by τ : V ⊗ W → W ⊗ V the switch isomorphism, τ(v ⊗ w) = w ⊗ v. +2. For a linear map ∆ : V → ⊗2V , we use Sweedler’s notation ∆(x) = � +(x) x1 ⊗ x2 for x ∈ V . +We will often omit the summation sign � +(x) to simplify the notations. +3. Denote by V ∗ = Hom(V, K) the linear dual of V . For ϕ ∈ V ∗ and u ∈ V , we write ⟨ϕ, u⟩ := +ϕ(u) ∈ K. +2 + +4. For a linear map φ : V → W, we define the map φ∗ : W ∗ → V ∗ by +⟨φ∗(ξ), v⟩ = ⟨ξ, φ(v)⟩, ∀v ∈ V, ξ ∈ W ∗. +(1.1) +5. For an element x in a mock-Lie algebra (A, •) and n ≥ 2, define the adjoint map L(x) : ⊗nA → +⊗nA by +L(x)(y1 ⊗ · · · ⊗ yn) = +n +� +i=1 +y1 ⊗ · · · ⊗ yi−1 ⊗ x • yi ⊗ yi+1 ⊗ · · · ⊗ yn +(1.2) +for all y1, . . . , yn ∈ A. Conversely, given Y = y1 ⊗ · · · ⊗ yn, we define L(Y ) : g → ⊗ng by +L(Y )(x) = L(x)(Y ), for x ∈ g. +2 +Preliminaries +In this section, we provide some preliminaries about mock-Lie algebras and left mock-pre-Lie alge- +bras. Our main references are [2,9,16]. +Definition 2.1. A mock-Lie algebra is a pair (A, •) consisting of a vector space A together with a +multiplication • : A ⊗ A → A satisfying +x • y = y • x, +(Commutativity), +(2.1) +x • (y • z) + y • (z • x) + z • (x • y) = 0, +(Jacobi identity), +(2.2) +for any x, y, z ∈ A. The Jacobi identity (2.2) is equivalent to +x • (y • z) = −(x • y) • z − y • (x • z). +(2.3) +In other words, the left multiplication L : A → End(A) defined by L(x)y = x • y, is anti-derivation on +A. Recall that a linear map D : A → A is called anti-derivation if for all x, y ∈ A, +D(x • y) = −D(x) • y − x • D(y). +Example 2.1. Let A be a 4-dimensional vector space with basis B = {e1, e2, e3, e4}. Then (A, •) is a +mock-Lie algebra where the product • is defined on the basis B by e1 • e1 = e2, e1 • e3 = e4. +Example 2.2. Recall that an anti-associative algebra is a pair (A, ⋆) consisting of a vector space A +together with a product ⋆ : A ⊗ A → A such that the anti-associator vanishes, i.e, +Aass(x, y, z) := (x ⋆ y) ⋆ z + x ⋆ (y ⋆ z) = 0, +∀x, y, z ∈ A. +(2.4) +Let (A, ⋆) be an anti-associative algebra. Then, (A, •) is a mock-Lie algebra, where x • y := x ⋆ y + y ⋆ +x, ∀x, y ∈ A. +Now, we recall the definition of representations of a mock-Lie algebra. +Definition 2.2. A representation of a mock-Lie algebra (A, •) is a pair (V, ρ) where V is a vector space +and ρ : A → End(V ) is a linear map such that the following equality holds, for all x, y ∈ A, +ρ(x • y) = −ρ(x)ρ(y) − ρ(y)ρ(x). +(2.5) +3 + +Example 2.3. Let (A, •) be a mock-Lie algebra. Then (A, L) is a representation of A on itself called +the adjoint representation. +An equivalent characterisation of representations on mock-Lie algebras is given in the following. +Proposition 2.4. Let (A, •) be a mock-Lie algebra, V be a vector space and ρ : A → End(V ) a +linear map. Then (V, ρ) be a representation of A if and only if the direct sum A ⊕ V together with the +multiplication defined by +(x + u) •A⊕V (y + v) = x • y + ρ(x)v + ρ(y)u, ∀x, y ∈ A, ∀u, v ∈ V, +(2.6) +is a mock-Lie algebra. This mock-Lie algebra is called the semi-direct product of A and V and it is +denoted by A ⋉ρ V . +Definition 2.3. Let (A, •) be a mock-Lie algebra and two representations (V1, ρ1) and (V2, ρ2). A linear +map φ : V1 → V2 is said to be a morphism of representations if +ρ2(x) ◦ φ = φ ◦ ρ1(x), ∀x ∈ A. +(2.7) +If φ is bejictive, then (V1, ρ1) and (V2, ρ2) are equivalent (isomorphic). +To relate matched pairs of mock-Lie algebras to mock-Lie bialgebras and Manin triples for mock- +Lie algebras in the next section, we need the notions of the coadjoint representation, which is the dual +representation of the adjoint representation. In the following we recall these facts. +Let (A, •) be a mock-Lie algebra and (V, ρ) be a representation of A. Let V ∗ be the dual vector +space of V . Define the linear map ρ∗ : A → End(V ∗) as +⟨ρ∗(x)u∗, v⟩ = ⟨u∗, ρ(x)v⟩, +∀x ∈ A, v ∈ V, u∗ ∈ V ∗, +(2.8) +where ⟨·, ·⟩ is the usual pairing between V and the dual space V ∗. With the above notations, we have the +following +Proposition 2.5. Let (V, ρ) be a representation of a mock-Lie algebra (A, •). Then (V ∗, ρ∗) is a repre- +sentation of A on V ∗. +Consider the case when V = A and define the linear map L∗ : A → End(A∗) by +⟨L∗(x)(ξ), y⟩ = ⟨ξ, L(x)y⟩, +∀x, y ∈ A, ξ ∈ A∗. +(2.9) +Then we have the following: +Corollary 2.6. Let (A, •) be a mock-Lie algebra and (A, L) be the adjoint representation of A. Then +(A∗, L∗) is a representation of (A, •) on A∗ which is called the coadjoint representation. +If there is a mock-Lie algebra structure on the dual space A∗, we denote the left multiplication by L. +Definition 2.4. Let (A, •) be a mock-Lie algebra and (V, ρ) be a representation. A linear map T : V → +A is called an O-operator associated to(V, ρ) if T satisfies +T(u) • T(v) = T +� +ρ(Tu)v + ρ(Tv)u +� +, +∀u, v ∈ V. +(2.10) +In the case (V, ρ) = (A, L), the O-operator T is called a Rota-Baxter operator (of weight zero). +4 + +Definition 2.5. A mock-pre-Lie algebra is a vector space A equipped with a linear map · : A ⊗ A → A +satisfying the following identity +Aass(x, y, z) = −Aass(y, x, z), +∀x, y, z ∈ A, +(2.11) +Recall that Aass(x, y, z) = (x · y) · z + x · (y · z). Therefore, Eq. (2.11) is equivalent to +(x ⋆ y) · y = (x · y) · z − y · (x · z), +where x ⋆ y = x · y + y · x, for all x, y ∈ A. +Note that if (A, ·) is a mock-pre-Lie algebra, then the product given by +x ⋆ y = x · y + y · x, +∀x, y ∈ A, +(2.12) +defines a mock-Lie algebra structure, which is called the sub-adjacent mock-Lie algebra of (A, ·), and +denoted by Aac. Furthermore, (A, ·) is called the compatible mock-pre-Lie algebra structure on Aac. +On the other hand, let Θ : A → End(A) defined by Θ(x)y = x · y, ∀x, y ∈ A. Then (A, Θ) is a +representation of the mock-Lie algebra Aac. +Proposition 2.7. Let (A, •) be a mock-Lie algebra and (V, ρ) be a representation of A. If T is an +O-operator associated to (V, ρ), then (V, ·) is a mock-pre-Lie algebra, where +u · v = ρ(Tu)v, +∀u, v ∈ V. +(2.13) +Proposition 2.8. Let (A, •) be a mock-Lie algebra. Then there is a compatible mock-pre-Lie algebra +if and only if there exists an invertible O-operator T : V → A associated to a representation (V, ρ). +Furthermore, the compatible mock-pre-Lie structure on A is given by +x · y = T +� +ρ(x)T −1(y) +� +, +∀x, y ∈ A. +(2.14) +3 +Matched pairs, Manin triples and mock-Lie bialgebras +In this section, we introduce the notions of Manin triple of a mock-Lie algebra and mock-Lie bialge- +bras. The equivalence between them is interpreted in terms of matched pairs of mock-Lie algebras. +We first recall the notion of matched pairs of mock-Lie algebras (see [22]). Let (A, •) and (H, ⋄) be +two mock-Lie algebras. Let ρ : A → End(H) and µ : H → End(A) be two linear maps. On the direct +sum A ⊕ H of the underlying vector spaces, define a linear map ◦ : ⊗2(A ⊕ H) → A ⊕ H by +(x + a) ◦ (y + b) = x • y + µ(b)x + µ(a)y + a ⋄ b + ρ(y)a + ρ(x)b, ∀x, y ∈ A, a, b ∈ H. +(3.1) +Theorem 3.1. Let (A, •) and (H, ⋄) be two mock-Lie algebras. Then (A ⊕ H, ◦) is a mock-Lie algebra +if and only if (H, ρ) and (A, µ) are representations of (A, •) and (H, ⋄) respectively, and for all x, y ∈ +A, a, b ∈ H, the following compatibility conditions are satisfied: +ρ(x)(a ⋄ b) + ρ(x)a ⋄ b + a ⋄ ρ(x)b + ρ(µ(a)x)b + ρ(µ(b)x)a = 0, +(3.2) +µ(a)(x • y) + µ(a)x • y + x • µ(a)y + µ(ρ(x)a)y + µ(ρ(y)a)x = 0. +(3.3) +Definition 3.1. A matched pair of mock-Lie algebras is a quadruple (A, H; ρ, µ) consisting of two +mock-Lie algebras (A, •) and (H, ⋄), together with representations ρ : A → End(H) and µ : H → +End(A) respectively, such that the compatibility conditions (3.2) and (3.3) are satisfied. +5 + +Remark 3.1. We denote the mock-Lie algebra defined by Eq. (3.1) by A ⊲⊳ H. It is straightforward to +show that every mock-Lie algebra which is a direct sum of the underlying vector spaces of two mock-Lie +subalgebras can be obtained from a matched pair of mock-Lie algebras as above. +Definition 3.2. A bilinear form ω on a mock-Lie algebra (A, •) is called invariant if it satisfies +ω(x • y, z) = ω(x, y • z), +∀x, y, z ∈ A. +(3.4) +Proposition 3.2. Let (A, •) be a mock-Lie algebra and (A, L) be the adjoint representation of A on +itself. Then (A, L) and (A∗, L∗) are equivalent as representations of the mock-Lie algebra (A, •) if and +only if there exists a nondegenerate symmetric invariant bilinear form ω on A. +Proof. Suppose that there exists a nondegenerate symmetric invariant bilinear form ω on A. Since ω is +nondegenerate, there exists a linear isomorphism φ : A → A∗ defined by +⟨φ(x), y⟩ = ω(x, y), +∀x, y ∈ A. +Hence for any x, y, z ∈ A, we have +⟨φ(L(x)(y)), z⟩ = ω(L(x)(y), z) = ω(x • y, z) = ω(y, x • z) += ⟨φ(y), x • z⟩ = ⟨L∗(x)φ(y), z⟩. +That is, (A, L) and (A∗, L∗) are equivalent. Conversely, by a similar way, we can get the conclusion. +Definition 3.3. A Manin triple of mock-Lie algebras is a triple of mock-Lie algebras (A, A+, A−) to- +gether with a nondegenerate symmetric invariant bilinear form ω on A such that the following conditions +are satisfied: +(a) A+, A− are mock-Lie subalgebras of A. +(b) A = A+ ⊕ A− as vector spaces. +(c) A+ and A− are isotropic with respect to ω, that is, ω(x+, y+) = ω(x−, y−) = 0, for any x+, y+ ∈ +A+, x−, y− ∈ A−. +A homomorphism between two Manin triples of mock-Lie algebras (A, A+, A−) and +(B, B+, B−) associated to two nondegenerate symmetric invariant bilinear forms ω1 and ω2 respectively, +is a homomorphism of mock-Lie algebras f : A → B such that +f(A+) ⊂ B+, +f(A−) ⊂ B−, +ω1(x, y) = ω2(f(x), f(y)), ∀x, y ∈ A. +If in addition, f is an isomorphism of vector spaces, then the two Manin triples are called isomorphic. +Definition 3.4. [22] Let (A, •) be a mock-Lie algebra. Suppose that there is a mock-Lie algebra structure +(A∗, ⋄) on the dual space A∗ of A and there is a mock-Lie algebra structure on the direct sum A ⊕ A∗ +of the underlying vector spaces A and A∗ such that (A, •) and (A∗, ⋄) are subalgebras and the natural +non-degerenate symmetric bilinear form on A ⊕ A∗ given by +ωd(x + ξ, y + η) := ⟨x, η⟩ + ⟨ξ, y⟩, +∀x, y ∈ A, ξ, η ∈ A∗, +(3.5) +is invariant, then (A ⊕ A∗, A, A∗) is called a standard Manin triple of mock-Lie algebra associated to +standard bilinear form ωd. +Obviously, a standard Manin triple of mock-Lie algebras is a Manin triple of mock-Lie algebras. +Conversely, we have +6 + +Proposition 3.3. Every Manin triple of mock-Lie algebras is isomorphic to a standard one. +Proof. Since A+ and A− are isotropic under the nondegenerate invariant bilinear form ω on A+ ⊕ +A−, then in this case A− and (A+)∗ are identified by ω and the mock-Lie algebra structure on A− is +transferred to (A+)∗. Hence the mock-Lie algebra structure on A+ ⊕ A− is transferred to A+ ⊕ (A+)∗. +Transfer the nondegenrate bilinear form ω to A+ ⊕ (A+)∗, we obtain the standard bilinear form given +by (3.5). Thus, (A, A+, A−) is isomorphic to the stansadrd Manin triple (A ⊕ A∗, A, A∗). +Proposition 3.4. +[22] Let (A, •) be a mock-Lie algebra. Suppose that there is a mock-Lie algebra +structure (A∗, ⋄) on A∗. Then there exists a mock-Lie algebra sructure on the vector space A ⊕ A∗ +such that (A ⊕ A∗, A, A∗) is a standard Manin triple of mock-Lie algebras with respect to ωd defined +by (3.5) if and only if (A, A∗; L∗, L∗) is a matched pair of mock-Lie algebras. Here L∗ is the coadjoint +representation of the mock-Lie algebra (A∗, ⋄). +Proposition 3.5. Let (A, •) be a mock-Lie algebra. Suppose that there is a mock-Lie algebra structure +(A∗, ⋄) on A∗. Then (A, A∗; L∗, L∗) is a matched pair of mock-Lie algebras if and only if for any +x, y ∈ A, ξ ∈ A∗, we have +L∗(ξ)(x • y) + (L∗(ξ)(x)) • y + x • (L∗(ξ)(y)) + L∗(L∗(x)(ξ))(y) + L∗(L∗(y)(ξ))(x) = 0. +(3.6) +Proof. Obiviously, Eq.(3.6) is exactly Eq. (3.3) in the case ρ = L∗, µ = L∗. In addition, for any +x, y ∈ A, ξ, η ∈ A∗, we have +⟨L∗(ξ)(x • y), η⟩ = ⟨x • y, L(ξ)(η)⟩ = ⟨L(x)(y), ξ ⋄ η⟩ = ⟨y, L∗(x)(ξ ⋄ η)⟩; +⟨(L∗(ξ)(x)) • y, η⟩ = ⟨L(L∗(ξ)(x))(y), η⟩ = ⟨y, L∗(L∗(ξ)(x))(η)⟩; +⟨x • (L∗(ξ)(y)), η⟩ = ⟨L(x)(L∗(ξ)(y)), η⟩ = ⟨L∗(ξ)(y), L∗(x)(η)⟩ = ⟨y, L(ξ)(L∗(x)(η))⟩ += ⟨y, ξ ⋄ (L∗(x)(η))⟩; +⟨L∗(L∗(x)(ξ))(y), η⟩ = ⟨y, L(L∗(x)(ξ))(η)⟩ = ⟨y, (L∗(x)(ξ)) ⋄ η⟩; +⟨L∗(L∗(y)(ξ))(x), η⟩ = ⟨x, L(L∗(y)(ξ))(η)⟩ = ⟨x, η ⋄ (L∗(y)(ξ))⟩ = ⟨x, L(η)((L∗(y)(ξ))⟩ += ⟨L∗(η)(x), L∗(y)(ξ)⟩ = ⟨L(L∗(η)(x))(y), ξ⟩ = ⟨y, L∗(L∗(η)(x))(ξ)⟩. +Then Eq. (3.2) holds if and only if Eq. (3.3) holds. Therefore the conclusion holds. +Theorem 3.6. Let (A, •) be a mock-Lie algebra. Suppose that there is a mock-Lie algebra structure ”⋄” +on its dual space A∗ given by a linear map ∆∗ : A∗ ⊗ A∗ → A∗, that is, ξ ⋄ η = ∆∗(ξ ⊗ η), for any +ξ, η ∈ A∗. Then (A, A∗; L∗, L∗) is a matched pair of mock-Lie algebras if and only if ∆ : A → A ⊗ A +satisfies the following condition: +∆(x • y) = − +� +L(x) ⊗ id + id ⊗ L(x) +� +∆(y) − +� +L(y) ⊗ id + id ⊗ L(y) +� +∆(x), +(3.7) +for any x, y ∈ A. +Proof. Using Proposition 3.5, we can prove that Eq. (3.7) is equivalent to Eq. (3.6). In fact, for any +x, y ∈ A, ξ, η ∈ A∗, we have +⟨L∗(ξ)(x • y), η⟩ = ⟨x • y, ξ ⋄ η⟩ = ⟨x • y, ∆∗(ξ ⊗ η)⟩ = ⟨∆(x • y), ξ ⊗ η⟩; +⟨(L∗(ξ)(x)) • y, η⟩ = ⟨L(y)(L∗(ξ)(x)), η⟩ = ⟨L∗(ξ)(x), L∗(y)(η)⟩ = ⟨x, L(ξ)(L∗(y)(η))⟩ += ⟨x, ξ ⋄ (L∗(y)(η))⟩ = ⟨(id ⊗ L(y))∆(x), ξ ⊗ η⟩; +⟨x • (L∗(ξ)(y)), η⟩ = ⟨L(x)(L∗(ξ)(y)), η⟩ = ⟨L∗(ξ)(y), L∗(x)(η)⟩ = ⟨y, L(ξ)(L∗(x)(η))⟩ +7 + += ⟨y, ξ ⋄ (L∗(x)(η))⟩ = ⟨(id ⊗ L(x))∆(y), ξ ⊗ η⟩; +⟨L∗(L∗(x)(ξ))(y), η⟩ = ⟨y, (L∗(x)(ξ)) ⋄ η⟩ = ⟨(L(x) ⊗ id)∆(y), ξ ⊗ η⟩; +⟨L∗(L∗(y)(ξ))(x), η⟩ = ⟨x, (L∗(y)(ξ)) ⋄ η⟩ = ⟨(L(y) ⊗ id)∆(x), ξ ⊗ η⟩. +Then Eq. (3.6) is equivalent to Eq. (3.7). Hence the conclusion holds. +Remark 3.2. From the symmetry of the mock-Lie algebras (A, •) and (A∗, ⋄) in the standard Manin +triple of mock-Lie algebras with respect to ωd, we also can consider a linear map γ : A∗ → A∗ ⊗ A∗ +such that γ∗ : A ⊗ A → A gives the mock-Lie algebra structure ” • ” on A. It is straightforward to show +that ∆ satisfies Eq. (3.7) if and only if γ satisfies +γ(ξ ⋄ η) = − +� +L(ξ) ⊗ id + id ⊗ L(ξ) +� +γ(η) − +� +L(η) ⊗ id + id ⊗ L(η) +� +γ(ξ), +(3.8) +for any ξ, η ∈ A∗. +Definition 3.5. Let (A, •) be a mock-Lie algebra. A mock-Lie bialgebra structure on A is a symmetric +linear map ∆ : A → A ⊗ A such that +1. ∆∗ : A∗ ⊗ A∗ → A∗ defines a mock-Lie algebra structure on A∗; +2. ∆ satifies Eq. (3.7), called the compatibility condition. +We denote it by (A, ∆) or (A, A∗). +We can unwrap the compatibility condition Eq. (3.7) as +∆(x • y) = −(x • y1) ⊗ y2 − y1 ⊗ (x • y2) − (y • x1) ⊗ x2 − x1 ⊗ (y • x2). +(3.9) +Remark 3.3. The compatibility condition Eq. (3.7) is, in fact, a cocycle condition in the zigzag cohomol- +ogy of mock-Lie algebra introduced in [10]. Indeed, we can regard A⊗2 as an A-module via the adjoint +action (1.2): +x · (y1 ⊗ y2) = L(x)(y1 ⊗ y2) = (x • y1) ⊗ y2 + y1 ⊗ (x • y2), +(3.10) +for x ∈ A and y1 ⊗ y2 ∈ A⊗2. Then we can think of the linear map ∆ : A → A⊗2 as a 1-cochain. Then +the differential on ∆ is given by +d1∆(x, y) = ∆(x • y) + x · ∆(y) + y · ∆(x) += ∆(x • y) + L(x)(∆(y)) + L(y)(∆(x)). +Therefore, Eq. (3.7) says exactly that ∆ ∈ C1(A, A⊗2) is a 1-cocycle. +Example 3.7. Let (A, A∗) be a mock-Lie bialgebra on a mock-Lie algebra (A, •). Then (A∗, γ)(or(A∗, A)) +is a mock-Lie bialgebra on the mock-Lie algebra (A∗, ⋄), where γ is given in Remark 3.2. +Definition 3.6. Let (A1, A∗ +1) and (A2, A∗ +2) be two mock-Lie bialgebras. A linear map ψ : A1 → A2 is a +homomorphism of mock-Lie bialgebras if ψ satifies, for any x, y ∈ A1 +ψ(x •1 y) = ψ(x) •2 ψ(y), +(ψ ⊗ ψ) ◦ ∆1 = ∆2 ◦ ψ. +(3.11) +Now, combining Proposition 3.4 and Theorem 3.6, we have the following conclusion. +Theorem 3.8. Let (A, •) be a mock-Lie algebra. Suppose that there is a mock-Lie algebra structure on +A∗ denoted by ” ⋄ ” which is defined as a linear map ∆ : A → A ⊗ A. Then the following conditions +are equivalent: +1. (A⊕A∗, A, A∗) is a standard Manin triple of mock-Lie algebras with respect to ωd defined by Eq. +(3.5). +2. (A, A∗; L∗, L∗) is a matched pair of mock-Lie algebras. +3. (A, A∗) is a mock-Lie bialgebra. +8 + +4 +Coboundary mock-Lie bialgebras and the mock-Lie Yang-Baxter equa- +tion +In this section, we consider a special class of mock-Lie bialgebras called coboundary mock-Lie +bialgebras and introduce the notion of mock-Lie Yang-Baxter equation. +Definition 4.1. A mock-Lie bialgebra (A, A∗) is called coboundary if there exists an element r ∈ A⊗A +such that for any x ∈ A, +∆(x) = +� +L(x) ⊗ id − id ⊗ L(x) +� +r. +(4.1) +Lemma 4.1. Let (A, •) be a mock-Lie algebra and r ∈ A ⊗ A. Suppose that the linear map ∆ : A → +A ⊗ A is defined by Eq. (4.1). Then ∆ satisfies the compatibility condition given by Eq. (3.7). +Proof. Let r = r1 ⊗ r2 ∈ A ⊗ A. Using the commutativity and Jacobi identity for mock-Lie algebras. +Then for any x, y ∈ A, we have +− +� +L(x) ⊗ id + id ⊗ L(x) +� +∆(y) − +� +L(y) ⊗ id + id ⊗ L(y) +� +∆(x) += − +� +L(x) ⊗ id + id ⊗ L(x) +�� +L(y) ⊗ id − id ⊗ L(y) +� +r +− +� +L(y) ⊗ id + id ⊗ L(y) +�� +L(x) ⊗ id − id ⊗ L(x) +� +r += − +� +L(x) ⊗ id + id ⊗ L(x) +�� +(y • r1) ⊗ r2 − r1 ⊗ (y • r2) +� +− +� +L(y) ⊗ id + id ⊗ L(y) +�� +(x • r1) ⊗ r2 − r1 ⊗ (x • r2) +� += − +� +x • (y • r1) ⊗ r2 − (x • r1) ⊗ (y • r2) + (y • r1) ⊗ (x • r2) − r1 ⊗ (x • (y • r2)) +� +− +� +y • (x • r1) ⊗ r2 − (y • r1) ⊗ (x • r2) + (x • r1) ⊗ (y • r2) − r1 ⊗ (y • (x • r2)) +� += +� +− x • (y • r1) − y • (x • r1) +� +⊗ r2 + r1 ⊗ +� +x • (y • r2) + y • (x • r2) +� += +� +(x • y) • r1 +� +⊗ r2 − r1 ⊗ +� +(x • y) • r2 +� += +� +L(x • y) ⊗ id − id ⊗ L(x • y) +� +r +=∆(x • y). +Hence the proof. +Let ∆ : A → A ⊗ A be a linear map and σ : A⊗3 → A⊗3 be defined as σ(x ⊗ y ⊗ z) = y ⊗ z ⊗ x, +for any x, y, z ∈ A. Let E∆ : A → A⊗3 be a linear map given by +E∆(x) = (id + σ + σ2) +� +(id ⊗ ∆)∆(x) +� +. +(4.2) +Lemma 4.2. Let A be a vector space and ∆ : A → A ⊗ A be a linear map. Then the product ” ⋄ ” in +A∗ given by ∆∗ : A∗ ⊗ A∗ → A∗ satisfies the Jacobi identity if and only if E∆ = 0. +Proof. For any ξ, η ∈ A∗, x ∈ A, we have +⟨ξ ⋄ η, x⟩ = ⟨∆∗(ξ ⊗ η), x⟩ = ⟨ξ ⊗ η, ∆(x)⟩. +(4.3) +Threrefore, for any ξ, η, ν ∈ A∗ and x ∈ A, the Jacobi identity satisfies +⟨J(ξ, η, ν), x⟩ +=⟨∆∗(id ⊗ ∆∗)(ξ ⊗ η ⊗ ν) + ∆∗(id ⊗ ∆∗)(η ⊗ ν ⊗ ξ) + ∆∗(id ⊗ ∆∗)(ν ⊗ ξ ⊗ η), x⟩ +=⟨∆∗(id ⊗ ∆∗) +� +id + σ + σ2� +(ξ ⊗ η ⊗ ν), x⟩ +=⟨ξ ⊗ η ⊗ ν, +� +id + σ + σ2� +((id ⊗ ∆)∆)(x)⟩. +Therefore J(ξ, η, ν) = 0, for any ξ, η, ν ∈ A∗ if and only if E∆ = 0. +9 + +Let (A, •) be a mock-Lie algebra and r = � +i ai ⊗ bi ∈ A ⊗ A. Set +r12 = +� +i +ai ⊗ bi ⊗ 1, +r13 = +� +i +ai ⊗ 1 ⊗ bi, +r23 = +� +i +1 ⊗ ai ⊗ bi, +(4.4) +where 1 is a unit element if (A, •) is unital or a symbol playing a similar role of the unit for the non-unital +cases. The operation between two rij is in obvious way. For example, +r12 • r13 = +� +ij +ai • aj ⊗ bi ⊗ bj, r13 • r23 = +� +ij +ai ⊗ aj ⊗ bi • bj, r23 • r12 = +� +ij +aj ⊗ ai • bj ⊗ bi. (4.5) +Note that the above elements are independents of the existence of the unit. A tensor r ∈ A ⊗ A +is called symmetric (resp. skew-symmetric) if r = τ(r) ( resp. r = −τ(r)). On the other hand, any +r ∈ A ⊗ A can be identified as a linear map from the dual space A∗ to A in the following way: +⟨ξ, r(η)⟩ = ⟨ξ ⊗ η, r⟩, +∀ξ, η ∈ A∗. +(4.6) +The tensor r ∈ A ⊗ A is called nondegenerate if the above induced linear map is invertible. +Proposition 4.3. Let (A, •) be a mock-Lie algebra. Define a linear map ∆ : A → A ⊗ A by Eq. (4.1) +with some r ∈ A ⊗ A satisfying +� +L(x) ⊗ id − id ⊗ L(x) +�� +r + τ(r) +� += 0, +(4.7) +for all x ∈ A. Then +E∆(x) + Q(x)[[r, r]] = 0, +(4.8) +where +[[r, r]] = r12 • r13 + r13 • r23 − r12 • r23, +(4.9) +and Q(x) = +� +L(x) ⊗ id ⊗ id + id ⊗ L(x) ⊗ id + id ⊗ id ⊗ L(x) +� +for any x ∈ A. +Proof. Let r = � +i ai ⊗ bi, the condition (4.7) is equivalent to +� +i +(x • ai) ⊗ bi − ai ⊗ (x • bi) + (x • bi) ⊗ ai − bi ⊗ (x • ai) = 0. +(4.10) +Note that E∆(x) is the sum of twelve terms and that Q(x)[[r, r]] is a sum of nine terms, but two +terms appear in both sums up to sign and hence are canceled. Thus E∆(x) + Q(x)[[r, r]] is a sum of +seventeen terms. After rearranging the terms suitably, we obtain +E∆(x) + Q(x)[[r, r]] += +� +i,j +� +− (x • bi) • aj ⊗ bj ⊗ ai + x • (ai • aj) ⊗ bi ⊗ bj + (x • bi) • bj ⊗ ai ⊗ aj +− (bi • bj) ⊗ (x • ai) ⊗ aj + (ai • aj) ⊗ (x • bi) ⊗ bj + (bi • aj) ⊗ bj ⊗ (x • ai) ++ (ai • aj) ⊗ bi ⊗ (x • bj) − ai ⊗ (x • bi) • aj ⊗ bj + ai ⊗ aj ⊗ (x • bi) • bj ++ bj ⊗ (x • ai) ⊗ (bi ⊗ aj) − bj ⊗ ai ⊗ (x • bi) • aj − aj ⊗ (bi • bj) ⊗ (x • ai) ++ aj ⊗ (x • bi) • bj ⊗ ai − ai ⊗ x • (bi • aj) ⊗ bj − ai ⊗ (bi • aj) ⊗ (x • bj) ++ ai ⊗ (x • aj) ⊗ (bi • bj) + ai ⊗ aj ⊗ x • (bi • bj) +� +. +Interchanging the indices i and j in the first term and using the Jacobi identity in A, the first term becomes +� +i,j +x • (bj • ai) ⊗ bi ⊗ aj + bj • (ai • x) ⊗ bi ⊗ aj. +10 + +Using the Eq. (4.10), the sum of bj • (ai • x) ⊗ bi ⊗ aj and the third and fourth terms is +� +i,j +� +L(bj) ⊗ id +�� +(ai • x) ⊗ bi + (x • bi) ⊗ ai − bi ⊗ (x • ai) +� +⊗ aj +� +j +� +L(bj) ⊗ id +� � +i +� +(ai • x) ⊗ bi + (x • bi) ⊗ ai − bi ⊗ (x • ai) +� +⊗ aj += +� +i,j +� +L(bj) ⊗ id +�� +ai ⊗ (x • bi) +� +⊗ aj += +� +i,j +(ai • bj) ⊗ (x • bi) ⊗ aj. +Similarly, the sum of (ai • bj) ⊗ (x • bi) ⊗ aj and the fifth term becomes +� +i,j +bj ⊗ (x • bi) ⊗ (ai • aj) + aj ⊗ (x • bi) ⊗ (ai • bj), +and the sum of the sixth and seventh terms is +� +i,j +aj ⊗ bi ⊗ x • (ai • bj) + bj ⊗ bi ⊗ x • (ai • aj). +Finally, the sum of x • (bj • ai) ⊗ bi ⊗ aj and the second term in the sum of the expression of E∆(x) + +Q(x)[[r, r]] becomes +� +i,j +bj ⊗ bi ⊗ ai • (x • aj) + aj ⊗ bi ⊗ ai • (x • bj). +Inserting these results, we find that the expression of E∆(x) + Q(x)[[r, r]] can be written in the form +� +i +� +ai ⊗ Ui + bi ⊗ Vi +� +: In fact, +Ui = +� +j +� +− (bj • bi) ⊗ (x • aj) − (bi • aj) ⊗ (x • bj) + bj ⊗ x • (aj • bi) ++ aj ⊗ x • (bi • bj) + (x • bj) ⊗ (aj • bi) + (x • aj) ⊗ (bi • bj) +− x • (bi • aj) ⊗ bj + (x • bj) • bi ⊗ aj − (x • bi) • aj ⊗ bj ++ aj ⊗ (x • bi) • bj + bj ⊗ (x • bi) • aj +� +. +On the right-hand side, the sum of the first four terms is zero by Eq. (4.10), and the sum of the next three +terms becomes +x • (bi • bj) ⊗ aj. +By the Jacobi identity in A, the sum of x • (bi • bj) ⊗ aj and the eighth term is +−bj • (x • bi) ⊗ aj. +Finally, the sum of −bj • (x • bi) ⊗ aj and the last three terms becomes +� +j +−bj • (x • bi) ⊗ aj − (x • bi) • aj ⊗ bj + aj ⊗ (x • bi) • bj + bj ⊗ (x • bi) • aj = 0, +if we replace x in Eq. (4.10) by x • bi: Hence, we get Ui = 0. Similarly, we can proves that +Vi = +� +j +� +(x • bj) ⊗ (aj • ai) + bj ⊗ x • (aj • ai) + bj ⊗ aj • (x • ai) +11 + ++ (x • aj) ⊗ (bj • ai) − aj ⊗ (x • bj) • ai +� +=0. +Hence the conclusion holds. +Using the above discussion, we have the following result. +Theorem 4.4. Let (A, •) be a mock-Lie algebra and r ∈ A⊗A. Define a bilinear map ⋄ : A∗⊗A∗ → A∗ +by +⟨ξ ⋄ η, x⟩ = ⟨∆∗(ξ ⊗ η), x⟩ = ⟨ξ ⊗ η, ∆(x)⟩, +where ∆ is defined by Eq. (4.1). Then (A∗, ⋄) is a mock-Lie algebra if and only if the following conditions +are satisfied: +(i) +� +L(x) ⊗ id − id ⊗ L(x) +�� +r + τ(r) +� += 0, +(ii) +Q(x)[[r, r]] = 0, +for all x ∈ A. Under these conditions, (A, A∗) is a coboundary mock-Lie bialgebra. +Proof. The bracket ⋄ is determined by the cobracket ∆(x) = +� +L(x) ⊗ id − id ⊗ L(x) +� +r. Hence (A∗, ⋄) +is a mock-Lie algebra if and only if ⋄ is symmetric and satisfies the Jacobi identity. +For any x ∈ A, ξ, η ∈ A∗, we have +⟨ξ ⋄ η − η ⋄ ξ, x⟩ = ⟨∆∗(ξ ⊗ η) − ∆∗(η ⊗ ξ), x⟩ = ⟨ξ ⊗ η, ∆(x) − τ ◦ ∆(x)⟩ += ⟨ξ ⊗ η, +� +L(x) ⊗ id − id ⊗ L(x) +� +r − τ ◦ ( +� +L(x) ⊗ id − id ⊗ L(x) +� +r)⟩ += ⟨ξ ⊗ η, L(x)r1 ⊗ r2 − r1 ⊗ L(x)r2 − r2 ⊗ L(x)r1 + L(x)r2 ⊗ r1⟩ += ⟨ξ ⊗ η, +� +L(x) ⊗ id − id ⊗ L(x) +�� +r + τ(r) +� +⟩. +Then r satisfies (i) if and only if ⋄ is symmetric. The proof that (ii) is equivalent to the condition +that ⋄ satisfies the Jacobi identity which follows from Lemma 4.2 and Proposition 4.3. Since ∆(x) = +� +L(x) ⊗ id − id ⊗ L(x) +� +r, the compatibility conditions for a mock-Lie bialgebra in Definition 3.5 hold +naturally. Therefore the conclusion follows. +Remark 4.1. An easy way to satisfy conditions (i) and (ii) in Theorem 4.4 is to assume that r is skew- +symmetric and +[[r, r]] = 0 +(4.11) +respectively. Eq. (4.11) is the mock-Lie Yang-Baxter equation in the mock-Lie algebra (A, •). A +quasitriangular mock-Lie bialgebra is a coboundary mock-Lie bialgebra, in which r is a solution of the +mock-Lie Yang- Baxter equation. A triangular mock-Lie bialgebra is a coboundary mock-Lie bialgebra, +in which r is a skew-symmetric solution of the mock-Lie Yang-Baxter equation. +A direct application of Theorem 4.4 is given as follows. +Theorem 4.5. Let (A, A∗) be a mock-Lie bialgebra. Then there is a canonical coboundary mock-Lie +bialgebra structure on A ⊕ A∗ such that both i1 : A → A ⊕ A∗ and i2 : A∗ → A ⊕ A∗ into the two +summands are homomorphisms of mock-Lie bialgebras. Here the mock-Lie bialgebra structure on A is +(A, −∆A) , where ∆A is given by Eq. (4.1). +Proof. Let r ∈ A ⊗ A∗ ⊂ (A ⊕ A∗) ⊗ (A ⊕ A∗) correspond to the identity map id : A → A. Let +{e1, · · · , en} be a basis of A and {f1, · · · , fn} be its dual basis. Then r = � +i ei ⊗ fi. Suppose that the +12 + +mock-Lie algebra structure ” ◦D(A) ” on A ⊕ A∗ is given by D(A) = A ⊲⊳ A∗. Then by Eq. (3.1), we +have +x ◦D(A) y = x • y, +a∗ ◦D(A) b∗ = a∗ ⋄ b∗, +x ◦D(A) a∗ = L∗(x)a∗ + L∗(a∗)x, +for any x, y ∈ A, a∗, b∗ ∈ A∗. Next we prove that r satisfies the two conditions in Theorem 4.4. If so, +then +∆D(A)(u) = (L◦D(A)(u) ⊗ idD(A) − idD(A) ⊗ L◦D(A)(u))r, +∀u ∈ D(A) +can induce a coboundary mock-Lie bialgebra structure on D(A). Since +⟨ +� +i +ei ⊗ fi, fs ⊗ et⟩ = ⟨et, fs⟩, +we have +⟨[[r, r]]D(A), (es + ft) ⊗ (ek + fl) ⊗ (ep + fq)⟩ += +� +ij +⟨ei ◦D(A) ej ⊗ fi ⊗ fj − ei ⊗ fi ◦D(A) ej ⊗ fj + ei ⊗ ej ⊗ fi ◦D(A) fj, (es + ft) ⊗ (ek + fl) ⊗ (ep + fq)⟩ += +� +ij +⟨ei • ej ⊗ fi ⊗ fj − ei ⊗ (L∗(fi)ej + L∗(ej)fi) ⊗ fj + ei ⊗ ej ⊗ fi ⋄ fj, (es + ft) ⊗ (ek + fl) ⊗ (ep + fq)⟩ += +� +ij +� +⟨ei • ej, ft⟩⟨fi, ek⟩⟨fj, ep⟩ − ⟨ei, ft⟩⟨ej, fi ⋄ fl⟩⟨fj, ep⟩ − ⟨ei, ft⟩⟨fi, ej • ek⟩⟨fj, ep⟩ ++ ⟨ei, ft⟩⟨ej, fl⟩⟨fi ⋄ fj, ep⟩ +� +=⟨ek • ep, ft⟩ − ⟨ep, ft ⋄ fl⟩ − ⟨ft, ep • ek⟩ + ⟨ft ⋄ fl, ep⟩ +=0, +we get [[r, r]]D(A) = 0. Similarly, we prove that +� +L◦D(A)(u) ⊗ idD(A) − idD(A) ⊗ L◦D(A)(u) +�� +r + τ(r) +� += 0, +for all u ∈ D(A). Hence there is a coboundary mock-Lie bialgebra structure on D(A) by Theorem 4.4. For +ei ∈ A, we have +∆D(A)(ei) = +� +j +� +ei • ej ⊗ fj − ej ⊗ ei ◦D(A) fj +� += +� +j +� +ei • ej ⊗ fj − ej ⊗ +� +L∗(ei)fj + L∗(fj)ei +�� += +� +j,m +� +ei • ej ⊗ fj − ⟨fj, ei • em⟩ej ⊗ fm − ⟨fj ⋄ fm, ei⟩ej ⊗ em +� += − +� +j,m +⟨fj ⋄ fm, ei⟩ej ⊗ em += −∆A(ei). +Therefore i1 : A → A ⊕ A∗ is a homomorphism of mock-Lie bialgebras. Similarly, i2 : A∗ → A ⊕ A∗ is also a +homomorphism of mock-Lie bialgebras since ∆D(A)(fi) = ∆A∗(fi). +Remark 4.2. With the above mock-Lie bialgebra structure given in Theorem 4.5, A ⊕ A∗ is called the +double of A. We denote it by D(A). +13 + +5 +O-operators of mock-Lie algebras and mock-Lie Yang-Baxter equation +In this section, we interpret solution of mock-Lie Yang-Baxter equation in term of O-operators (see +[26]). Let V be a vector space. For any r ∈ V ⊗ V , r can be regarded as a map from V ∗ to V in the +following way: +⟨u∗, r(v∗)⟩ = ⟨u∗ ⊗ v∗, r⟩, +∀u∗, v∗ ∈ V ∗, +(5.1) +where ⟨·, ·⟩ is the ordinary pairing between the vector space V and the dual space V ∗. The tensor +r ∈ V ⊗V is called nondegenerate if the above induced linear map is invertible. Moreover, any invertible +linear map T : V ∗ → V induces a nondegenerate bilinear form ω(, ) on V by +ω(u, v) = ⟨T −1(u), v⟩, +∀u, v ∈ V. +(5.2) +Definition 5.1. [9] A symplectic form on a mock-Lie algebra (A, ·) is a skew-symmetric non-degenerate +bilinear form ω satisfying +ω(x • y, z) + ω(y • z, x) + ω(z • x, y) = 0, +∀x, y, z ∈ A. +(5.3) +A mock-Lie algebra is called symplectic if it is endowed with a symplectic form. +Proposition 5.1. Let (A, •) be a mock-Lie algebra and r ∈ A ⊗ A be skew-symmetric. Then r is a +solution of mock-Lie YBE in A if and only if r satisfies +r(ξ) • r(η) = r +� +L∗(r(ξ))η + L∗(r(η))ξ +� +, +∀ξ, η ∈ A∗. +(5.4) +Proof. Let {e1, · · · , en} be a basis of A and {e∗ +1, · · · , e∗ +n} be the dual basis. Since r is skew-symmetric, +we can set r = � +1≤i,j≤n aijei ⊗ ej, aij = −aji. Suppose that ei • ej = �n +k=1 Ck +ijek, where Ck +ij’s are +the structure coefficients of mock-Lie algebra A on the basis {e1, · · · , en}, then we get +r12 • r13 = +� +� +1≤i,j≤n +aijei ⊗ ej ⊗ 1 +� +• +� +� +1≤p,q≤n +apqep ⊗ 1 ⊗ eq +� += +� +1≤i,j,p,q,k≤n +Ck +ipaijapqek ⊗ ej ⊗ eq; +r13 • r23 = +� +1≤i,j,p,q,k≤n +Ck +jqaijapqei ⊗ ep ⊗ ek; +r12 • r23 = +� +1≤i,j,p,q,k≤n +Ck +jpaijapqei ⊗ ek ⊗ eq. +Then r is a solution of mock-Lie YBE in A if and only if +� +1≤i,p≤n +� +Ck +ipaijapq + Cq +piakpaji − Cj +ipakiapq +� +ek ⊗ ej ⊗ eq. +On the other hand, by Eq. (5.1), we get r(e∗ +j) = �n +i=1 aijei = − �n +i=1 ajiei, 1 ≤ j ≤ n. If we take +ξ = e∗ +j, η = e∗ +q and by Eq. (5.4), we get +� +1≤i,p≤n +� +Ck +ipaijapq + Cq +piakpaji − Cj +ipakiapq +� +ek = 0. +Therefore, it is easy to see that r is a solution of mock-Lie YBE in A if and only if r satisfies Eq. +(5.4). +14 + +Example 5.2. Let (A, •) be a mock-Lie algebra. Then a Rota-Baxter operator (of weight zero) is an O- +operator of A associated to the adjoint representation (A, L) and a skew-symmetric solution of mock-Lie +YBE in A is an O-operator of A associated to the representation (A∗, L∗). +Corollary 5.3. Let (A, •) be a mock-Lie algebra and r ∈ A ⊗ A be skew-symmetric. Suppose that there +is a symmetric nondegenerate invariant bilinear form ω on A. Let φ : A → A∗ a linear map given +by ⟨φ(x), y⟩ = ω(x, y) for any x, y ∈ A. Then r is a solution of mock-Lie YBE if and only if rφ is a +Rota-Baxter operator (of weight zero) on A. +Proof. For any x, y ∈ A, we have φ(L(x)y) = L∗(x)φ(y) since +⟨φ(L(x)y), z⟩ = ω(x • y, z) = ω(y • x, z) += ω(y, x • z) = ⟨L∗(x)φ(y), z⟩, +∀x, y, z ∈ A. +That is, the representations (A, L) and (A∗, L∗) are isomorphic. Let ξ = φ(x), η = φ(y), then by +Proposition 5.1, r is a solution of mock-Lie YBE in A if and only if +rφ(x) • rφ(y) = r(ξ) • r(η) = r +� +L∗(r(ξ))η + L∗(r(η))ξ +� += rφ +� +rφ(x) • y + x • rφ(y) +� +. +Therefore the conclusion holds. +Let (A, •) be a mock-Lie algebra. Let (V, ρ) be a representation of A and ρ∗ : A → gl(V ∗) be +the dual representation. A linear map T : V → A can be identified as an element in A ⊗ V ∗ ⊂ +(A ⋉ρ∗ V ∗) ⊗ (A ⋉ρ∗ V ∗) as follows: Let {e1, · · · , en} be a basis of A, let {v1, · · · , vm} be a basis of +V and {v∗ +1, · · · , v∗ +m} be its dual space of V ∗. We set +T(vi) = +n +� +k=1 +aikek, i = 1, · · · , m. +Since as vector space, Hom(V, A) ∼= A ⊗ V ∗, then we have +T = +m +� +i=1 +T(vi) ⊗ v∗ +i = +m +� +i=1 +n +� +k=1 +aikek ⊗ v∗ +i ∈ A ⊗ V ∗ ⊂ (A ⋉ρ∗ V ∗) ⊗ (A ⋉ρ∗ V ∗). +(5.5) +Theorem 5.4. With the above notations, r = T − τ(T) is a skew-symmetric solution of the mock-Lie +YBE in the semi-direct product mock-Lie algebra (A⋉ρ∗ V ∗) if and only if T is an O-operator associated +to (V, ρ). +Proof. We have +r = T − τ(T) = +m +� +i=1 +T(vi) ⊗ v∗ +i − +m +� +i=1 +v∗ +i ⊗ T(vi), +thus we obtain +r12 • r13 = +� +1≤i,j≤m +� +Tvi • Tvj ⊗ v∗ +i ⊗ v∗ +j − ρ∗(Tvi)v∗ +j ⊗ v∗ +i ⊗ Tvj − ρ∗(Tvj)v∗ +i ⊗ Tvi ⊗ v∗ +j +� +; +r12 • r23 = +� +1≤i,j≤m +� +− v∗ +i ⊗ Tvi • Tvj ⊗ v∗ +j + Tvi ⊗ ρ∗(Tvj)v∗ +i ⊗ v∗ +j + v∗ +i ⊗ ρ∗(Tvi)v∗ +j ⊗ Tvj +� +; +15 + +r13 • r23 = +� +1≤i,j≤m +� +v∗ +i ⊗ v∗ +j ⊗ Tvi • Tvj − Tvi ⊗ v∗ +j ⊗ ρ∗(Tvj)v∗ +i − v∗ +i ⊗ Tvj ⊗ ρ∗(Tvi)v∗ +j +� +. +By the definition of dual representation, we know +ρ∗(Tvj)v∗ +i = +m +� +p=1 +⟨v∗ +i , ρ(Tvj)vp⟩v∗ +p. +Then +� +1≤i,j≤m +Tvi ⊗ ρ∗(Tvj)v∗ +i ⊗ v∗ +j = +� +1≤i,j,p≤m +⟨v∗ +p, ρ(Tvj)vi⟩Tvp ⊗ v∗ +i ⊗ v∗ +j += +� +1≤i,j≤m +T +� +⟨v∗ +p, ρ(Tvj)vi⟩vp +� +⊗ v∗ +i ⊗ v∗ +j = +� +1≤i,j≤m +T +� +ρ(Tvj)vi +� +⊗ v∗ +i ⊗ v∗ +j . +Then we get +r12 • r13 = +� +1≤i,j≤m +� +Tvi • Tvj ⊗ v∗ +i ⊗ v∗ +j − v∗ +i ⊗ v∗ +j ⊗ T(ρ(Tvj)vi) − v∗ +i ⊗ T(ρ(Tvj)vi) ⊗ v∗ +j +� +; +− r12 • r23 = +� +1≤i,j≤m +� +v∗ +i ⊗ Tvi • Tvj ⊗ v∗ +j − T(ρ(Tvj)vi) ⊗ v∗ +i ⊗ v∗ +j − v∗ +i ⊗ v∗ +j ⊗ T(ρ(Tvi)vj) +� +; +r13 • r23 = +� +1≤i,j≤m +� +v∗ +i ⊗ v∗ +j ⊗ Tvi • Tvj − T(ρ(Tvi)vj) ⊗ v∗ +i ⊗ v∗ +j − v∗ +i ⊗ T(ρ(Tvi)vj) ⊗ v∗ +j +� +. +Hence, we get +r12 • r13 + r13 • r23 − r12 • r23 += +� +1≤i,j≤m +�� +Tvi • Tvj − T(ρ(Tvi)vj) − T(ρ(Tvj)vi) +� +⊗ v∗ +i ⊗ v∗ +j ++ v∗ +i ⊗ v∗ +j ⊗ +� +Tvi • Tvj − T(ρ(Tvi)vj) − T(ρ(Tvj)vi) +� ++ v∗ +i ⊗ +� +Tvi • Tvj − T(ρ(Tvi)vj) − T(ρ(Tvj)vi) +� +⊗ v∗ +j +� +. +So r is a solution of the mock-Lie YBE in the semi-direct product mock-Lie algebra (A ⋉ρ∗ V ∗) if and +only if T is an O-operator associated to (V, ρ). +Combining Proposition 5.1 and Theorem 5.4, we have the following conclusion. +Corollary 5.5. Let (A, •) be a mock-Lie algebra and (V, ρ) be a representation of A. Set �A = A⋉ρ∗ V ∗. +Let T : V → A be a linear map. Then the following conditions are equivalent: +1. T is an O-operator of A associated to (V, ρ). +2. T − τ(T) is a skew-symmetric solution of the mock-Lie YBE in the Jordan algebra �A. +3. T − τ(T) is an O-operator of the mock-Lie algebra �A associated to ( �A∗, L∗ +� +A). +Remark 5.1. The equivalence between the above (1) and (3) can be obtained by a straightforward proof +and then Theorem 5.4 follows from this equivalence and Proposition 5.1. +16 + +The following conclusion reveals the relationship between mock-pre-Lie algebras and the mock-Lie +algebras with a symplectic form: +Proposition 5.6. Let (A, •) be a mock-Lie algebra with a symplectic form ω. Then there exists a com- +patible pre-mock-Lie algebra structure ” · ” on A given by +ω(x · y, z) = ω(y, x • z), +∀x, y, z ∈ A. +(5.6) +Proof. Define a linear map T : A → A∗ by ⟨T(x), y⟩ = ω(x, y) for any x, y ∈ A. For any ξ, η, γ ∈ A∗, +since T is invertible, there exist x, y, z ∈ A such that Tx = ξ, Ty = η, Tz = γ. Then T −1 : A∗ → A is +an O-operator of A associated to (A∗, L∗) since for any x, y, z ∈ A, we have +⟨T(x • y), z⟩ = ω(x • y, z) = ω(y, x • z) + ω(x, y • z) += ⟨L∗(x)T(y), z⟩ + ⟨L∗(y)T(x), z⟩. +By Proposition 2.8, there is a compatible mock-pre-Lie algebra structure ” · ” on A given by +x · y = T −1� +L∗(x)T(y) +� +, +∀x, y ∈ A, +which implies that +ω(x · y, z) = ⟨T(x · y), z⟩ = ⟨L∗(x)T(y), z⟩ += ⟨T(y), x • z⟩ = ω(y, x • z), +∀x, y, z ∈ A. +Hence the proof. +The following conclusion provides a construction of solutions of mock-Lie YBE in certain mock-Lie +algebras from mock-pre-Lie algebras. +Corollary 5.7. Let (A, ·) be a mock-pre-Lie algebra. Let {e1, · · · , en} be a basis of A and {e∗ +1, · · · , e∗ +n} +the dual basis. Then +r = +n +� +i=1 +(ei ⊗ e∗ +i − e∗ +i ⊗ ei) +(5.7) +is a skew-symmetric solution of the mock-Lie YBE in the mock-Lie algebra (Aac) ⋉Θ∗ (Aac)∗. +Proof. It follows from Theorem 5.4 and the fact that the identity map id is an O-operator of the sub- +adjacent mock-Lie algebra Aac of a mock-pre-Lie algebra associated to the representation (A, Θ). +References +[1] M. AGUIAR, Infinitesimal Hopf algebras, Contemporary Mathematics 267, Amer. Math. Soc., +(2000) 1-29. 2 +[2] S. ATTAN, Representations and O-operators of Hom-(pre)-Jacobi-Jordan algebras. arXiv +preprint arXiv:2105.14650 (2021). 3 +[3] C. BAI, +Left-symmetric bialgebras and an analogue of the classical Yang–Baxter equation, +Communications in Contemporary Mathematics, 10(02), (2008), 221-260. 2 +[4] C. BAI, Double constructions of Frobenius algebras, Connes cocycles and their duality, J. Non- +commut. Geom., 4, (2010), pp. 475-530. 2 +17 + +[5] C. BAI, A unified algebraic approach to the classical Yang-Baxter equation,J. Phys. A: Math. +Theor. 40 (2007), 11073-11082. 2 +[6] +C. BAI, L. GUO AND T. 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ZUMANOVICH, Special and exceptional mock-Lie algebras, Linear Algebra Appl. 518 +(2017),79-96. 1 +19 + diff --git a/J9FOT4oBgHgl3EQfyjQh/content/tmp_files/load_file.txt b/J9FOT4oBgHgl3EQfyjQh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb5183f36aeec4f6b1388204850e6f1c18ecab31 --- /dev/null +++ b/J9FOT4oBgHgl3EQfyjQh/content/tmp_files/load_file.txt @@ -0,0 +1,874 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf,len=873 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='12928v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='RA] 27 Dec 2022 Mock-Lie bialgebras and mock-Lie analogue of the classical Yang-Baxter equation K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Benali1 *, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Chtioui1 †, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hajjaji1 ‡, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Mabrouk2 § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' University of Sfax, Faculty of Sciences, BP 1171, 3038 Sfax, Tunisia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' University of Gafsa, Faculty of Sciences, 2112 Gafsa, Tunisia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Abstract The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The study of a special case called coboundary mock-Lie bialgebra leads to the introduction the mock-Lie Yang-Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang-Baxter equation on a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Note that a skew-symmetric solution of mock-Lie Yang-Baxter equation gives a mock-Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Finally, the notation of O-operators are studied to construct skew-symmetric solution of mock-Lie Yang-Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Key words: mock-Lie algebra, mock-Lie bialgebra, Matched pair, Manin triple, mock-Lie Yang-Baxter equation, O-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' C (2020):16W10, 16T10, 16T15, 16T25, 17B38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Preliminaries 3 3 Matched pairs, Manin triples and mock-Lie bialgebras 5 4 Coboundary mock-Lie bialgebras and the mock-Lie Yang-Baxter equation 9 5 O-operators of mock-Lie algebras and mock-Lie Yang-Baxter equation 14 1 Introduction A while ago, a new class of algebras emerged in the literature the so called mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' These are commutative algebras satisfying the Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' They were appeared for the first time in [16] and since then a lot of works are done on this subject, note for example [23, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' These algebras live a dual life: as member of a very particular class of Jordan algebras and as strange cousins of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The theory of Lie bialgebra and Poisson Lie groups dates back to the early 80s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' E-mail: karimabenali172@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='fr †E-mail: chtioui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='taoufik@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='fr ‡E-mail: atefhajjaji100@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='com §E-mail: mabrouksami00@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='fr (Corresponding author) 1 Poisson Lie groups are Lie groups equipped with an additional structure, a Poisson bracket satisfy- ing a compatibility condition with the group multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The infinitesimal object associated with a poisson Lie group is the tangent vector space at the origin of the group, which is in a naturel way a Lie algebra g, see for instance [14, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='The Poisson structure on the group induces on the Lie algebra an additional structure which is nothing but a Lie algebra structure on the dual vector space g∗ satisfying a compatibility condition with the Lie bracket on g itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Such a Lie algebra together with its additional structure is called a Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' So a bialgebra structure on a given algebra is obtained by a corre- sponding set of comultiplication together with the set of compatibility conditions between multiplication and comultiplication [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For example take a finite dimensional vector space V with a given algebraic structure, this can be acheived by equipping the dual space V ∗ with the same algebraic structure and a set of compatibility conditions between the structures on V and those on V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Among the well-known bial- gebra structures, we have the associative bialgebra and infinitesimal bialgebra introduced in [1,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Note that these two structures have the same associative multiplications on V and V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' They are distinguished only by the compatibility conditions, with the comultiplication acting as an homomorphism (respectively a derivation) on the multiplication for the associative bialgebra (respectively the infinitesimal bialgebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' In general, it is quite common to have multiple bialgebra structures that differ only by their compati- bility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A good compatibility condition is prescribed on one hand by a strong motivation and potential applications, and on the other hand by a rich structure theory and effective constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' See also [4,11,15,17–21,27,28,32–34] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' One reason for the usefulness of the Lie bialgebra is that it has a coboundary theory, which leads to the construction of Lie bialgebras from solutions of the classical Yang-Baxter equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The origin of the Yang-Baxter-equations is purely physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' They were first introduced by Baxter, McGuire, and Yang in [12, 13, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Later on, this equation attracts the attention of scientists and becomes one of the most basic equation in mathematical physics [6,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Namely it plays a crucial role for introducing the theory of quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' This exceptional importance can be seen in many other domaines like: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The various forms of the Yang- Baxter-equation and some of their uses in physics are summarized in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Many scientists have found solutions for the Yang-Baxter equation, however the full classification of its solutions remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' In the theory of Lie bialgebras, it is essential to consider the coboundary case, which is related to the theory of the classical Yang-Baxter equation [3,5,7,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We aim to have an anlogue in th mock-Lie case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' This paper is organized as follows: In Section 2 we recall some basic definitions and constructions about mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Section 3 deals with matched pairs, manin triple and mock-Lie bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' In Section 4, we introduce and develop the notion of coboundary mock-Lie bialgebras and the mock- Lie Yang-Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' In Section 5, we give the O-operators of mock-Lie algebras and construct a solution mock-Lie Yang-Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Unless otherwise specified, all the vector spaces and algebras are finite dimensional over a field K of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let V and W be two vector spaces: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Denote by τ : V ⊗ W → W ⊗ V the switch isomorphism, τ(v ⊗ w) = w ⊗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For a linear map ∆ : V → ⊗2V , we use Sweedler’s notation ∆(x) = � (x) x1 ⊗ x2 for x ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We will often omit the summation sign � (x) to simplify the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Denote by V ∗ = Hom(V, K) the linear dual of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For ϕ ∈ V ∗ and u ∈ V , we write ⟨ϕ, u⟩ := ϕ(u) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For a linear map φ : V → W, we define the map φ∗ : W ∗ → V ∗ by ⟨φ∗(ξ), v⟩ = ⟨ξ, φ(v)⟩, ∀v ∈ V, ξ ∈ W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For an element x in a mock-Lie algebra (A, •) and n ≥ 2, define the adjoint map L(x) : ⊗nA → ⊗nA by L(x)(y1 ⊗ · · · ⊗ yn) = n � i=1 y1 ⊗ · · · ⊗ yi−1 ⊗ x • yi ⊗ yi+1 ⊗ · · · ⊗ yn (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2) for all y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' , yn ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Conversely, given Y = y1 ⊗ · · · ⊗ yn, we define L(Y ) : g → ⊗ng by L(Y )(x) = L(x)(Y ), for x ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 2 Preliminaries In this section, we provide some preliminaries about mock-Lie algebras and left mock-pre-Lie alge- bras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Our main references are [2,9,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A mock-Lie algebra is a pair (A, •) consisting of a vector space A together with a multiplication • : A ⊗ A → A satisfying x • y = y • x, (Commutativity), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1) x • (y • z) + y • (z • x) + z • (x • y) = 0, (Jacobi identity), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2) for any x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The Jacobi identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2) is equivalent to x • (y • z) = −(x • y) • z − y • (x • z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3) In other words, the left multiplication L : A → End(A) defined by L(x)y = x • y, is anti-derivation on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Recall that a linear map D : A → A is called anti-derivation if for all x, y ∈ A, D(x • y) = −D(x) • y − x • D(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let A be a 4-dimensional vector space with basis B = {e1, e2, e3, e4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A, •) is a mock-Lie algebra where the product • is defined on the basis B by e1 • e1 = e2, e1 • e3 = e4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Recall that an anti-associative algebra is a pair (A, ⋆) consisting of a vector space A together with a product ⋆ : A ⊗ A → A such that the anti-associator vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='e, Aass(x, y, z) := (x ⋆ y) ⋆ z + x ⋆ (y ⋆ z) = 0, ∀x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4) Let (A, ⋆) be an anti-associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then, (A, •) is a mock-Lie algebra, where x • y := x ⋆ y + y ⋆ x, ∀x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Now, we recall the definition of representations of a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A representation of a mock-Lie algebra (A, •) is a pair (V, ρ) where V is a vector space and ρ : A → End(V ) is a linear map such that the following equality holds, for all x, y ∈ A, ρ(x • y) = −ρ(x)ρ(y) − ρ(y)ρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5) 3 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A, L) is a representation of A on itself called the adjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' An equivalent characterisation of representations on mock-Lie algebras is given in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra, V be a vector space and ρ : A → End(V ) a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (V, ρ) be a representation of A if and only if the direct sum A ⊕ V together with the multiplication defined by (x + u) •A⊕V (y + v) = x • y + ρ(x)v + ρ(y)u, ∀x, y ∈ A, ∀u, v ∈ V, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6) is a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' This mock-Lie algebra is called the semi-direct product of A and V and it is denoted by A ⋉ρ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and two representations (V1, ρ1) and (V2, ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A linear map φ : V1 → V2 is said to be a morphism of representations if ρ2(x) ◦ φ = φ ◦ ρ1(x), ∀x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) If φ is bejictive, then (V1, ρ1) and (V2, ρ2) are equivalent (isomorphic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' To relate matched pairs of mock-Lie algebras to mock-Lie bialgebras and Manin triples for mock- Lie algebras in the next section, we need the notions of the coadjoint representation, which is the dual representation of the adjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' In the following we recall these facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and (V, ρ) be a representation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let V ∗ be the dual vector space of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Define the linear map ρ∗ : A → End(V ∗) as ⟨ρ∗(x)u∗, v⟩ = ⟨u∗, ρ(x)v⟩, ∀x ∈ A, v ∈ V, u∗ ∈ V ∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='8) where ⟨·, ·⟩ is the usual pairing between V and the dual space V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' With the above notations, we have the following Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (V, ρ) be a representation of a mock-Lie algebra (A, •).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (V ∗, ρ∗) is a repre- sentation of A on V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Consider the case when V = A and define the linear map L∗ : A → End(A∗) by ⟨L∗(x)(ξ), y⟩ = ⟨ξ, L(x)y⟩, ∀x, y ∈ A, ξ ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='9) Then we have the following: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and (A, L) be the adjoint representation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A∗, L∗) is a representation of (A, •) on A∗ which is called the coadjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' If there is a mock-Lie algebra structure on the dual space A∗, we denote the left multiplication by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and (V, ρ) be a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A linear map T : V → A is called an O-operator associated to(V, ρ) if T satisfies T(u) • T(v) = T � ρ(Tu)v + ρ(Tv)u � , ∀u, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='10) In the case (V, ρ) = (A, L), the O-operator T is called a Rota-Baxter operator (of weight zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 4 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A mock-pre-Lie algebra is a vector space A equipped with a linear map · : A ⊗ A → A satisfying the following identity Aass(x, y, z) = −Aass(y, x, z), ∀x, y, z ∈ A, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='11) Recall that Aass(x, y, z) = (x · y) · z + x · (y · z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='11) is equivalent to (x ⋆ y) · y = (x · y) · z − y · (x · z), where x ⋆ y = x · y + y · x, for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Note that if (A, ·) is a mock-pre-Lie algebra, then the product given by x ⋆ y = x · y + y · x, ∀x, y ∈ A, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='12) defines a mock-Lie algebra structure, which is called the sub-adjacent mock-Lie algebra of (A, ·), and denoted by Aac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Furthermore, (A, ·) is called the compatible mock-pre-Lie algebra structure on Aac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' On the other hand, let Θ : A → End(A) defined by Θ(x)y = x · y, ∀x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A, Θ) is a representation of the mock-Lie algebra Aac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and (V, ρ) be a representation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' If T is an O-operator associated to (V, ρ), then (V, ·) is a mock-pre-Lie algebra, where u · v = ρ(Tu)v, ∀u, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='13) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then there is a compatible mock-pre-Lie algebra if and only if there exists an invertible O-operator T : V → A associated to a representation (V, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Furthermore, the compatible mock-pre-Lie structure on A is given by x · y = T � ρ(x)T −1(y) � , ∀x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='14) 3 Matched pairs, Manin triples and mock-Lie bialgebras In this section, we introduce the notions of Manin triple of a mock-Lie algebra and mock-Lie bialge- bras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The equivalence between them is interpreted in terms of matched pairs of mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We first recall the notion of matched pairs of mock-Lie algebras (see [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) and (H, ⋄) be two mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let ρ : A → End(H) and µ : H → End(A) be two linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' On the direct sum A ⊕ H of the underlying vector spaces, define a linear map ◦ : ⊗2(A ⊕ H) → A ⊕ H by (x + a) ◦ (y + b) = x • y + µ(b)x + µ(a)y + a ⋄ b + ρ(y)a + ρ(x)b, ∀x, y ∈ A, a, b ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) and (H, ⋄) be two mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A ⊕ H, ◦) is a mock-Lie algebra if and only if (H, ρ) and (A, µ) are representations of (A, •) and (H, ⋄) respectively, and for all x, y ∈ A, a, b ∈ H, the following compatibility conditions are satisfied: ρ(x)(a ⋄ b) + ρ(x)a ⋄ b + a ⋄ ρ(x)b + ρ(µ(a)x)b + ρ(µ(b)x)a = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2) µ(a)(x • y) + µ(a)x • y + x • µ(a)y + µ(ρ(x)a)y + µ(ρ(y)a)x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3) Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A matched pair of mock-Lie algebras is a quadruple (A, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ρ, µ) consisting of two mock-Lie algebras (A, •) and (H, ⋄), together with representations ρ : A → End(H) and µ : H → End(A) respectively, such that the compatibility conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 5 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We denote the mock-Lie algebra defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1) by A ⊲⊳ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' It is straightforward to show that every mock-Lie algebra which is a direct sum of the underlying vector spaces of two mock-Lie subalgebras can be obtained from a matched pair of mock-Lie algebras as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A bilinear form ω on a mock-Lie algebra (A, •) is called invariant if it satisfies ω(x • y, z) = ω(x, y • z), ∀x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and (A, L) be the adjoint representation of A on itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A, L) and (A∗, L∗) are equivalent as representations of the mock-Lie algebra (A, •) if and only if there exists a nondegenerate symmetric invariant bilinear form ω on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that there exists a nondegenerate symmetric invariant bilinear form ω on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Since ω is nondegenerate, there exists a linear isomorphism φ : A → A∗ defined by ⟨φ(x), y⟩ = ω(x, y), ∀x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence for any x, y, z ∈ A, we have ⟨φ(L(x)(y)), z⟩ = ω(L(x)(y), z) = ω(x • y, z) = ω(y, x • z) = ⟨φ(y), x • z⟩ = ⟨L∗(x)φ(y), z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' That is, (A, L) and (A∗, L∗) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Conversely, by a similar way, we can get the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A Manin triple of mock-Lie algebras is a triple of mock-Lie algebras (A, A+, A−) to- gether with a nondegenerate symmetric invariant bilinear form ω on A such that the following conditions are satisfied: (a) A+, A− are mock-Lie subalgebras of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (b) A = A+ ⊕ A− as vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (c) A+ and A− are isotropic with respect to ω, that is, ω(x+, y+) = ω(x−, y−) = 0, for any x+, y+ ∈ A+, x−, y− ∈ A−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A homomorphism between two Manin triples of mock-Lie algebras (A, A+, A−) and (B, B+, B−) associated to two nondegenerate symmetric invariant bilinear forms ω1 and ω2 respectively, is a homomorphism of mock-Lie algebras f : A → B such that f(A+) ⊂ B+, f(A−) ⊂ B−, ω1(x, y) = ω2(f(x), f(y)), ∀x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' If in addition, f is an isomorphism of vector spaces, then the two Manin triples are called isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' [22] Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that there is a mock-Lie algebra structure (A∗, ⋄) on the dual space A∗ of A and there is a mock-Lie algebra structure on the direct sum A ⊕ A∗ of the underlying vector spaces A and A∗ such that (A, •) and (A∗, ⋄) are subalgebras and the natural non-degerenate symmetric bilinear form on A ⊕ A∗ given by ωd(x + ξ, y + η) := ⟨x, η⟩ + ⟨ξ, y⟩, ∀x, y ∈ A, ξ, η ∈ A∗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5) is invariant, then (A ⊕ A∗, A, A∗) is called a standard Manin triple of mock-Lie algebra associated to standard bilinear form ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Obviously, a standard Manin triple of mock-Lie algebras is a Manin triple of mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Conversely, we have 6 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Every Manin triple of mock-Lie algebras is isomorphic to a standard one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Since A+ and A− are isotropic under the nondegenerate invariant bilinear form ω on A+ ⊕ A−, then in this case A− and (A+)∗ are identified by ω and the mock-Lie algebra structure on A− is transferred to (A+)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence the mock-Lie algebra structure on A+ ⊕ A− is transferred to A+ ⊕ (A+)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Transfer the nondegenrate bilinear form ω to A+ ⊕ (A+)∗, we obtain the standard bilinear form given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Thus, (A, A+, A−) is isomorphic to the stansadrd Manin triple (A ⊕ A∗, A, A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' [22] Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that there is a mock-Lie algebra structure (A∗, ⋄) on A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then there exists a mock-Lie algebra sructure on the vector space A ⊕ A∗ such that (A ⊕ A∗, A, A∗) is a standard Manin triple of mock-Lie algebras with respect to ωd defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5) if and only if (A, A∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' L∗, L∗) is a matched pair of mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Here L∗ is the coadjoint representation of the mock-Lie algebra (A∗, ⋄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that there is a mock-Lie algebra structure (A∗, ⋄) on A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A, A∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' L∗, L∗) is a matched pair of mock-Lie algebras if and only if for any x, y ∈ A, ξ ∈ A∗, we have L∗(ξ)(x • y) + (L∗(ξ)(x)) • y + x • (L∗(ξ)(y)) + L∗(L∗(x)(ξ))(y) + L∗(L∗(y)(ξ))(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Obiviously, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6) is exactly Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3) in the case ρ = L∗, µ = L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' In addition, for any x, y ∈ A, ξ, η ∈ A∗, we have ⟨L∗(ξ)(x • y), η⟩ = ⟨x • y, L(ξ)(η)⟩ = ⟨L(x)(y), ξ ⋄ η⟩ = ⟨y, L∗(x)(ξ ⋄ η)⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ⟨(L∗(ξ)(x)) • y, η⟩ = ⟨L(L∗(ξ)(x))(y), η⟩ = ⟨y, L∗(L∗(ξ)(x))(η)⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ⟨x • (L∗(ξ)(y)), η⟩ = ⟨L(x)(L∗(ξ)(y)), η⟩ = ⟨L∗(ξ)(y), L∗(x)(η)⟩ = ⟨y, L(ξ)(L∗(x)(η))⟩ = ⟨y, ξ ⋄ (L∗(x)(η))⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ⟨L∗(L∗(x)(ξ))(y), η⟩ = ⟨y, L(L∗(x)(ξ))(η)⟩ = ⟨y, (L∗(x)(ξ)) ⋄ η⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ⟨L∗(L∗(y)(ξ))(x), η⟩ = ⟨x, L(L∗(y)(ξ))(η)⟩ = ⟨x, η ⋄ (L∗(y)(ξ))⟩ = ⟨x, L(η)((L∗(y)(ξ))⟩ = ⟨L∗(η)(x), L∗(y)(ξ)⟩ = ⟨L(L∗(η)(x))(y), ξ⟩ = ⟨y, L∗(L∗(η)(x))(ξ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2) holds if and only if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Therefore the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that there is a mock-Lie algebra structure ”⋄” on its dual space A∗ given by a linear map ∆∗ : A∗ ⊗ A∗ → A∗, that is, ξ ⋄ η = ∆∗(ξ ⊗ η), for any ξ, η ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A, A∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' L∗, L∗) is a matched pair of mock-Lie algebras if and only if ∆ : A → A ⊗ A satisfies the following condition: ∆(x • y) = − � L(x) ⊗ id + id ⊗ L(x) � ∆(y) − � L(y) ⊗ id + id ⊗ L(y) � ∆(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) for any x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5, we can prove that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' In fact, for any x, y ∈ A, ξ, η ∈ A∗, we have ⟨L∗(ξ)(x • y), η⟩ = ⟨x • y, ξ ⋄ η⟩ = ⟨x • y, ∆∗(ξ ⊗ η)⟩ = ⟨∆(x • y), ξ ⊗ η⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ⟨(L∗(ξ)(x)) • y, η⟩ = ⟨L(y)(L∗(ξ)(x)), η⟩ = ⟨L∗(ξ)(x), L∗(y)(η)⟩ = ⟨x, L(ξ)(L∗(y)(η))⟩ = ⟨x, ξ ⋄ (L∗(y)(η))⟩ = ⟨(id ⊗ L(y))∆(x), ξ ⊗ η⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ⟨x • (L∗(ξ)(y)), η⟩ = ⟨L(x)(L∗(ξ)(y)), η⟩ = ⟨L∗(ξ)(y), L∗(x)(η)⟩ = ⟨y, L(ξ)(L∗(x)(η))⟩ 7 = ⟨y, ξ ⋄ (L∗(x)(η))⟩ = ⟨(id ⊗ L(x))∆(y), ξ ⊗ η⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ⟨L∗(L∗(x)(ξ))(y), η⟩ = ⟨y, (L∗(x)(ξ)) ⋄ η⟩ = ⟨(L(x) ⊗ id)∆(y), ξ ⊗ η⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ⟨L∗(L∗(y)(ξ))(x), η⟩ = ⟨x, (L∗(y)(ξ)) ⋄ η⟩ = ⟨(L(y) ⊗ id)∆(x), ξ ⊗ η⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6) is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' From the symmetry of the mock-Lie algebras (A, •) and (A∗, ⋄) in the standard Manin triple of mock-Lie algebras with respect to ωd, we also can consider a linear map γ : A∗ → A∗ ⊗ A∗ such that γ∗ : A ⊗ A → A gives the mock-Lie algebra structure ” • ” on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' It is straightforward to show that ∆ satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) if and only if γ satisfies γ(ξ ⋄ η) = − � L(ξ) ⊗ id + id ⊗ L(ξ) � γ(η) − � L(η) ⊗ id + id ⊗ L(η) � γ(ξ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='8) for any ξ, η ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A mock-Lie bialgebra structure on A is a symmetric linear map ∆ : A → A ⊗ A such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ∆∗ : A∗ ⊗ A∗ → A∗ defines a mock-Lie algebra structure on A∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ∆ satifies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7), called the compatibility condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We denote it by (A, ∆) or (A, A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We can unwrap the compatibility condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) as ∆(x • y) = −(x • y1) ⊗ y2 − y1 ⊗ (x • y2) − (y • x1) ⊗ x2 − x1 ⊗ (y • x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='9) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The compatibility condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) is, in fact, a cocycle condition in the zigzag cohomol- ogy of mock-Lie algebra introduced in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Indeed, we can regard A⊗2 as an A-module via the adjoint action (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2): x · (y1 ⊗ y2) = L(x)(y1 ⊗ y2) = (x • y1) ⊗ y2 + y1 ⊗ (x • y2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='10) for x ∈ A and y1 ⊗ y2 ∈ A⊗2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then we can think of the linear map ∆ : A → A⊗2 as a 1-cochain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then the differential on ∆ is given by d1∆(x, y) = ∆(x • y) + x · ∆(y) + y · ∆(x) = ∆(x • y) + L(x)(∆(y)) + L(y)(∆(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) says exactly that ∆ ∈ C1(A, A⊗2) is a 1-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, A∗) be a mock-Lie bialgebra on a mock-Lie algebra (A, •).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A∗, γ)(or(A∗, A)) is a mock-Lie bialgebra on the mock-Lie algebra (A∗, ⋄), where γ is given in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A1, A∗ 1) and (A2, A∗ 2) be two mock-Lie bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A linear map ψ : A1 → A2 is a homomorphism of mock-Lie bialgebras if ψ satifies, for any x, y ∈ A1 ψ(x •1 y) = ψ(x) •2 ψ(y), (ψ ⊗ ψ) ◦ ∆1 = ∆2 ◦ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='11) Now, combining Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6, we have the following conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that there is a mock-Lie algebra structure on A∗ denoted by ” ⋄ ” which is defined as a linear map ∆ : A → A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (A⊕A∗, A, A∗) is a standard Manin triple of mock-Lie algebras with respect to ωd defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (A, A∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' L∗, L∗) is a matched pair of mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (A, A∗) is a mock-Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 8 4 Coboundary mock-Lie bialgebras and the mock-Lie Yang-Baxter equa- tion In this section, we consider a special class of mock-Lie bialgebras called coboundary mock-Lie bialgebras and introduce the notion of mock-Lie Yang-Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A mock-Lie bialgebra (A, A∗) is called coboundary if there exists an element r ∈ A⊗A such that for any x ∈ A, ∆(x) = � L(x) ⊗ id − id ⊗ L(x) � r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and r ∈ A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that the linear map ∆ : A → A ⊗ A is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then ∆ satisfies the compatibility condition given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let r = r1 ⊗ r2 ∈ A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Using the commutativity and Jacobi identity for mock-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then for any x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' y ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(x) ⊗ id + id ⊗ L(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='∆(y) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(y) ⊗ id + id ⊗ L(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='∆(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='= − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(x) ⊗ id + id ⊗ L(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(y) ⊗ id − id ⊗ L(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(y) ⊗ id + id ⊗ L(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(x) ⊗ id − id ⊗ L(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='= − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(x) ⊗ id + id ⊗ L(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='(y • r1) ⊗ r2 − r1 ⊗ (y • r2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(y) ⊗ id + id ⊗ L(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='(x • r1) ⊗ r2 − r1 ⊗ (x • r2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='= − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='x • (y • r1) ⊗ r2 − (x • r1) ⊗ (y • r2) + (y • r1) ⊗ (x • r2) − r1 ⊗ (x • (y • r2)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='y • (x • r1) ⊗ r2 − (y • r1) ⊗ (x • r2) + (x • r1) ⊗ (y • r2) − r1 ⊗ (y • (x • r2)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='− x • (y • r1) − y • (x • r1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='⊗ r2 + r1 ⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='x • (y • r2) + y • (x • r2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='(x • y) • r1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='⊗ r2 − r1 ⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='(x • y) • r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='L(x • y) ⊗ id − id ⊗ L(x • y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='=∆(x • y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let ∆ : A → A ⊗ A be a linear map and σ : A⊗3 → A⊗3 be defined as σ(x ⊗ y ⊗ z) = y ⊗ z ⊗ x, for any x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let E∆ : A → A⊗3 be a linear map given by E∆(x) = (id + σ + σ2) � (id ⊗ ∆)∆(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let A be a vector space and ∆ : A → A ⊗ A be a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then the product ” ⋄ ” in A∗ given by ∆∗ : A∗ ⊗ A∗ → A∗ satisfies the Jacobi identity if and only if E∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For any ξ, η ∈ A∗, x ∈ A, we have ⟨ξ ⋄ η, x⟩ = ⟨∆∗(ξ ⊗ η), x⟩ = ⟨ξ ⊗ η, ∆(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3) Threrefore, for any ξ, η, ν ∈ A∗ and x ∈ A, the Jacobi identity satisfies ⟨J(ξ, η, ν), x⟩ =⟨∆∗(id ⊗ ∆∗)(ξ ⊗ η ⊗ ν) + ∆∗(id ⊗ ∆∗)(η ⊗ ν ⊗ ξ) + ∆∗(id ⊗ ∆∗)(ν ⊗ ξ ⊗ η), x⟩ =⟨∆∗(id ⊗ ∆∗) � id + σ + σ2� (ξ ⊗ η ⊗ ν), x⟩ =⟨ξ ⊗ η ⊗ ν, � id + σ + σ2� ((id ⊗ ∆)∆)(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Therefore J(ξ, η, ν) = 0, for any ξ, η, ν ∈ A∗ if and only if E∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 9 Let (A, •) be a mock-Lie algebra and r = � i ai ⊗ bi ∈ A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Set r12 = � i ai ⊗ bi ⊗ 1, r13 = � i ai ⊗ 1 ⊗ bi, r23 = � i 1 ⊗ ai ⊗ bi, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4) where 1 is a unit element if (A, •) is unital or a symbol playing a similar role of the unit for the non-unital cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The operation between two rij is in obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For example, r12 • r13 = � ij ai • aj ⊗ bi ⊗ bj, r13 • r23 = � ij ai ⊗ aj ⊗ bi • bj, r23 • r12 = � ij aj ⊗ ai • bj ⊗ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5) Note that the above elements are independents of the existence of the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A tensor r ∈ A ⊗ A is called symmetric (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' skew-symmetric) if r = τ(r) ( resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' r = −τ(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' On the other hand, any r ∈ A ⊗ A can be identified as a linear map from the dual space A∗ to A in the following way: ⟨ξ, r(η)⟩ = ⟨ξ ⊗ η, r⟩, ∀ξ, η ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6) The tensor r ∈ A ⊗ A is called nondegenerate if the above induced linear map is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Define a linear map ∆ : A → A ⊗ A by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1) with some r ∈ A ⊗ A satisfying � L(x) ⊗ id − id ⊗ L(x) �� r + τ(r) � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then E∆(x) + Q(x)[[r, r]] = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='8) where [[r, r]] = r12 • r13 + r13 • r23 − r12 • r23, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='9) and Q(x) = � L(x) ⊗ id ⊗ id + id ⊗ L(x) ⊗ id + id ⊗ id ⊗ L(x) � for any x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let r = � i ai ⊗ bi, the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) is equivalent to � i (x • ai) ⊗ bi − ai ⊗ (x • bi) + (x • bi) ⊗ ai − bi ⊗ (x • ai) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='10) Note that E∆(x) is the sum of twelve terms and that Q(x)[[r, r]] is a sum of nine terms, but two terms appear in both sums up to sign and hence are canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Thus E∆(x) + Q(x)[[r, r]] is a sum of seventeen terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' After rearranging the terms suitably,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' we obtain E∆(x) + Q(x)[[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' r]] = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='j � − (x • bi) • aj ⊗ bj ⊗ ai + x • (ai • aj) ⊗ bi ⊗ bj + (x • bi) • bj ⊗ ai ⊗ aj − (bi • bj) ⊗ (x • ai) ⊗ aj + (ai • aj) ⊗ (x • bi) ⊗ bj + (bi • aj) ⊗ bj ⊗ (x • ai) + (ai • aj) ⊗ bi ⊗ (x • bj) − ai ⊗ (x • bi) • aj ⊗ bj + ai ⊗ aj ⊗ (x • bi) • bj + bj ⊗ (x • ai) ⊗ (bi ⊗ aj) − bj ⊗ ai ⊗ (x • bi) • aj − aj ⊗ (bi • bj) ⊗ (x • ai) + aj ⊗ (x • bi) • bj ⊗ ai − ai ⊗ x • (bi • aj) ⊗ bj − ai ⊗ (bi • aj) ⊗ (x • bj) + ai ⊗ (x • aj) ⊗ (bi • bj) + ai ⊗ aj ⊗ x • (bi • bj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Interchanging the indices i and j in the first term and using the Jacobi identity in A, the first term becomes � i,j x • (bj • ai) ⊗ bi ⊗ aj + bj • (ai • x) ⊗ bi ⊗ aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 10 Using the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='10), the sum of bj • (ai • x) ⊗ bi ⊗ aj and the third and fourth terms is � i,j � L(bj) ⊗ id �� (ai • x) ⊗ bi + (x • bi) ⊗ ai − bi ⊗ (x • ai) � ⊗ aj � j � L(bj) ⊗ id � � i � (ai • x) ⊗ bi + (x • bi) ⊗ ai − bi ⊗ (x • ai) � ⊗ aj = � i,j � L(bj) ⊗ id �� ai ⊗ (x • bi) � ⊗ aj = � i,j (ai • bj) ⊗ (x • bi) ⊗ aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Similarly, the sum of (ai • bj) ⊗ (x • bi) ⊗ aj and the fifth term becomes � i,j bj ⊗ (x • bi) ⊗ (ai • aj) + aj ⊗ (x • bi) ⊗ (ai • bj), and the sum of the sixth and seventh terms is � i,j aj ⊗ bi ⊗ x • (ai • bj) + bj ⊗ bi ⊗ x • (ai • aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Finally, the sum of x • (bj • ai) ⊗ bi ⊗ aj and the second term in the sum of the expression of E∆(x) + Q(x)[[r, r]] becomes � i,j bj ⊗ bi ⊗ ai • (x • aj) + aj ⊗ bi ⊗ ai • (x • bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Inserting these results, we find that the expression of E∆(x) + Q(x)[[r, r]] can be written in the form � i � ai ⊗ Ui + bi ⊗ Vi � : In fact, Ui = � j � − (bj • bi) ⊗ (x • aj) − (bi • aj) ⊗ (x • bj) + bj ⊗ x • (aj • bi) + aj ⊗ x • (bi • bj) + (x • bj) ⊗ (aj • bi) + (x • aj) ⊗ (bi • bj) − x • (bi • aj) ⊗ bj + (x • bj) • bi ⊗ aj − (x • bi) • aj ⊗ bj + aj ⊗ (x • bi) • bj + bj ⊗ (x • bi) • aj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' On the right-hand side, the sum of the first four terms is zero by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='10), and the sum of the next three terms becomes x • (bi • bj) ⊗ aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' By the Jacobi identity in A, the sum of x • (bi • bj) ⊗ aj and the eighth term is −bj • (x • bi) ⊗ aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Finally, the sum of −bj • (x • bi) ⊗ aj and the last three terms becomes � j −bj • (x • bi) ⊗ aj − (x • bi) • aj ⊗ bj + aj ⊗ (x • bi) • bj + bj ⊗ (x • bi) • aj = 0, if we replace x in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='10) by x • bi: Hence, we get Ui = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Similarly, we can proves that Vi = � j � (x • bj) ⊗ (aj • ai) + bj ⊗ x • (aj • ai) + bj ⊗ aj • (x • ai) 11 + (x • aj) ⊗ (bj • ai) − aj ⊗ (x • bj) • ai � =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Using the above discussion, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and r ∈ A⊗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Define a bilinear map ⋄ : A∗⊗A∗ → A∗ by ⟨ξ ⋄ η, x⟩ = ⟨∆∗(ξ ⊗ η), x⟩ = ⟨ξ ⊗ η, ∆(x)⟩, where ∆ is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then (A∗, ⋄) is a mock-Lie algebra if and only if the following conditions are satisfied: (i) � L(x) ⊗ id − id ⊗ L(x) �� r + τ(r) � = 0, (ii) Q(x)[[r, r]] = 0, for all x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Under these conditions, (A, A∗) is a coboundary mock-Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The bracket ⋄ is determined by the cobracket ∆(x) = � L(x) ⊗ id − id ⊗ L(x) � r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence (A∗, ⋄) is a mock-Lie algebra if and only if ⋄ is symmetric and satisfies the Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For any x ∈ A, ξ, η ∈ A∗, we have ⟨ξ ⋄ η − η ⋄ ξ, x⟩ = ⟨∆∗(ξ ⊗ η) − ∆∗(η ⊗ ξ), x⟩ = ⟨ξ ⊗ η, ∆(x) − τ ◦ ∆(x)⟩ = ⟨ξ ⊗ η, � L(x) ⊗ id − id ⊗ L(x) � r − τ ◦ ( � L(x) ⊗ id − id ⊗ L(x) � r)⟩ = ⟨ξ ⊗ η, L(x)r1 ⊗ r2 − r1 ⊗ L(x)r2 − r2 ⊗ L(x)r1 + L(x)r2 ⊗ r1⟩ = ⟨ξ ⊗ η, � L(x) ⊗ id − id ⊗ L(x) �� r + τ(r) � ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then r satisfies (i) if and only if ⋄ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The proof that (ii) is equivalent to the condition that ⋄ satisfies the Jacobi identity which follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Since ∆(x) = � L(x) ⊗ id − id ⊗ L(x) � r, the compatibility conditions for a mock-Lie bialgebra in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5 hold naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Therefore the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' An easy way to satisfy conditions (i) and (ii) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4 is to assume that r is skew- symmetric and [[r, r]] = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='11) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='11) is the mock-Lie Yang-Baxter equation in the mock-Lie algebra (A, •).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A quasitriangular mock-Lie bialgebra is a coboundary mock-Lie bialgebra, in which r is a solution of the mock-Lie Yang- Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A triangular mock-Lie bialgebra is a coboundary mock-Lie bialgebra, in which r is a skew-symmetric solution of the mock-Lie Yang-Baxter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A direct application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4 is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, A∗) be a mock-Lie bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then there is a canonical coboundary mock-Lie bialgebra structure on A ⊕ A∗ such that both i1 : A → A ⊕ A∗ and i2 : A∗ → A ⊕ A∗ into the two summands are homomorphisms of mock-Lie bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Here the mock-Lie bialgebra structure on A is (A, −∆A) , where ∆A is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let r ∈ A ⊗ A∗ ⊂ (A ⊕ A∗) ⊗ (A ⊕ A∗) correspond to the identity map id : A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let {e1, · · · , en} be a basis of A and {f1, · · · , fn} be its dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then r = � i ei ⊗ fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that the 12 mock-Lie algebra structure ” ◦D(A) ” on A ⊕ A∗ is given by D(A) = A ⊲⊳ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1), we have x ◦D(A) y = x • y, a∗ ◦D(A) b∗ = a∗ ⋄ b∗, x ◦D(A) a∗ = L∗(x)a∗ + L∗(a∗)x, for any x, y ∈ A, a∗, b∗ ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Next we prove that r satisfies the two conditions in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' If so, then ∆D(A)(u) = (L◦D(A)(u) ⊗ idD(A) − idD(A) ⊗ L◦D(A)(u))r, ∀u ∈ D(A) can induce a coboundary mock-Lie bialgebra structure on D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Since ⟨ � i ei ⊗ fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' fs ⊗ et⟩ = ⟨et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' fs⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' we have ⟨[[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' r]]D(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (es + ft) ⊗ (ek + fl) ⊗ (ep + fq)⟩ = � ij ⟨ei ◦D(A) ej ⊗ fi ⊗ fj − ei ⊗ fi ◦D(A) ej ⊗ fj + ei ⊗ ej ⊗ fi ◦D(A) fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (es + ft) ⊗ (ek + fl) ⊗ (ep + fq)⟩ = � ij ⟨ei • ej ⊗ fi ⊗ fj − ei ⊗ (L∗(fi)ej + L∗(ej)fi) ⊗ fj + ei ⊗ ej ⊗ fi ⋄ fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (es + ft) ⊗ (ek + fl) ⊗ (ep + fq)⟩ = � ij � ⟨ei • ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ft⟩⟨fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ek⟩⟨fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ep⟩ − ⟨ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ft⟩⟨ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' fi ⋄ fl⟩⟨fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ep⟩ − ⟨ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ft⟩⟨fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ej • ek⟩⟨fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ep⟩ + ⟨ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ft⟩⟨ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' fl⟩⟨fi ⋄ fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ep⟩ � =⟨ek • ep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ft⟩ − ⟨ep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ft ⋄ fl⟩ − ⟨ft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ep • ek⟩ + ⟨ft ⋄ fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' ep⟩ =0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' we get [[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' r]]D(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Similarly, we prove that � L◦D(A)(u) ⊗ idD(A) − idD(A) ⊗ L◦D(A)(u) �� r + τ(r) � = 0, for all u ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence there is a coboundary mock-Lie bialgebra structure on D(A) by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For ei ∈ A, we have ∆D(A)(ei) = � j � ei • ej ⊗ fj − ej ⊗ ei ◦D(A) fj � = � j � ei • ej ⊗ fj − ej ⊗ � L∗(ei)fj + L∗(fj)ei �� = � j,m � ei • ej ⊗ fj − ⟨fj, ei • em⟩ej ⊗ fm − ⟨fj ⋄ fm, ei⟩ej ⊗ em � = − � j,m ⟨fj ⋄ fm, ei⟩ej ⊗ em = −∆A(ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Therefore i1 : A → A ⊕ A∗ is a homomorphism of mock-Lie bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Similarly, i2 : A∗ → A ⊕ A∗ is also a homomorphism of mock-Lie bialgebras since ∆D(A)(fi) = ∆A∗(fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' With the above mock-Lie bialgebra structure given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5, A ⊕ A∗ is called the double of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We denote it by D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 13 5 O-operators of mock-Lie algebras and mock-Lie Yang-Baxter equation In this section, we interpret solution of mock-Lie Yang-Baxter equation in term of O-operators (see [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let V be a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For any r ∈ V ⊗ V , r can be regarded as a map from V ∗ to V in the following way: ⟨u∗, r(v∗)⟩ = ⟨u∗ ⊗ v∗, r⟩, ∀u∗, v∗ ∈ V ∗, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1) where ⟨·, ·⟩ is the ordinary pairing between the vector space V and the dual space V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The tensor r ∈ V ⊗V is called nondegenerate if the above induced linear map is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Moreover, any invertible linear map T : V ∗ → V induces a nondegenerate bilinear form ω(, ) on V by ω(u, v) = ⟨T −1(u), v⟩, ∀u, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2) Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' [9] A symplectic form on a mock-Lie algebra (A, ·) is a skew-symmetric non-degenerate bilinear form ω satisfying ω(x • y, z) + ω(y • z, x) + ω(z • x, y) = 0, ∀x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3) A mock-Lie algebra is called symplectic if it is endowed with a symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and r ∈ A ⊗ A be skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then r is a solution of mock-Lie YBE in A if and only if r satisfies r(ξ) • r(η) = r � L∗(r(ξ))η + L∗(r(η))ξ � , ∀ξ, η ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let {e1, · · · , en} be a basis of A and {e∗ 1, · · · , e∗ n} be the dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Since r is skew-symmetric, we can set r = � 1≤i,j≤n aijei ⊗ ej, aij = −aji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that ei • ej = �n k=1 Ck ijek, where Ck ij’s are the structure coefficients of mock-Lie algebra A on the basis {e1, · · · , en}, then we get r12 • r13 = � � 1≤i,j≤n aijei ⊗ ej ⊗ 1 � � � 1≤p,q≤n apqep ⊗ 1 ⊗ eq � = � 1≤i,j,p,q,k≤n Ck ipaijapqek ⊗ ej ⊗ eq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' r13 • r23 = � 1≤i,j,p,q,k≤n Ck jqaijapqei ⊗ ep ⊗ ek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' r12 • r23 = � 1≤i,j,p,q,k≤n Ck jpaijapqei ⊗ ek ⊗ eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then r is a solution of mock-Lie YBE in A if and only if � 1≤i,p≤n � Ck ipaijapq + Cq piakpaji − Cj ipakiapq � ek ⊗ ej ⊗ eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' On the other hand, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1), we get r(e∗ j) = �n i=1 aijei = − �n i=1 ajiei, 1 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' If we take ξ = e∗ j, η = e∗ q and by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4), we get � 1≤i,p≤n � Ck ipaijapq + Cq piakpaji − Cj ipakiapq � ek = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Therefore, it is easy to see that r is a solution of mock-Lie YBE in A if and only if r satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 14 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then a Rota-Baxter operator (of weight zero) is an O- operator of A associated to the adjoint representation (A, L) and a skew-symmetric solution of mock-Lie YBE in A is an O-operator of A associated to the representation (A∗, L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and r ∈ A ⊗ A be skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Suppose that there is a symmetric nondegenerate invariant bilinear form ω on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let φ : A → A∗ a linear map given by ⟨φ(x), y⟩ = ω(x, y) for any x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then r is a solution of mock-Lie YBE if and only if rφ is a Rota-Baxter operator (of weight zero) on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For any x, y ∈ A, we have φ(L(x)y) = L∗(x)φ(y) since ⟨φ(L(x)y), z⟩ = ω(x • y, z) = ω(y • x, z) = ω(y, x • z) = ⟨L∗(x)φ(y), z⟩, ∀x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' That is, the representations (A, L) and (A∗, L∗) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let ξ = φ(x), η = φ(y), then by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1, r is a solution of mock-Lie YBE in A if and only if rφ(x) • rφ(y) = r(ξ) • r(η) = r � L∗(r(ξ))η + L∗(r(η))ξ � = rφ � rφ(x) • y + x • rφ(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Therefore the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (V, ρ) be a representation of A and ρ∗ : A → gl(V ∗) be the dual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' A linear map T : V → A can be identified as an element in A ⊗ V ∗ ⊂ (A ⋉ρ∗ V ∗) ⊗ (A ⋉ρ∗ V ∗) as follows: Let {e1, · · · , en} be a basis of A, let {v1, · · · , vm} be a basis of V and {v∗ 1, · · · , v∗ m} be its dual space of V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We set T(vi) = n � k=1 aikek, i = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Since as vector space, Hom(V, A) ∼= A ⊗ V ∗, then we have T = m � i=1 T(vi) ⊗ v∗ i = m � i=1 n � k=1 aikek ⊗ v∗ i ∈ A ⊗ V ∗ ⊂ (A ⋉ρ∗ V ∗) ⊗ (A ⋉ρ∗ V ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5) Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' With the above notations, r = T − τ(T) is a skew-symmetric solution of the mock-Lie YBE in the semi-direct product mock-Lie algebra (A⋉ρ∗ V ∗) if and only if T is an O-operator associated to (V, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' We have r = T − τ(T) = m � i=1 T(vi) ⊗ v∗ i − m � i=1 v∗ i ⊗ T(vi), thus we obtain r12 • r13 = � 1≤i,j≤m � Tvi • Tvj ⊗ v∗ i ⊗ v∗ j − ρ∗(Tvi)v∗ j ⊗ v∗ i ⊗ Tvj − ρ∗(Tvj)v∗ i ⊗ Tvi ⊗ v∗ j � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' r12 • r23 = � 1≤i,j≤m � − v∗ i ⊗ Tvi • Tvj ⊗ v∗ j + Tvi ⊗ ρ∗(Tvj)v∗ i ⊗ v∗ j + v∗ i ⊗ ρ∗(Tvi)v∗ j ⊗ Tvj � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 15 r13 • r23 = � 1≤i,j≤m � v∗ i ⊗ v∗ j ⊗ Tvi • Tvj − Tvi ⊗ v∗ j ⊗ ρ∗(Tvj)v∗ i − v∗ i ⊗ Tvj ⊗ ρ∗(Tvi)v∗ j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' By the definition of dual representation, we know ρ∗(Tvj)v∗ i = m � p=1 ⟨v∗ i , ρ(Tvj)vp⟩v∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then � 1≤i,j≤m Tvi ⊗ ρ∗(Tvj)v∗ i ⊗ v∗ j = � 1≤i,j,p≤m ⟨v∗ p, ρ(Tvj)vi⟩Tvp ⊗ v∗ i ⊗ v∗ j = � 1≤i,j≤m T � ⟨v∗ p, ρ(Tvj)vi⟩vp � ⊗ v∗ i ⊗ v∗ j = � 1≤i,j≤m T � ρ(Tvj)vi � ⊗ v∗ i ⊗ v∗ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then we get r12 • r13 = � 1≤i,j≤m � Tvi • Tvj ⊗ v∗ i ⊗ v∗ j − v∗ i ⊗ v∗ j ⊗ T(ρ(Tvj)vi) − v∗ i ⊗ T(ρ(Tvj)vi) ⊗ v∗ j � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' − r12 • r23 = � 1≤i,j≤m � v∗ i ⊗ Tvi • Tvj ⊗ v∗ j − T(ρ(Tvj)vi) ⊗ v∗ i ⊗ v∗ j − v∗ i ⊗ v∗ j ⊗ T(ρ(Tvi)vj) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' r13 • r23 = � 1≤i,j≤m � v∗ i ⊗ v∗ j ⊗ Tvi • Tvj − T(ρ(Tvi)vj) ⊗ v∗ i ⊗ v∗ j − v∗ i ⊗ T(ρ(Tvi)vj) ⊗ v∗ j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence, we get r12 • r13 + r13 • r23 − r12 • r23 = � 1≤i,j≤m �� Tvi • Tvj − T(ρ(Tvi)vj) − T(ρ(Tvj)vi) � ⊗ v∗ i ⊗ v∗ j + v∗ i ⊗ v∗ j ⊗ � Tvi • Tvj − T(ρ(Tvi)vj) − T(ρ(Tvj)vi) � + v∗ i ⊗ � Tvi • Tvj − T(ρ(Tvi)vj) − T(ρ(Tvj)vi) � ⊗ v∗ j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' So r is a solution of the mock-Lie YBE in the semi-direct product mock-Lie algebra (A ⋉ρ∗ V ∗) if and only if T is an O-operator associated to (V, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Combining Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4, we have the following conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra and (V, ρ) be a representation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Set �A = A⋉ρ∗ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let T : V → A be a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' T is an O-operator of A associated to (V, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' T − τ(T) is a skew-symmetric solution of the mock-Lie YBE in the Jordan algebra �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' T − τ(T) is an O-operator of the mock-Lie algebra �A associated to ( �A∗, L∗ � A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The equivalence between the above (1) and (3) can be obtained by a straightforward proof and then Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4 follows from this equivalence and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' 16 The following conclusion reveals the relationship between mock-pre-Lie algebras and the mock-Lie algebras with a symplectic form: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, •) be a mock-Lie algebra with a symplectic form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then there exists a com- patible pre-mock-Lie algebra structure ” · ” on A given by ω(x · y, z) = ω(y, x • z), ∀x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Define a linear map T : A → A∗ by ⟨T(x), y⟩ = ω(x, y) for any x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' For any ξ, η, γ ∈ A∗, since T is invertible, there exist x, y, z ∈ A such that Tx = ξ, Ty = η, Tz = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then T −1 : A∗ → A is an O-operator of A associated to (A∗, L∗) since for any x, y, z ∈ A, we have ⟨T(x • y), z⟩ = ω(x • y, z) = ω(y, x • z) + ω(x, y • z) = ⟨L∗(x)T(y), z⟩ + ⟨L∗(y)T(x), z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='8, there is a compatible mock-pre-Lie algebra structure ” · ” on A given by x · y = T −1� L∗(x)T(y) � , ∀x, y ∈ A, which implies that ω(x · y, z) = ⟨T(x · y), z⟩ = ⟨L∗(x)T(y), z⟩ = ⟨T(y), x • z⟩ = ω(y, x • z), ∀x, y, z ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Hence the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' The following conclusion provides a construction of solutions of mock-Lie YBE in certain mock-Lie algebras from mock-pre-Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let (A, ·) be a mock-pre-Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Let {e1, · · · , en} be a basis of A and {e∗ 1, · · · , e∗ n} the dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Then r = n � i=1 (ei ⊗ e∗ i − e∗ i ⊗ ei) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='7) is a skew-symmetric solution of the mock-Lie YBE in the mock-Lie algebra (Aac) ⋉Θ∗ (Aac)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' It follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content='4 and the fact that the identity map id is an O-operator of the sub- adjacent mock-Lie algebra Aac of a mock-pre-Lie algebra associated to the representation (A, Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' AGUIAR, Infinitesimal Hopf algebras, Contemporary Mathematics 267, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FOT4oBgHgl3EQfyjQh/content/2301.12928v1.pdf'} +page_content=' Soc.' metadata={'source': 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implementation in FEniCSx +T. Lavigne, St´ephane Urcun, Pierre-Yves Rohan, Giuseppe Scium`e, Davide +Baroli, St´ephane P.A. Bordas +• Implementation of a poro-elastic formulation within FEniCSx. +• Single and double compartment columns are modeled. +• Elastic and Hyper-elastic solid scaffold are computed. +• Fast and accurate computation is obtained. +arXiv:2301.11256v1 [q-bio.TO] 26 Jan 2023 + +Multi-compartment poroelastic models of perfused +biological soft tissues: implementation in FEniCSx +T. Lavignea,b,c, St´ephane Urcuna, Pierre-Yves Rohanb, Giuseppe Scium`ec, +Davide Barolid,, St´ephane P.A. Bordasa,e,f +aInstitute of Computational Engineering, Department of Engineering, University of +Luxembourg, 6, avenue de la Fonte, Esch-sur-Alzette, L-4364, Luxembourg +bArts et Metiers Institute of Technology, IBHGC, 151 bd de l’hopital, Paris, 75013, +France +cArts et Metiers Institute of Technology, Univ. of Bordeaux, CNRS, Bordeaux INP, +INRAE, I2M Bordeaux, Avenue d’Aquitaine, Pessac, 33607, France +dUniversit`a della Svizzera Italiana, Euler Institute, Lugano, Swiss +eClyde Visiting Fellow, Department of Mechanical Engineering, The University of Utah, +Salt Lake City, United States +fDepartment of Medical Research, China Medical University Hospital, China Medical +University, Taichung, Taiwan +Abstract +Soft biological tissues demonstrate strong time-dependent and strain-rate me- +chanical behavior, arising from their intrinsic viscoelasticity and fluid-solid +interactions especially at sufficiently large time scales. The time-dependent +mechanical properties of soft tissues influence their physiological functions +and are linked to several pathological processes. Poroelastic modelling rep- +resents a promising approach because it allows to integrate multiscale/mul- +tiphysics data to probe biologically relevant phenomena at a smaller scale +and embed the relevant mechanisms at the larger scale. The implementation +of multiphasic flow poroelastic models howver is a complex undertaking, re- +quiring extensive knowledge. The FEniCSx Project provides a novel tool for +the automated solution of partial differential equations by the finite element +method. This paper aims to provide the required tools to model the mixed +formulation of poro-elasticity, from the theory to the implementation, within +FEniCSx. Several benchmark cases are studied. A column under confined +compression conditions is compared to the Terzaghi analytical solution, us- +ing the L2-norm. An implementation of poro-hyper-elasticity is proposed. +A bi-compartment column is compared to previously published results af a +Cast3m implementation. For all cases, accurate results are obtained in terms + +of a normalized Root Mean Square Error (RMSE). Furthermore, the FEn- +iCSx computation is found three times faster than the legacy FEniCS one. +The benefits of parallel computation are also highlighted. +Keywords: +Mixed Space, Poro-elasticity, Bi-compartment, FEniCSx +1. Introduction +Numerous problems of relevance in biomechanics, have at their core, the +presence of a deformable solid matrix which experiences flow-induced strain +: skin (brain (Budday et al. [1], Hosseini-Farid et al. [2], Franceschini et al. +[3], Urcun et al. [4]), muscle tissue (Lavigne et al. [5]), tumour (Scium`e et al. +[6], Scium`e [7], Oftadeh et al. [8]), artciular cartilage (Ateshian [9]) and lum- +bar intervertebral discs (Argoubi and Shirazi-Adl [10]) just to name a few). +The time-dependent mechanical properties of soft tissues influence their phys- +iological functions and are linked to several pathological processes. Although +a fluid-structure interaction (FSI) problem, the number and range of fluid +flows is generally so vast that the direct approach of a defined boundary be- +tween fluid and solid is impossible to apply. In these cases, homogenisation +and statistical treatment of the material-fluid system is possibly the only way +forward. A prominent technique of this type is that of poroelasticity. +Extensive studies have shown that poroelastic models can accuratley re- +produce the time-dependent behavior of soft tissues under different loading +conditions (Gimnich et al. [11], Argoubi and Shirazi-Adl [10], Peyrounette +et al. [12], Siddique et al. [13], Hosseini-Farid et al. [2], Franceschini et al. +[3], Lavigne et al. [5]). Compared to a visco-(hyper)-elastic formulation (Van +Loocke et al. [14], Simms et al. [15], Wheatley et al. [16], Vaidya and Wheat- +ley [17]), the poroelastic properties are independant of the sample size (Urcun +et al. [4]). Furthermore, a poroelastic approach can integrate multiscale/mul- +tiphysics data to probe biologically relevant phenomena at a smaller scale +and embed the relevant mechanisms at the larger scale (in particular, bio- +chemistry of oxygen and of inflammatory signalling pathways), allowing the +interpretation of the different time characteristics (Urcun et al. [18], Scium`e +et al. [6], Scium`e [7], Gray and Miller [19], Mascheroni et al. [20]). +Email address: davide.baroli@usi.ch (Davide Baroli ) +Preprint submitted to Elsevier +January 27, 2023 + +In most commercially available FE software packages used in research in +biomecahnics (ABAQUS, ANSYS, RADIOSS, etc), pre-programmed mate- +rial models for soft biological tissues are available. The disadvantage of these +pre-programmed models is that they are presented to the user as a black box. +Therefore, many researchers turn to implementing their own material formu- +lations through user subroutines (the reader is refered, for example, to the +excellent tutorial of Fehervary et al. [21] on the implementation of a nonlinear +hyperelastic material model using user subroutines in ABAQUS). This task, +however, is complex. When documentation is available, these only provide +expressions, without any derivations, lack details and background informa- +tion, making the implementation complex and error-prone. In addition, in +case of a custom formulation or the introduction of bio-chemical equations +for example, specific computational skills are required making the task even +more challenging. In the end, the use of commercially available FE software +packages limit the straightforward reproducibility of the research by other +teams. +The interest for open-source tools has skyrocketed to increase the impact +of the studies within the community. For Finite Element modeling, the FEn- +iCS project (Alnæs et al. [22]) is an OpenAccess software which has proven +its efficiency in biomechanics (Mazier et al. [23]). Based on a Python/C++ +coding interface and the Unified Form Language, it allows to easily solve +a defined variational form. Furthermore, its compatibility to open-source +meshers like GMSH make its use appealing. The project has already shown +is capacity solving large deformation problems (Mazier et al. [24]) and mixed +formulations (Urcun et al. [18, 4], Bulle [25]). Previous work provided the +implementation of poro-mechanics within the FEniCS project (Haagenson +et al. [26], Joodat et al. [27]). However, the FEniCS project is legacy and +has been replaced by the FEniCSx project in August 2022 (Alnæs et al. +[28], Scroggs et al. [29, 30]). +The aim of this paper is to propose a step-by-step explanation on how +to implement several poro-mechanical models in FEniCSx with a special at- +tention to parallel computation. First, an instantaneous uni-axial confined +compression of a porous elastic medium is proposed. This example corre- +sponds to an avascular tissue. Then, the same single-compartment model +is computed for a hyper-elastic solid scaffold followed by a 3D confined bi- +compartment modeling. +3 + +2. Confined compression of a column: geometrical definition +The time-dependant response of soft tissues are often assessed based on +confined compression creep and stress relaxation test data (Budday et al. +[1], Hosseini-Farid et al. [2], Franceschini et al. [3], Urcun et al. [4]). In this +section, therefore, all the benchmark examples focus on uni-axial confined +compression of a column sample as shown in figure 1. +Both 2D and 3D +geometries are studied. The column is described by its width (0.1*h) and +height (h) in 2D and its length in 3D. +h +p0 +us · x = 0 +us · y = 0 +pl = 0 (pb = 0) +y +x +Figure 1: Load (red), Boundary conditions (blue) and mesh (gray) of the uni-axial confined +compression of a porous 2D column of height h. +All the proposed codes were computed within a docker image of FEniCSx. +The dolfinx version used in this paper is v0.5.2. FEniCSx is a proficient plat- +form for parallel computation. All described codes here-under are compatible +with multi-kernel computation. The corresponding terminal command is: +mpirun +-n python3 + +Where is the number of threads to use and <filename> is the python +code of the problem. +Within the FEniCSx software, the domain (geometry) is discretized to +match with the Finite Element (FE) method. The space is thus divided in +4 + +nx × ny = 2 × 40 elements in 2D and nx × ny × nz = 2 × 40 × 2 elements +in 3D. The choice of the number of elements is further discussed section +3.5.2. In this article, the meshes are directly created within the FEniCSx +environment. However, as a strong compatibility exists with the GMSH api +(Geuzaine and Remacle [31]), it is recommended to use GMSH for this step. +An example of the use of GMSH API for a more complex geometry is given +section Appendix B. It is worth noting that we identify all the boundaries +of interest at this step for the future declaration of boundary conditions. +2.1. 2D mesh +Working in the python environment requires to import the working li- +braries. To create a 2D mesh, the first step is to import the following li- +braries: +import +dolfinx +import +numpy as np +from +dolfinx.mesh +import +create_rectangle , CellType , locate_entities , +meshtags +from +mpi4py +import +MPI +Then, the domain of resolution (mesh) is computed with: +Width , Height = 1e-5, 1e-4 #[m] +nx , ny += 2, 40 +#[ ] +mesh += create_rectangle (MPI.COMM_WORLD , np.array ([[0 ,0] ,[ Width , Height ]]) , +[nx ,ny], cell_type=CellType. quadrilateral ) +Once the mesh object has been created, its boundaries are identified us- +ing couples of (marker, locator) to tag with a marker value the elements of +dimension fdim fulfilling the locator requirements. +For the 2D mesh, the (marker, locator) couples are given by +# identifiers: 1 , 2, 3, 4 = bottom , right , top , left +boundaries = [(1, lambda x: np.isclose(x[1], 0)), +(2, lambda x: np.isclose(x[0], Width)), +(3, lambda x: np.isclose(x[1], Height)), +(4, lambda x: np.isclose(x[0], 0))] +Finally the entities are marked by: +facet_indices , facet_markers = [], [] +# dimension +of the +elements +we are +looking +for +fdim = mesh.topology.dim - 1 +for (marker , locator) in +boundaries: +facets = locate_entities (mesh , fdim , locator) +facet_indices .append(facets) +facet_markers .append(np.full_like(facets , marker)) +facet_indices = np.hstack( facet_indices ).astype(np.int32) +facet_markers = np.hstack( facet_markers ).astype(np.int32) +sorted_facets = np.argsort( facet_indices ) +5 + +# the +meshtags () +function +requires +sorted +facet_indices +facet_tag = meshtags(mesh , fdim , facet_indices [ sorted_facets ], facet_markers +[ sorted_facets ]) +2.2. 3D mesh +The method for a 3D mesh is similar to the 2D case. First the libraries +are imported and the geometry is created using a 3D function. The (marker, +locator) tuples are completed to describe all the boundaries of the domain. +The same tagging routine is used. +## libraries +import +dolfinx +import +numpy +from +dolfinx.mesh +import +create_box , CellType , locate_entities , +meshtags +from +mpi4py +import +MPI +## Mesh +generation +Length , Height , Width = 0.1, 1, 0.1 #[m] +nx , ny , nz = 2, 40, 2 +mesh += create_box(MPI.COMM_WORLD , numpy.array ([[0.0 ,0.0 ,0.0] ,[ Length , +Height , Width ]]) , [nx , ny , nz], cell_type=CellType.hexahedron ) +## Define +the +boundaries +of the +domain: +# 1, 2, 3, 4, 5, 6 = bottom , right , top , left , back , front +boundaries = [(1, lambda x: numpy.isclose(x[1], 0)), +(2, lambda x: numpy.isclose(x[0], Length)), +(3, lambda x: numpy.isclose(x[1], Height)), +(4, lambda x: numpy.isclose(x[0], 0)), +(5, lambda x: numpy.isclose(x[2], Width)), +(6, lambda x: numpy.isclose(x[2], 0))] +facet_indices , facet_markers = [], [] +fdim = mesh.topology.dim - 1 +for (marker , locator) in +boundaries: +facets = locate_entities (mesh , fdim , locator) +facet_indices .append(facets) +facet_markers .append(numpy.full_like(facets , marker)) +facet_indices = numpy.hstack( facet_indices ).astype(numpy.int32) +facet_markers = numpy.hstack( facet_markers ).astype(numpy.int32) +sorted_facets = numpy.argsort( facet_indices ) +facet_tag = meshtags(mesh , fdim , facet_indices [ sorted_facets ], facet_markers +[ sorted_facets ]) +3. Single-compartment porous medium +We propose to reproduce the instantaneous uni-axial confined compres- +sion at the top surface of a single-compartment porous column of height h, +Figure 1, described by a 2D elastic or a 3D hyper-elastic solid scaffold. Re- +garding the 2D elastic case, the column has a height of h = 100µm, the +instantaneous load p0 has a magnitude of 100Pa and is applied during 6 s. +Regarding the 3D hyper-elastic case, the column has a height of h = 1m, the +6 + +instantaneous load p0 has a magnitude of p0 = 0.3MPa and is applied during +100000 s. The mechanical parameters are respectively given Table 1 and +Table 2. To assess the reliability of our results, we compare our computed +solutions to the Terzaghi’s analytical solution and to the results of Selvadurai +and Suvorov [32], for the elastic and hyper-elastic scaffolds respectively. +Parameter +Symbol +Value +Unit +Young modulus +E +5000 +Pa +Poisson ratio +ν +0.4 +- +Intrinsic permeability +kε +1.8 × 10−15 +m2 +Biot coefficient +β +1 +- +Density of phase α +ρα +- +kg m−3 +IF viscosity +µl +1 × 10−3 +Pa s +Porosity +εl +0.5 +- +Solid grain Bulk modulus +Ks +1. × 1010 +Pa +Fluid Bulk modulus +Kl +2.2 × 109 +Pa +Table 1: Elastic mechanical parameters to compare with the Terzaghi solution +Parameter +Symbol +Value +Unit +Young modulus +E +600000 +Pa +Poisson ratio +ν +0.3 +- +Bulk modulus +K +500000 +Pa +Intrinsic permeability +kε +3 × 10−14 +m2 +IF viscosity +µl +1 × 10−3 +Pa s +Porosity +εl +0.2 +- +Solid grain Bulk modulus +Ks +1. × 1010 +Pa +Fluid Bulk modulus +Kl +2.2 × 109 or 5 × 105 +Pa +Biot coefficient +β +1 − K +Ks ≈ 1 +- +Table 2: Hyper-elastic mechanical parameters from Selvadurai and Suvorov [32]. In the +absence of information on the porosity, solid grain bulk modulus and fluid bulk modulus, +the parameter are arbitrarily chosen. +3.1. Terzaghi’s Analytical solution +The Terzaghi consolidation problem is often used for benchmarking porous +media mechanics, as an analytical solution of this problem exists. An im- +plementation of this experiment was proposed by Haagenson et al. [26], +7 + +within the legacy FEniCS project. +The Terzaghi problem consists in an +uni-directional confined compression experiment of a column (see Figure +1). Assuming small and uni-directional strains, incompressible homogeneous +phases and constant mechanical properties, the analytical expression of the +pore pressure is given in terms of series in Equation 1. +pl = 4p0 +π ++∞ +� +k=1 +(−1)k−1 +2k − 1 cos +� +(2k − 1)π +2 +y +h +� +exp +� +−(2k − 1)2π2 +4 +cvt +h2 +� +(1) +cv = +kε +µl(Sβ + β2 +M ) +(2) +M = 3Ks(1 − ν) +(1 + ν) +(3) +Sβ = β − εl +0 +Ks ++ εl +0 +Kl +(4) +Where p0=timposed ·n is the full applied load, y is the altitude, h is the initial +height of the sample , cv is the consolidation coefficient defined by (Equation +2), M the longitudinal modulus (Equation 3), Sβ the inverse of the Biot +Modulus (Equation 4) and εl +0 is the initial porosity. +3.2. Governing equations +Let one consider a bi-phasic structure composed of a solid scaffold filled +with interstitial fluid (IF). The medium is assumed fully saturated. In this +section, to set up the governing equations, we make the hypothesis of a Biot +coefficient equal to 1. The following convention is assumed: •s denotes the +solid phase and •l denotes the fluid phase (IF). The primary variables of +the problem are the pressure applied in the pores of the porous medium, +namely pl, and the displacement of the solid scaffold, namely us. (Equation +5) constrains the different volume fractions. The volume fraction of the phase +α is defined by (Equation 6). εl is called the porosity of the medium. +εs + εl = 1 +(5) +εα = +Volumeα +Volumetotal +(6) +8 + +Assuming that there is no inter-phase mass transport, the continuity +equations (mass conservation) of the liquid and solid phases are respectively +given by Equation 7 and Equation 8. +∂ +∂t(ρlεl) + ∇ · (ρlεlvl) = 0 +(7) +∂ +∂t(ρs(1 − εl)) + ∇ · (ρs(1 − εl)vs) = 0 +(8) +Regarding the distributivity of the divergence term, with a scalar and V +vector, +∇ · (aV) = a∇ · (V) + ∇a · V +(9) +Applied to 7 and Equation 8, and considering the definition of the material +derivative, Ds +Dtf = ∂f +∂t + ∇f · vs, the continuity equations are given by: +Ds +Dt(ρs(1 − εl)) + ρs(1 − εl)∇ · vs = 0 +(10) +Ds +Dt(ρlεl) + ∇ · (ρlεl(vl − vs)) + ρlεl∇ · vs = 0 +(11) +For the fluid phase, the Darcy’s law (Equation 12) is used to evaluate the +fluid flow in the porous medium. +εl(vl − vs) = −kε +µl (∇pl − ρlg) +(12) +Where kε is the intrinsic permeability (m2), µl is the dynamic viscosity (Pa s) +and g the gravity. +Introducing the state law +1 +ρα +Dsρα +Dt += +1 +Kα +Dpα +Dt , Kα being the bulk modulus +of the phase alpha, the Darcy’s law and summing 10 and Equation 11, we +obtain: +� εl +Kl + 1 − εl +Ks +� Dspl +Dt + ∇ · vs − ∇ · +�kε +µl ∇pl +� += 0 +(13) +Where S = +� +εl +Kl + 1−εl +Ks +� +is called the storativity coefficient. +9 + +Once the continuity equations are settled, one can define the quasi-static +momentum balance of the porous medium, Equation 14. +∇ · ttot = 0 +(14) +Where ttot is the total Cauchy stress tensor. We introduce an effective stress +tensor denoted teff, responsible for all deformation of the solid scaffold. Then, +ttot can be expressed as: +ttot = teff − βplId +(15) +Where Id is the identity matrix and β is the Biot coefficient. +Finally, the governing equations of this single compartment porous medium +are: +� εl +Kl + 1 − εl +Ks +� Dspl +Dt + ∇ · vs − ∇ · +�kε +µl ∇pl +� += 0 on Ω +(16) +∇ · ttot = 0 on Ω +(17) +Three boundaries are defined: the first one, Γu has imposed displacement +(Equation 18), the second one Γs has imposed external forces (Equation 19) +and Γp is submitted to an imposed pressure (fluid leakage condition (Equation +20)). We obtain: +teff = timposed on Γs +(18) +us = uimposed on Γu +(19) +p = 0 on Γp +(20) +According to Figure 1, Γp = Γs is the top surface and Γu covers the lateral +and bottom surfaces. +3.3. Effective stress +Two type of solid constitutive laws are considered: an elastic scaffold and +a hyper-elastic one. +10 + +3.3.1. Linear elasticity +In case of a Elastic scaffold, the effective stress tensor is defined as follows: +ϵ(u) = 1 +2(∇u + ∇uT) +(21) +teff = 2µϵ(us) + λtr(ϵ(us))Id +(22) +Where Id is the identity matrix and (λ, µ) the Lame coefficients. +3.3.2. Hyper-elasticity +In case of an hyper-elastic scaffold, other quantities are required. Let one +introduce the deformation gradient F: +F = Id + ∇us +(23) +Then, J is the determinant of F: +J = det(F) +(24) +According to the classic formulation of a finite element procedure, we +introduce C the right Cauchy-Green stress tensor and its first invariant I1. +By definition: +C = FTF +(25) +I1 = Tr(C) +(26) +The theory of hyper-elasticity defines a potential of elastic energy W(F). +The generalized Neo-Hookean potential (Equation 27) introduced by Treloar +[33], implemented in Abaqus and used by Selvadurai and Suvorov [32] is +evaluated in this article. +W(F) = µ +2 (J−2/3I1 − tr(Id)) + +�λ +2 + µ +3 +� +∗ (J − 1)2 +(27) +However, other potential were developed. It was shown that the hyper- +elastic potential can be expressed as the combination of a isochoric com- +ponent and a volumetric component (Marino [34], Horgan and Saccomandi +11 + +[35]). We define the lame coefficients by µ = +E +2(1−ν) and λ = +Eν +(1+ν)(1−2ν). For +a Neo-Hookean material, we further have: +W(F) = ˜W(I1, J) + U(J) +(28) +Where ˜W(I1, J) is the isochoric part and U(J) the volumetric one. The +study of Selvadurai and Suvorov [32] presented a compressible case (ν = 0.3) +reaching high deformation. Therefore, a compressible formulation of the Neo- +Hookean strain-energy potential from Pence and Gou [36], Horgan and Sac- +comandi [35] is also computed for comparison. Therefore, the implemented +isochoric part of the strain energy potential is: +˜W1(I1, J) = µ +2 (I1 − tr(Id) − 2 log[J]) +(29) +Two different volumetric parts (U1 and U2) which were proposed in Doll +and Schweizerhof [37] are implemented, +U1(J) = λ +2 log[J]2 +(30) +U2(J) = λ +2(J − 1)2 +(31) +Finally, from the potential (Equation 28 or 27) derives the first Piola- +Kirchhoff stress tensor as the effective stress such that: +teff = ∂W +∂F +(32) +3.4. Variational formulation +For the computation of the Finite Element (FE) model, the variational +form of Equation 16 and Equation 17 is introduced. Let one consider (q,v) +the test functions defined in the mixed space L2 +0(Ω) × [H1(Ω)]2. +With a first order approximation in time, Equation 16 gives: +S +dt +� +Ω +(pl − pl +n)qdΩ + 1 +dt +� +Ω +∇ · (us − us +n)qdΩ ++kε +µl +� +Ω +∇pl∇qdΩ = 0, ∀ q ∈ L2 +0(Ω) +(33) +12 + +Similarly, by integrating by part Equation 17, and including the Neumann +boundary conditions, we get: +� +Ω +teff : ∇vdΩ − +� +Ω +βpl∇ · vdΩ − +� +Γs +timposed · n · vdΓs = 0, +∀ v ∈ [H1(Ω)]2 +(34) +The first order approximation in time impose to define the initial condi- +tions which are fixed according to Table 3. +Parameter +Symbol +Value +Unit +Displacement +us +0 +m +Displacement at previous step +us +n +0 +m +IF pressure +pl +timposed · n +Pa +IF pressure at previous step +pl +n +0 +Pa +Table 3: Initial conditions for the single compartment model +Finally, the problem to solve is: Find (pl, us) ∈ L2 +0(Ω) × [H1(Ω)]2 such +that Equation 33 and Equation 34 are verified. +3.5. 2D linear elastic solid scaffold +3.5.1. FEniCSx implementation +This section aims to provide a possible implementation of a 2D elastic +problem and its comparison with the Terzaghi analytical solution. Conversely +to the former FEniCS project, Dolfinx is based on a more explicit use of the +libraries and requires to import them in the FEniCSx environment separately. +Therefore, each function used in the following implementation of the problem +needs to be imported as a first step. +import +numpy as np +from +dolfinx +import +nls +from +dolfinx.fem.petsc +import +NonlinearProblem +from +ufl +import +VectorElement , FiniteElement , MixedElement , +TestFunctions , TrialFunction +from +ufl +import +Measure , FacetNormal +from +ufl +import +nabla_div , dx , dot , inner , grad , derivative , +split +from +petsc4py.PETSc +import +ScalarType +from +mpi4py +import +MPI +from +dolfinx.fem +import (Constant , dirichletbc , Function , +FunctionSpace , locate_dofs_topological ) +from +dolfinx.io +import +XDMFFile +13 + +Then, the time parametrization is introduced, the load value T such that +timposed = p0 · n with n the outward normal to the mesh, and the material +parameters which are defined as ufl constants over the mesh. +## Time +parametrization +t += 0 +# Start +time +Tf += 6. +# End +time +num_steps = 1000 +# Number of time +steps +dt += (Tf -t)/num_steps # Time +step +size +## Material +parameters +E += Constant(mesh , ScalarType (5000)) +nu += Constant(mesh , ScalarType (0.4)) +lambda_m += Constant(mesh , ScalarType(E.value*nu.value /((1+ nu.value) +*(1 -2*nu.value)))) +mu += Constant(mesh , ScalarType (E.value /(2*(1+ nu.value)))) +rhos += Constant(mesh , ScalarType (1)) +kepsilon += Constant(mesh , ScalarType (1.8e -15)) +mul += Constant(mesh , ScalarType (1e -2)) +rhol += Constant(mesh , ScalarType (1)) +beta += Constant(mesh , ScalarType (1)) +epsilonl += Constant(mesh , ScalarType (0.2)) +Kf += Constant(mesh , ScalarType (2.2 e9)) +Ks += Constant(mesh , ScalarType (1 e10)) +S += (epsilonl/Kf)+(1- epsilonl)/Ks +## Mechanical +loading +pinit = 100 #[Pa] +T += Constant(mesh ,ScalarType (-pinit)) +The surface element for integration based on the tags and the normals of +the mesh are computed. +# Create +the +surfacic +element +ds = Measure("ds", domain=mesh , subdomain_data =facet_tag) +# compute +the +mesh +normals +to +express t^imposed = T.normal +normal = FacetNormal (mesh) +Two type of elements are defined for displacement and pressure, then +combined to obtain the mixed space (MS) of the solution. +displacement_element += VectorElement ("CG", mesh.ufl_cell (), 2) +pressure_element += FiniteElement ("CG", mesh.ufl_cell (), 1) +MS += FunctionSpace (mesh , MixedElement ([ +displacement_element , pressure_element ])) +The space of resolution being defined, we can introduce the Dirichlet +boundary conditions according to Equation 19, Equation 20 and Figure 1. +# 1 = bottom: uy=0, 2 = right: ux=0, 3= top: pl=0 drainage , 4= left: ux=0 +bcs += [] +fdim = mesh.topology.dim - 1 +# uy=0 +facets = facet_tag.find (1) +dofs += locate_dofs_topological (MS.sub (0).sub (1) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (1))) +# ux=0 +facets = facet_tag.find (2) +14 + +dofs += locate_dofs_topological (MS.sub (0).sub (0) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (0))) +# ux=0 +facets = facet_tag.find (4) +dofs += locate_dofs_topological (MS.sub (0).sub (0) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (0))) +# leakage p=0 +facets = facet_tag.find (3) +dofs += locate_dofs_topological (MS.sub (1) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (1))) +The problem depends on the time Equation 33. Initial conditions in dis- +placement and pressure are required. Therefore, we defined X0 the unknown +function and Xn the solution at the previous step. Giving the collapse() +function, we identify the initial displacement function Un and its mapping +within the Xn solution. Then, its values are set to 0 and reassigned in Xn +using the map. Xn.x.scatter forward() allows to update the values of Xn in +case of parallel computation. The same method is used to set up the initial +pressure field. To fit with the studied problems, the load is instantaneously +applied. Therefore, the initial pore pressure of the sample is assumed equal +to p0. +# X0 , Xn: Solution +and +previous +functions +of space +X0 = Function(MS) +Xn = Function(MS) +# Initial +values +# Solid +Displacement +Un_ , Un_to_MS = MS.sub (0).collapse () +FUn_ = Function(Un_) +with +FUn_.vector.localForm () as +initial_local : +initial_local .set(ScalarType (0.0)) +# Assign in Xn and +broadcast +to all the +threads +Xn.x.array[Un_to_MS] = FUn_.x.array +Xn.x. scatter_forward () +# IF +Pressure +Pn_ , Pn_to_MS = MS.sub (1).collapse () +FPn_ = Function(Pn_) +with +FPn_.vector.localForm () as +initial_local : +initial_local .set(ScalarType(pinit)) +# Assign in Xn and +broadcast +to all the +threads +Xn.x.array[Pn_to_MS] = FPn_.x.array +Xn.x. scatter_forward () +The deformation and effective stress given Equation 21 and Equation 22 +are defined by the following function: +def +teff_Elastic (u,lambda_m ,mu): +from +ufl +import sym , grad , nabla_div , Identity +## Deformation +epsilon = sym(grad(u)) +## Stress +return +lambda_m * nabla_div(u) * Identity(u. geometric_dimension ()) + 2* +mu*epsilon +15 + +Finally, splitting the two functions X0, Xn, and introducing the test func- +tions, the weak form is implemented as follows. +u,p +=split(X0) +u_n ,p_n=split(Xn) +# Set up the +test +functions +v,q = TestFunctions (MS) +# Equation +33 +F += (1/dt)*nabla_div(u-u_n)*q*dx + (kepsilon/mul)*dot(grad(p),grad(q))*dx ++ ( S/dt )*(p-p_n)*q*dx +# Equation +34 +F += inner(grad(v),teff(u))*dx - beta * p * nabla_div(v)*dx - T*inner(v, +normal)*ds (3) +Introducing the trial function of the mixed space dX0, we define the non- +linear problem based on the variational form, the unknown, the boundary +conditions and the Jacobian: +dX0 += TrialFunction (MS) +Js += derivative(F, X0 , dX0) +Problem = NonlinearProblem (F, X0 , bcs = bcs , J = Js) +3.5.2. Solving and results +To solve the non-linear problem defined here-above, a Newton solver is +tuned. +solver += nls.petsc. NewtonSolver (mesh.comm , Problem) +# Absolute +tolerance +solver.atol = 5e -10 +# relative +tolerance +solver.rtol = 1e -11 +solver. convergence_criterion = " incremental " +The parameters were set according to Table 1. During the resolution, we +computed for each step the error in L2-norm in pressure defined Equation 35. +These formulations are easily evaluated within the FEniCSx environment by +defining the following functions: +E(pl) = +�� +Ω(pl − pex)2dx +�� +Ω(pex)2dx +(35) +Where pex is the exact solution, computed from the Terzaghi’s analytical +formula. +def +terzaghi_p(x): +kmax =1e3 +p0 ,L=pinit ,Height +cv = kepsilon.value/mul.value *( lambda_m.value +2* mu.value) +pression =0 +16 + +for k in range (1,int(kmax)): +pression +=p0 *4/ np.pi*( -1) **(k -1) /(2*k -1)*np.cos ((2*k -1) *0.5* np.pi*(x +[1]/L))*np.exp ( -(2*k-1) **2*0.25* np.pi **2* cv*t/L**2) +pl=pression +return +pl +def +L2_error_p(mesh ,pressure_element ,__p): +V2 = FunctionSpace (mesh , pressure_element ) +pex = Function(V2) +pex. interpolate (terzaghi_p ) +L2_errorp , L2_normp = form(inner(__p - pex , __p - pex) * dx), form(inner +(pex , pex) * dx) +error_localp = assemble_scalar (L2_errorp)/ assemble_scalar (L2_normp) +error_L2p = np.sqrt(mesh.comm.allreduce(error_localp , op=MPI.SUM)) +return +error_L2p +To get a code suitable for parallel computation, the solutions needed to +be gathered on a same processor using the MPI.allreduce() function. Once +the error functions were defined, the problem is solved within the time loop: +# Create an output +xdmf +file to store +the +values +xdmf = XDMFFile(mesh.comm , "./ terzaghi.xdmf", "w") +xdmf.write_mesh(mesh) +# Solve +the +problem +and +evaluate +values of +interest +t = 0 +L2_p = np.zeros(num_steps , dtype=PETSc. ScalarType ) +for n in range(num_steps): +t += dt +num_its , converged = solver.solve(X0) +X0.x. scatter_forward () +# Update +Value +Xn.x.array [:] = X0.x.array +Xn.x. scatter_forward () +__u , __p = X0.split () +# Export +the +results +__u.name = " Displacement " +__p.name = "Pressure" +xdmf. write_function (__u ,t) +xdmf. write_function (__p ,t) +# Compute +L2 norm +for +pressure +error_L2p += L2_error_p (mesh ,pressure_element ,__p) +L2_p[n] = error_L2p +# Solve +tracking +if mesh.comm.rank == 0: +print(f"Time +step {n}, Number of +iterations {num_its}, Load {T.value +}, L2 -error p {error_L2p :.2e}") +xdmf.close () +The results obtained for pressure and displacements are provided Figure +2. The code to evaluate the pressure at given points is provided Appendix +C. +The curves show the efficiency of the simulation to reproduce the an- +alytical solution. +The accuracy of the simulation was also supported by +the estimation of the error based on the L2-norm of the pressure equal to +17 + +(a) +(b) +Figure 2: Comparison of the computed pore pressure against the analytical solution: in +(a) time and (b) space. The pressure was well recovered based on the evaluation of the +L2-norm error (3.57 ± 2.46) × 10−3. +(3.57 ± 2.46) × 10−3 which is deemed satisfactory. The same problem was +solved using the legacy FEniCS version. The proposed FEniCSx implementa- +tion was faster. It was computed in 9.48 seconds compared to the previously +31.82 seconds. +To show the efficiency of the parallel computation, the 3D case Appendix +A is considered. For a given spatio-temporal discretisation, a larger com- +putational time of 1 hour 4 minutes 29 seconds is needed using FEniCSx. +To reduce the time, the code naturally supports parallel computation. The +same code was run for several number of threads. Computed on 2 threads, +the code required 53 minutes 27 seconds. For 4 threads, the running time +was further reduced to 46 minutes 27 seconds. Finally, using 8 threads, the +computation time was reduced up to 28 minutes 9 seconds. +Finally, a convergence analysis on the meshing of the column was carried +out. The L2 error metric was used and its evolution for a nx × ny discretized +mesh is given Figure 3. As we could have expected from the 1D behavior of +a confined compression Terzaghi case, the error is almost independent from +the nx choice. Figure 3(a) shows that a ny ≥ 10 gives better estimations. +According to Figure 3(b), a balance between precision and computation time +must be considered. The more elements, the higher the computation time. +To ensure obtaining a reliable solution, a mesh of nx × ny = 2 × 40 was used. +18 + +1e2 +1.0 +Analytic y=0 +Analytic y=h/2 +0.8 +FEniCSx y=0 +Pressure (Pa) +FEniCSxy=h/2 +0.6 +0.4 +0.2 +0 +1 +2 +3 +4 +5 +6 +time (s)1e-4 +1.0 +Analytic +FEniCSx +0.8 +(m) +0.6 +Height ( +0.4 +t=4.8s +t=2.4s +t=1.2s +0.2 +0.0 +0 +10 +20 +30 +40 +50 +60 +70 +Pressure(Pa)(a) +(b) +Figure 3: Convergence analysis for a nx × ny discretized mesh: L-2 norm (a) and Compu- +tation time (b) +3.6. 3D hyper-elastic scaffold +3.6.1. FEniCSx implementation +The implementation method of the 3D case is the same. However, spe- +cial attention must be placed on the boundary. Indeed, moving from 2D +to 3D introduces two more boundaries. Therefore, the Dirichlet boundary +conditions definition is completed with: +# uz=0 +facets = facet_tag.find (5) +dofs += locate_dofs_topological (MS.sub (0).sub (2) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (2))) +# uz=0 +facets = facet_tag.find (6) +dofs += locate_dofs_topological (MS.sub (0).sub (2) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (2))) +The effective stress tensor is also different. As an example, the stress +tensor resulting from the potential W(F) = ˜W1(I1, J) + U1(J) is defined in +FEniCSx by: +def +teff(u,lambda_m ,mu): +from +ufl +import +variable , Identity , grad , det , tr , ln , diff +## Deformation +gradient +F = variable(Identity(len(u)) + grad(u)) +J += variable(det(F)) +## Right +Cauchy -Green +tensor +C = variable(F.T * F) +##Invariants +of +deformation +tensors +Ic = variable(tr(C)) +## Potential +W = (mu / 2) * (Ic - 3) - mu * ln(J) + (lambda_m / 2) * (ln(J))**2 +return +diff(W, F) +19 + +0.15 +0.075 +D +0.050 +rr +0.025 +0 +10 +4 +6 +hy +20 +8 +xu +30 +10Computation time (s) +40 +30 +30 +20 +20 +30 +20 +2 +10 +4 +6 +nx +8 +10 +0All other developed potential are available in the supplementary material. +3.6.2. Results +The same solver options as for the 2D case were used. +To limit the +computation time, the time step was made variable: dt=500 for t ∈ [0, 20000], +dt=1000 for t ∈ [20000, 60000] and dt=10000 for t ∈ [60000, 100000]. A total +of 84 time steps was then considered. +The parameters were set according to Table 2. The results for the previ- +ously defined strain-energy potential are given Figure 4. Each finite element +problem was computed in 23.6 ± 4.3 seconds on 8 threads. Independently +from the choice of the potential, the consolidated pressure was retrieved. On +the contrary, the resulting displacement depends on the chosen potential but +a same order of magnitude is found for all the cases and describe well the +observations proposed in Selvadurai and Suvorov [32]. +In the absence of information about the porosity or the fluid bulk mod- +ulus in the referent study, two fluid bulk modulus were considered. In case +where the fluid bulk modulus is made close to the water one (Kf = 2.2×109), +the hyper-elastic material well recovers the expected values. However, mis- +matches appear for a linear scaffold. This can result from the use of a elastic +law for large deformations. In case of a lower value of the fluid bulk modulus +Kf = 5×105 (i.e., this can correspond to a non-constant value of the perme- +ability and the porosity), the elastic behavior was recovered but differences +on the hyper-elastic formulation were obtained. +We believe that these differences result from a permeability depending on +the stress state of the column which has not been developed in the referent +paper (’Initial values of the permeability and viscosity are the same for all +three materials.’ from Selvadurai and Suvorov [32]). +20 + +(a) +(b) +(c) +(d) +Figure 4: Kf = 2.2 × 109: (a) Displacement of the to surface points and (b) pressure +at the bottom of the column. Kf = 5 × 105: (c) Displacement of the to surface points +and (d) pressure at the bottom of the column. The computed Linear Elastic (LE) and +Neo-Hookean (NH) for both volumetric functions and the found calibrated parameters are +super-imposed with the expected values from Selvadurai and Suvorov [32]. +4. Confined bi-compartment porous-elastic medium +Sections 3 proposed a poro-mechanical modeling of a single-compartment +porous medium (suitable for an avascularised tissue for instance). In case of +in vivo modeling, at least one more fluid phase is required: the blood. A +3D confined compression example of a column of height 100 µm is proposed, +based on the here-after variational formulation and Scium`e [7] study. The +load is applied as a sinusoidal ramp up to the magnitude of 100 Pa during 5 +seconds. Then, the load is sustained for 125 seconds. +For more complex geometries, a gmsh example of a rectangle geometry +indented by a cylindrical beam on its top surface and the corresponding local +21 + +le- +Displacement (m) +-2 +3 +0 +20000 +40000 +60000 +80000 +100000 +time (s)1e5 +3.0 +LE +(l1-3-2log())+log()2 +2.5 +(l1-32log() +( -1)2 +2.0 +(-2/3/1 -3) + (+)* (- 1)2 +Linear Elastic [29] +1.5 +Neo-Hooke [29] +1.0 +0.5 +0.0 +0 +20000 +40000 +60000 +80000 +100000 +time (s)le- +Displacement (m) +-2 +3 +0 +20000 +40000 +60000 +80000 +100000 +time (s)1e5 +3.0 +2.5 + (Pa) +2.0 +Pressure +1.5 +1.0 +0.5 +0.0 +0 +20000 +40000 +60000 +80000 +100000 +time (s)refinement are proposed Appendix B. +Parameter +Symbol +Value +Unit +Young modulus +E +5000 +Pa +Poisson ratio +ν +0.2 +- +IF viscosity +µl +1 +Pa s +Intrinsic permeability +kε +1. × 10−14 +m2 +Biot coefficient +β +1 +- +Density of phase α +ρα +- +kg m−3 +Porosity +εl +0.5 +- +Vessel Bulk modulus +Kν +1 × 103 +Pa +vessel Intrinsic permeability +kε +b +2 × 10−16 or 4 × 10−16 +m2 +Blood viscosity +µb +4.0 × 10−3 +Pa s +Initial vascular porosity +εb +0 +0% or 2% or 4% +- +Vascular porosity +εb +Equation 48 +- +Table 4: Mechanical parameters for the bi-compartment model +4.1. Governing Equations +Let one consider a vascular multi-compartment structure composed of a +solid scaffold filled with interstitial fluid (IF) and blood. The medium is +assumed fully saturated. The following convention is assumed: •s denotes +the solid phase, •l denotes the interstitial fluid phase (IF) and •b denotes the +vascular part. The primary variables of the problem are the pressure applied +in the pores of the extra-vascular part of the porous medium, namely pl, the +blood pressure, namely pb, and the displacement of the solid scaffold, namely +us. (Equation 36) links the different volume fractions. The volume fraction +of the phase α is defined by (Equation 6). εl is called the extra-vascular +porosity of the medium. +εs + εl + εb = 1 +(36) +Assuming that there is no inter-phase mass transport (i.e. the IF and the +blood are assumed pure phases), the continuity equations (mass conservation) +of the solid, the IF and the blood phases are respectively given by Equation +37, 38, 39. +22 + +∂ +∂t(ρs(1 − εl − εb)) + ∇ · (ρs(1 − εl − εb)vs) = 0 +(37) +∂ +∂t(ρlεl) + ∇ · (ρlεlvl) = 0 +(38) +∂ +∂t(ρbεb) + ∇ · (ρbεbvb) = 0 +(39) +According to section 3.2, and dividing each equation by the corresponding +density, the continuity equations can be re-expressed as: +Ds +Dt(1 − εl − εb) + (1 − εl − εb)∇ · vs = 0 +(40) +Dsεl +Dt + ∇ · (εl(vl − vs)) + εl∇ · vs = 0 +(41) +Dsεb +Dt + ∇ · (εb(vb − vs)) + εb∇ · vs = 0 +(42) +For the fluid phase, Darcy’s law (Equation 43, 44) is used to evaluate the +fluid flow in the porous medium. +εl(vl − vs) = −kε +µl (∇pl − ρlg) +(43) +εb(vb − vs) = −kb +µb(∇pb − ρbg) +(44) +Where kε, kb are the intrinsic permeabilities (m2), µl, µb are the dynamic +viscosities (Pa s), pl, pb the pressures and g the gravity. +Equation 39 gives the following relationship: +Dsεl +Dt = −Dsεb +Dt + (1 − εl − εb)∇ · vs +(45) +Considering Equations 43, 45, Equation 41 becomes: +−Dsεb +Dt − ∇ · (kε +µl ∇pl) + (1 − εb)∇ · vs = 0 +(46) +23 + +Then, reading Equation 44, Equation 42 gives: +Dsεb +Dt − ∇ · (kb +µb∇pb) + εb∇ · vs = 0 +(47) +Considering a vascular tissue, we assume that the blood vessels are mostly +surrounded by IF so they have weak direct interaction with the solid scaf- +fold. Furthermore, the vessels are assumed compressible. Therefore, a state +equation for the volume fraction of blood is introduced Equation 48. +εb = εb +0 · +� +1 − pl − pb +Kν +� +(48) +Where εb +0 denotes the blood volume fraction when pl = pb, Kν is the vessel +compressibility. +It follows that Equations 46, 47 can be re-written as: +− εb +0 +Kν +�Dspl +Dt − Dspb +Dt +� +− ∇ · (kε +µl ∇pl) + (1 − εb)∇ · vs = 0 +(49) +εb +0 +Kν +�Dspl +Dt − Dspb +Dt +� +− ∇ · (kb +µb∇pb) + εb∇ · vs = 0 +(50) +Once the continuity equations are settled, one can define the quasi-static +momentum balance of the porous medium, Equation 51. +∇ · ttot = 0 +(51) +Where ttot is the total Cauchy stress tensor. We introduce an effective stress +tensor denoted teff, responsible for all deformation of the solid scaffold. Then, +ttot can be expressed as: +ttot = teff − (1 − ζ)plId − ζpbId +(52) +ϵ(u) = 1 +2(∇u + ∇uT) +(53) +teff = 2µϵ(us) + λtr(ϵ(us))Id +(54) +ζ = εb +0 +� +1 − 2pl − pb +Kν +� +(55) +24 + +Four boundaries are defined: the first one, Γu has imposed displacement +(Equation 56), the second one Γs has imposed external forces (Equation 57) +and Ωp has imposed pressure (fluid leakage condition (Equation 58, 59)). We +obtain: +teff = timposed on Γs +(56) +us = uimposed on Γu +(57) +pl = 0 on Γp +(58) +pb = 0 on Γp +(59) +The initial conditions are given Table 5. +Parameter +Symbol +Value +Unit +Displacement +us +0 +m +Displacement at previous step +us +n +0 +m +IF pressure +pl +0 +Pa +IF pressure at previous step +pl +n +0 +Pa +Blood pressure +pb +0 +Pa +Blood pressure at previous time step +pb +0 +Pa +Vascular porosity +εb +εb +0 +- +Table 5: Initial conditions for the bi-compartment model +4.2. Variational Form +For the computation of the FE model, the variational form of Equation 49- +51 must be introduced. Let one consider (ql,qb,v) the test functions defined in +the mixed space L2 +0(Ω) × L2 +0(Ω) × [H1(Ω)]3. With a first order approximation +in time, Equation 49, 50 gives: +25 + +− εb +0 +Kν +1 +dt +� +Ω +(pb − pb +n − pl + pl +n)qldΩ + 1 − εb +dt +� +Ω +∇ · (us − us +n)qldΩ ++kε +µl +� +Ω +∇pl∇qldΩ = 0, ∀ ql ∈ L2 +0(Ω) +(60) +εb +Kν +1 +dt +� +Ω +(pb − pb +n − pl + pl +n)qbdΩ + εb +dt +� +Ω +∇ · (us − us +n)qbdΩ ++kb +µb +� +Ω +∇pb∇qbdΩ = 0, ∀ qb ∈ L2 +0(Ω) +(61) +Similarly, by integrating by part Equation 51, and including the Neumann +boundary conditions, we get: +� +Ω +teff : ∇vdΩ − +� +Ω +(1 − ζ)pl∇ · vdΩ +− +� +Ω +ζpb∇ · vdΩ +− +� +Γs +timposed · vdΓs = 0, ∀ v ∈ [H1(Ω)]3 +(62) +4.3. FEniCSx Implementation +This section provides the code of a multi-compartment 3D column in +confined compression. In order to evaluate the FEniCSx implementation, +this case is similar to the Cast3m solution proposed in Scium`e [7]. 3 cases +are studied: avascular tissue, vascular porosity of 2% and vascular porosity of +4%. The load is applied as a sine ramp during 5 seconds and then sustained +during 125 seconds. +The time discretisation is introduced. +t, t_ramp , t_sust = 0, 5, 125 +# Start +time +Tf += t_ramp+t_sust +# End +time +num_steps += 1301 +# Number of time +steps +dt += (Tf -t)/num_steps +# Time +step +size +We then introduce the material parameters according to Table 5. The +three cases of vascularisation and Equation 55 are defined. +E += Constant(mesh , ScalarType (5000)) +nu += Constant(mesh , ScalarType (0.2)) +kepsilon_l = Constant(mesh , ScalarType (1e -14)) +mu_l += Constant(mesh , ScalarType (1)) +26 + +lambda_m += Constant(mesh , ScalarType(E.value*nu.value /((1+ nu.value) +*(1 -2*nu.value)))) +mu += Constant(mesh , ScalarType (E.value /(2*(1+ nu.value)))) +Knu += Constant(mesh , ScalarType (1000)) +# compressibility +of the +vessels +mu_b += Constant(mesh , ScalarType (0.004)) #dynamic +mu_l of the +blood +case =1 +if case +==0: +epsilon_b_0=Constant(mesh , ScalarType (0.00)) #initial +vascular +porosity +k_b=Constant(mesh , ScalarType (2e -16)) +#intrinsic +permeability +of +vessels +def +zeta(pl ,pb): +return +Constant(mesh ,ScalarType (0.)) +elif +case +==1: +epsilon_b_0=Constant(mesh , ScalarType (0.02)) #initial +vascular +porosity +k_b=Constant(mesh , ScalarType (2e -16)) +#intrinsic +permeability +of +vessels +def +zeta(pl ,pb): +return +epsilon_b_0.value *(1 -2*(pl -pb)/Knu.value) +elif +case +==2: +epsilon_b_0 = Constant(mesh , ScalarType (0.04)) +#initial +vascular +porosity +k_b = Constant(mesh , ScalarType (4e -16)) +#intrinsic +permeability +of +vessels +def +zeta(pl ,pb): +return +epsilon_b_0.value *(1 -2*(pl -pb)/Knu.value) +Then, the integration space, boundary and initial conditions are set up +for the displacement, the IF pressure and the blood pressure. +## Mechanical +loading (Terzaghi) +pinit = 200 #[Pa] +T += Constant(mesh ,ScalarType (-pinit)) +## Define +Mixed +Space (R2 ,R, R) -> (u,pl , pb) +element += VectorElement ("CG", mesh.ufl_cell (), 2) +pressure_element = FiniteElement ("CG", mesh.ufl_cell (), 1) +MS += FunctionSpace (mesh , MixedElement ([ element , +pressure_element , pressure_element ])) +# Create +the +solution +and +initial +spaces +X0 = Function(MS) +Xn = Function(MS) +# Create +the +surfacic +element +ds = Measure("ds", domain=mesh , subdomain_data =facet_tag) +# compute +the +normals +normal = FacetNormal (mesh) +# Define +the +Dirichlet +conditions +bcs += [] +# uy=0 +facets = facet_tag.find (1) +dofs += locate_dofs_topological (MS.sub (0).sub (1) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (1))) +# ux=0 +facets = facet_tag.find (2) +dofs += locate_dofs_topological (MS.sub (0).sub (0) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (0))) +# ux=0 +facets = facet_tag.find (4) +dofs += locate_dofs_topological (MS.sub (0).sub (0) , fdim , facets) +27 + +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (0))) +# uz=0 +facets = facet_tag.find (5) +dofs += locate_dofs_topological (MS.sub (0).sub (2) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (2))) +# uz=0 +facets = facet_tag.find (6) +dofs += locate_dofs_topological (MS.sub (0).sub (2) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (2))) +# leakage +pl=pb=0 +facets = facet_tag.find (3) +dofs += locate_dofs_topological (MS.sub (1) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (1))) +dofs += locate_dofs_topological (MS.sub (2) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (2))) +# Set +Initial +values +# Displacement +Un_ , Un_to_MS = MS.sub (0).collapse () +FUn_ = Function(Un_) +with +FUn_.vector.localForm () as +initial_local : +initial_local .set(ScalarType (0.0)) +# Update Xn for all +threads +Xn.x.array[Un_to_MS] = FUn_.x.array +Xn.x. scatter_forward () +# IF +Pressure +Pn_ , Pn_to_MS = MS.sub (1).collapse () +FPn_ = Function(Pn_) +with +FPn_.vector.localForm () as +initial_local : +initial_local .set(ScalarType (0)) +# Update Xn for all +threads +Xn.x.array[Pn_to_MS] = FPn_.x.array +Xn.x. scatter_forward () +# Blood +Pressure +Pbn_ , Pbn_to_MS = MS.sub (2).collapse () +FPbn_ = Function(Pbn_) +with +FPbn_.vector.localForm () as +initial_local : +initial_local .set(ScalarType (0)) +# Update Xn for all +threads +Xn.x.array[Pbn_to_MS] = FPbn_.x.array +Xn.x. scatter_forward () +Internal variables are required. The vessels are compressible so we include +the evolution of the vascular porosity as a function representing Equation 48. +# Internal +variables: vascular +porosity +Poro_space += FunctionSpace (mesh , pressure_element ) +poro_b += Function(Poro_space ) # vascular +porosity +# Initialize +with +poro_b.vector.localForm () as +initial_local : +initial_local .set(ScalarType( epsilon_b_0 .value)) +# Update +poro_b.x. scatter_forward () +poro_b.name="poro_b" +A xdmf file is opened to store the results. +xdmf = XDMFFile(mesh.comm , "terzaghi.xdmf", "w") +xdmf.write_mesh(mesh) +28 + +The test functions as well as the variational form are introduced according +to Equations 60, 61, 62. +u, pl , pb += split(X0) +u_n , pl_n , pb_n = split(Xn) +v, ql , qb = TestFunctions (MS) +dx = Measure("dx", metadata ={" quadrature_degree ": 4}) +F = (1- poro_b)*(1/ dt)*nabla_div(u-u_n)*ql*dx + ( kepsilon_l /( mu_l) )*dot( +grad(pl),grad(ql) )*dx - ( epsilon_b_0 /Knu)*( (1/ dt)*(pb -pb_n -pl+pl_n) )* +ql*dx +F += +poro_b *(1/ dt)*nabla_div(u-u_n)*qb*dx + ( k_b /( mu_b) )*dot( grad(pb), +grad(qb) )*dx + ( epsilon_b_0 /Knu)*( (1/ dt)*(pb -pb_n -pl+pl_n) )*qb*dx +F += inner(grad(v),teff(u))*dx - (1-zeta(pl ,pb))*pl*nabla_div(v)*dx - zeta( +pl ,pb)*pb*nabla_div(v)*dx - T*inner(v,normal)*ds (3) +Finally, the problem to be solved is defined and a Newton method is used +for each time step, the vascular porosity is updated and the results are stored +in the xdmf file. +dX0 += TrialFunction (MS) +J += derivative(F, X0 , dX0) +Problem = NonlinearProblem (F, X0 , bcs = bcs , J = J) +solver += nls.petsc. NewtonSolver (mesh.comm , Problem) +# Set +Newton +solver +options +solver.atol = 5e -10 +solver.rtol = 1e -11 +solver. convergence_criterion = " incremental " +t = 0 +for n in range(num_steps): +t += dt +if t < t_ramp: +f1 = 0.5 * (1 - np.cos(np.pi*t/t_ramp)) +else: +f1 = 1 +T.value = +-200*f1 +num_its , converged = solver.solve(X0) +X0.x. scatter_forward () +# Update +Value +Xn.x.array [:] = X0.x.array +Xn.x. scatter_forward () +# Update +porosity +poro_b.x.array [:] = epsilon_b_0 .value *(1 -(1/ Knu.value)*(X0.x.array[ +Pn_to_MS]-X0.x.array[Pbn_to_MS ])) +poro_b.x. scatter_forward () +# Save +data +__u , __pl , __pb = X0.split () +__u.name = " Displacement " +__pl.name = "Pressure +IF" +__pb.name = "Pressure +blood" +xdmf. write_function (__u ,t) +xdmf. write_function (__pl ,t) +xdmf. write_function (__pb ,t) +xdmf. write_function (poro_b ,t) +xdmf.close () +29 + +4.4. Results +(a) +(b) +(c) +Figure 5: Comparison of the results obtained using FEniCSx against Scium`e [7] results. All +results were shifted to obtain similar figures. The solid, doted and dashed lines respectively +represent the 0%, 2% 4% initial vascular porosity. (a) Evolution of the pressure at the +bottom points. (b) Displacement of the top points. (c) Vascular porosity at the bottom +points. The behaviour was well retrieved for all the cases with a NRMSE lower than 10% +for all variables according to Table 6. +The evolution of the vascular and interstitial pressures at the bottom +points and the vertical displacement at the top points are provided Figure +5. Each solution was obtained in 6 ± 2 minutes on 8 threads. The overall +behavior of the interstitial fluid pressure, the blood pressure and the solid +displacement were retrieved. To quantitatively assess the reliability of our +implemented model, The normalized root mean square error (NRMSE, Equa- +tion 63) was computed for each case with the results obtained with Cast3m +in Scium`e [7], Table 6. +30 + +200 +Load +0% +150 +p'atthe bottom points +100 +50 +pbatthe bottom points +0 +-10 +-5 +0 +5 +10 +15 +20 +25 +30 +time (s)1e-6 +1.0 +0.5 +Vertical Displacementoftoppoints +0.0 +0.5 +(w) +-1.0 +0% +-1.5 +-2.0 +-2.5 +-3.0 +-10 +0 +10 +20 +30 +time (s)1e-2 +4 +2% +3 +4% +1 +Vascular porosity at the bottom points +0 +0 +20 +40 +60 +80 +100 +120 +time (s)Load +Sciume et al. (2021), g = 0 +Sciume et al. (2021), g = 2 +Sciume et al. (2021), eg = 4NRMSE(x, xref) = +� +1 +N +� +i∈[1,N](x − xref)2 +mean(xref) +(63) +Parameter +0% +2% +4% +pl +1.4 % +3.1 % +5.1 % +uy +0.3 % +2.2 % +3.7 % +pb +- +4.7 % +8.8 % +εb +0 +- +0.4 % +0.6 % +Table 6: NRMSE computed for each studied variable. +The NRMSE was found lower than 10% for all unknowns. The differences +are assumed to result from the method of resolution which differs between +Cast3m and FEniCSx. Indeed, the Cast3m procedure relies on a staggered +solver whereas our results were obtained using a monolithic solver. +The +order of magnitudes of the NRMSE made us however consider our solution +as trustworthy. +5. Conclusion +The objective of this paper was to propose a step-by-step explanation +on how to implement several poro-mechanical models in FEniCSx with a +special attention to parallel computation. +Several benchmark cases for a +mixed formulation were evaluated. First, a confined column was simulated +under compression. Accurate results according to the L2-norm were found +compared to the analytical solution. Furthermore, the code was computed 3 +times faster than in the legacy FEniCS environment. Then, a possible im- +plementation of an hyper-elastic formulation was proposed. The model was +validated using Selvadurai and Suvorov [32] values. Finally, a confined bi- +compartment sample was simulated. The results were compared to Scium`e +[7] data. +Small differences were observed due to the choice of the solver +(staggered or monolithic) but remained acceptable. The authors hope that +this paper will contribute to facilitate the use of poroelasticity in the biome- +chanical engineering community. This article and its supplementary material +constitute a starting point to implement their own material models at a pre- +ferred level of complexity. +31 + +6. Supplementary material +The python codes corresponding to the workflows and the docker file of +this article are made available for 2D and 3D cases on the following link: +https://github.com/Th0masLavigne/Dolfinx_Porous_Media.git. +7. Declaration of Competing Interest +Authors have no conflicts of interest to report. +8. Acknowledgment +This research was funded in whole, or in part, by the Luxembourg Na- +tional Research Fund (FNR),grant reference No. 17013182. For the purpose +of open access, the author has applied a Creative Commons Attribution 4.0 +International (CC BY 4.0) license to any Author Accepted Manuscript ver- +sion arising from this submission. The present project is also supported by +the National Research Fund, Luxembourg, under grant No. C20/MS/14782078/QuaC. +32 + +Appendix A. 3D Terzaghi example +Here-after is proposed a minimal working code corresponding to the 2D +case included within the text. +import +numpy as np +import +csv +from +petsc4py +import +PETSc +import +dolfinx +from +dolfinx +import +nls +from +dolfinx.io +import +XDMFFile +from +dolfinx.mesh +import +CellType , create_box , locate_entities_boundary +, locate_entities , meshtags +from +dolfinx.fem +import (Constant , dirichletbc , Function , +FunctionSpace , locate_dofs_topological , form , assemble_scalar ) +from +dolfinx.fem.petsc +import +NonlinearProblem +from +dolfinx.geometry +import +BoundingBoxTree , compute_collisions , +compute_colliding_cells +from +petsc4py.PETSc +import +ScalarType +from +mpi4py +import +MPI +from +ufl +import (FacetNormal , Identity , Measure , TestFunctions +, TrialFunction , VectorElement , FiniteElement , dot , dx , inner , grad , +nabla_div , div , sym , MixedElement , derivative , split) +# +def +epsilon(u): +return +sym(grad(u)) +# +def +teff(u): +return +lambda_m * nabla_div(u) * Identity(u. geometric_dimension ()) + 2* +mu*epsilon(u) +# +kmax =1e3 +def +terzaghi_p(x): +p0 ,L=pinit ,Height +cv = permeability .value/viscosity.value *( lambda_m.value +2* mu.value) +pression =0 +for k in range (1,int(kmax)): +pression +=p0 *4/ np.pi*( -1) **(k -1) /(2*k -1)*np.cos ((2*k -1) *0.5* np.pi*(x +[1]/L))*np.exp ( -(2*k-1) **2*0.25* np.pi **2* cv*t/L**2) +pl=pression +return +pl +# +def +L2_error_p(mesh ,pressure_element ,__p): +V2 = FunctionSpace (mesh , pressure_element ) +pex = Function(V2) +pex. interpolate (terzaghi_p ) +L2_errorp , L2_normp = form(inner(__p - pex , __p - pex) * dx), form(inner +(pex , pex) * dx) +error_localp = assemble_scalar (L2_errorp)/ assemble_scalar (L2_normp) +error_L2p = np.sqrt(mesh.comm.allreduce(error_localp , op=MPI.SUM)) +return +error_L2p +# +## Create +the +domain / mesh +Height = 1e-4 #[m] +Width += 1e-5 #[m] +Length = 1e-5 #[m] +33 + +mesh += create_box(MPI.COMM_WORLD , np.array ([[0.0 ,0.0 ,0.0] ,[ Length , Width , +Height ]]) , [8, 8, 20], +cell_type=CellType. tetrahedron ) +# +## Define +the +boundaries: +# 1 = bottom , 2 = right , 3=top , 4=left , 5=back , 6= front +boundaries = [(1, lambda x: np.isclose(x[2], 0)), +(2, lambda x: np.isclose(x[0], Length)), +(3, lambda x: np.isclose(x[2], Height)), +(4, lambda x: np.isclose(x[0], 0)), +(5, lambda x: np.isclose(x[1], Width)), +(6, lambda x: np.isclose(x[1], 0))] +# +facet_indices , facet_markers = [], [] +fdim = mesh.topology.dim - 1 +for (marker , locator) in +boundaries: +facets = locate_entities (mesh , fdim , locator) +facet_indices .append(facets) +facet_markers .append(np.full_like(facets , marker)) +facet_indices = np.hstack( facet_indices ).astype(np.int32) +facet_markers = np.hstack( facet_markers ).astype(np.int32) +sorted_facets = np.argsort( facet_indices ) +facet_tag = meshtags(mesh , fdim , facet_indices [ sorted_facets ], facet_markers +[ sorted_facets ]) +# +## Time +parametrization +t += 0 +# Start +time +Tf += 6 +# End +time +num_steps = 1000 +# Number of time +steps +dt += (Tf -t)/num_steps # Time +step +size +# +## Material +parameters +E += Constant(mesh , ScalarType (5000)) +nu += Constant(mesh , ScalarType (0.4)) +lambda_m += Constant(mesh , ScalarType(E.value*nu.value /((1+ nu.value) +*(1 -2*nu.value)))) +mu += Constant(mesh , ScalarType (E.value /(2*(1+ nu.value)))) +rhos += Constant(mesh , ScalarType (1)) +permeability = Constant(mesh , ScalarType (1.8e -15)) +viscosity += Constant(mesh , ScalarType (1e -2)) +rhol += Constant(mesh , ScalarType (1)) +beta += Constant(mesh , ScalarType (1)) +porosity += Constant(mesh , ScalarType (0.2)) +Kf += Constant(mesh , ScalarType (2.2 e9)) +Ks += Constant(mesh , ScalarType (1 e10)) +S += (porosity/Kf)+(1- porosity)/Ks +# +## Mechanical +loading +pinit = 100 #[Pa] +T += Constant(mesh ,ScalarType (-pinit)) +# +# Create +the +surfacic +element +ds = Measure("ds", domain=mesh , subdomain_data =facet_tag) +normal = FacetNormal (mesh) +# +# Define +Mixed +Space (R2 ,R) -> (u,p) +displacement_element += VectorElement ("CG", mesh.ufl_cell (), 2) +pressure_element += FiniteElement ("CG", mesh.ufl_cell (), 1) +34 + +MS += FunctionSpace (mesh , MixedElement ([ +displacement_element , pressure_element ])) +# +# Define +the +Dirichlet +condition +# 1 = bottom: uy=0, 2 = right: ux=0, 3= top: pl=0 drainage , 4= left: ux=0 +bcs += [] +# uz=0 +facets = facet_tag.find (1) +dofs += locate_dofs_topological (MS.sub (0).sub (2) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (2))) +# ux=0 +facets = facet_tag.find (2) +dofs += locate_dofs_topological (MS.sub (0).sub (0) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (0))) +# ux=0 +facets = facet_tag.find (4) +dofs += locate_dofs_topological (MS.sub (0).sub (0) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (0))) +# uy=0 +facets = facet_tag.find (5) +dofs += locate_dofs_topological (MS.sub (0).sub (1) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (1))) +# uy=0 +facets = facet_tag.find (6) +dofs += locate_dofs_topological (MS.sub (0).sub (1) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (0).sub (1))) +# drainage p=0 +facets = facet_tag.find (3) +dofs += locate_dofs_topological (MS.sub (1) , fdim , facets) +bcs.append(dirichletbc(ScalarType (0) , dofs , MS.sub (1))) +# +# Create +the +initial +and +solution +spaces +X0 = Function(MS) +Xn = Function(MS) +# +# Initial +values +# +Un_ , Un_to_MS = MS.sub (0).collapse () +FUn_ = Function(Un_) +with +FUn_.vector.localForm () as +initial_local : +initial_local .set(ScalarType (0.0)) +# +# Update Xn +Xn.x.array[Un_to_MS] = FUn_.x.array +Xn.x. scatter_forward () +# +Pn_ , Pn_to_MS = MS.sub (1).collapse () +FPn_ = Function(Pn_) +with +FPn_.vector.localForm () as +initial_local : +initial_local .set(ScalarType(pinit)) +# +# Update Xn +Xn.x.array[Pn_to_MS] = FPn_.x.array +Xn.x. scatter_forward () +# +# Variational +form +# Identify +the +unknowns +from +the +function +35 + +u,p +=split(X0) +u_n ,p_n=split(Xn) +# Set up the +test +functions +v,q = TestFunctions (MS) +# Equation +17 +F += (1/dt)*nabla_div(u-u_n)*q*dx + ( permeability /viscosity)*dot(grad(p), +grad(q))*dx ++ ( S/dt )*(p-p_n)*q*dx +# Equation +18 +F += inner(grad(v),teff(u))*dx - beta * p * nabla_div(v)*dx - T*inner(v, +normal)*ds (3) +# Non +linear +problem +definition +dX0 += TrialFunction (MS) +J += derivative(F, X0 , dX0) +Problem = NonlinearProblem (F, X0 , bcs = bcs , J = J) +# set up the non -linear +solver +solver += nls.petsc. NewtonSolver (mesh.comm , Problem) +# Absolute +tolerance +solver.atol = 5e -10 +# relative +tolerance +solver.rtol = 1e -11 +solver. convergence_criterion = " incremental " +# +t = 0 +L2_p = np.zeros(num_steps , dtype=PETSc. ScalarType ) +for n in range(num_steps): +t += dt +num_its , converged = solver.solve(X0) +X0.x. scatter_forward () +# Update +Value +Xn.x.array [:] = X0.x.array +Xn.x. scatter_forward () +__u , __p = X0.split () +# Compute +L2 norm +for +pressure +error_L2p += L2_error_p (mesh ,pressure_element ,__p) +L2_p[n] = error_L2p +# Solve +tracking +if mesh.comm.rank == 0: +print(f"Time +step {n}, Number of +iterations {num_its}, Load {T.value +}, L2 -error p {error_L2p :.2e}") +if mesh.comm.rank == 0: +print(f"L2 error p, min {np.min(L2_p):.2e}, mean {np.mean(L2_p):.2e}, +max {np.max(L2_p):.2e}, std {np.std(L2_p):.2e}") +Appendix B. Local refinement +A 3D geometry can be meshed using the GMSH API of python (Geuzaine +and Remacle [31]). This allows to represent complex geometries including +circle arcs. An optimized and locally refined mesh can be therefore obtained. +This example uses the method proposed in the FEniCS project tutorial 1 pro- +1see https://docs.fenicsproject.org/dolfinx/main/python/demos/demo_gmsh. +html +36 + +vided by J. Dokken and G. Wells. An alternative procedure in the FEniCSx +environment with local refinement is then proposed in Appendix B.2. +Appendix B.1. Meshing using GMSH API +First, the environment is initialized and the physical variables required +for the box/cylinder creation are defined. +import +gmsh +import +numpy as np +# +gmsh.initialize () +# +# box +parameters +[Length , Width , Height] = [6e-4, 2.5e-4, 4e -5] +# cylinder +parameters +xc ,yc ,zc ,dx ,dy ,dz , r = 6e-4/2 , 0, 0, 0, 0, 4e-5, 1.5e-4 +# expected +dimension +of the +mesh +gdim = 3 +The geometries are created using built-in functions of GMSH; potential +duplicates are removed. +# create +the +geometry +box += gmsh.model.occ.addBox (0, 0, 0, Length , Width , Height) +cylinder = gmsh.model.occ.addCylinder (xc ,yc ,zc ,dx ,dy ,dz , r,tag =1000 , angle=np +.pi) +gmsh.model.occ. synchronize () +# Remove +duplicate +entities +and +synchronize +gmsh.model.occ. removeAllDuplicates () +gmsh.model.occ. synchronize () +Physical groups are defined: the volumes for the 3D meshing and the +surfaces for tagging. Surface groups were identified based on the coordinates +of the center of mass of each surface. +surfaces , volumes = [gmsh.model. getEntities (d) for d in [ gdim -1, gdim ]] +print(volumes) +# Volumes +gmsh.model. addPhysicalGroup (volumes [0][0] , [volumes [0][1]] , +-1) +gmsh.model. setPhysicalName (volumes [0][0] , +-1, ’Half_Cylinder ’) +gmsh.model. addPhysicalGroup (volumes [1][0] , [volumes [1][1]] , +-1) +gmsh.model. setPhysicalName (volumes [1][0] , +-1, ’Box ’) +# 1 = loading , 2 = top +minus +loading , 3 = bottom , 4 = left , 5 = right , 6 = +Front , 7 = back +bottom_marker , front_marker , back_marker , left_marker , right_marker , +top_marker , indenter_marker = 3, 6, 7, 4, 5, 2, 1 +bottom , front , back , left , right , top , indenter = [] ,[] ,[] ,[] ,[] ,[] ,[] +boundaries = gmsh.model. getBoundary (volumes , oriented=False) +for +boundary +in +boundaries: +center_of_mass = gmsh.model.occ. getCenterOfMass (boundary [0], +boundary +[1]) +if np.isclose( center_of_mass [1], Width): +back.append(boundary [1]) +elif np.isclose( center_of_mass [1], 0): +37 + +front.append(boundary [1]) +elif np.isclose( center_of_mass [0], 0): +left.append(boundary [1]) +elif np.isclose( center_of_mass [0], Length): +right.append(boundary [1]) +elif np.isclose( center_of_mass [2], 0): +bottom.append(boundary [1]) +elif np.isclose( center_of_mass [2], Height) and +center_of_mass [1]> Width +/3: +top.append(boundary [1]) +else: +indenter.append(boundary [1]) +# mark +the +surfaces +gmsh.model. addPhysicalGroup (boundaries [0][0] , bottom , bottom_marker ) +gmsh.model. setPhysicalName (boundaries [0][0] , +bottom_marker , ’bottom ’) +gmsh.model. addPhysicalGroup (boundaries [0][0] , front , front_marker ) +gmsh.model. setPhysicalName (boundaries [0][0] , +front_marker , ’front ’) +gmsh.model. addPhysicalGroup (boundaries [0][0] , back , back_marker ) +gmsh.model. setPhysicalName (boundaries [0][0] , +back_marker , ’back ’) +gmsh.model. addPhysicalGroup (boundaries [0][0] , left , left_marker ) +gmsh.model. setPhysicalName (boundaries [0][0] , +left_marker , ’left ’) +gmsh.model. addPhysicalGroup (boundaries [0][0] , right , right_marker ) +gmsh.model. setPhysicalName (boundaries [0][0] , +right_marker , ’right ’) +gmsh.model. addPhysicalGroup (boundaries [0][0] , top , top_marker ) +gmsh.model. setPhysicalName (boundaries [0][0] , +top_marker , ’top ’) +gmsh.model. addPhysicalGroup (boundaries [0][0] , +indenter , indenter_marker ) +gmsh.model. setPhysicalName (boundaries [0][0] , +indenter_marker , ’indenter ’) +gmsh.model.occ. synchronize () +# Write a geo +file +for +verification +in the +GMSH +GUI +gmsh.write(’Geom_2reelle_8EP . geo_unrolled ’) +Then, a threshold function is defined over a distance field to mesh the +circular area. This allows for creating an adaptive mesh: coarse far from the +circular area, refine close to it. +indenter_interface = surfaces [0][1] +distance = gmsh.model.mesh.field.add("Distance") +gmsh.model.mesh.field.setNumbers (distance , "FacesList", [ indenter_interface +]) +# A threshold +function +is +defined: +resolution = r/10 +threshold = gmsh.model.mesh.field.add("Threshold") +gmsh.model.mesh.field.setNumber(threshold , "IField", distance) +gmsh.model.mesh.field.setNumber(threshold , "LcMin", resolution ) +gmsh.model.mesh.field.setNumber(threshold , "LcMax", 5* resolution) +gmsh.model.mesh.field.setNumber(threshold , "DistMin", 0.6*r) +gmsh.model.mesh.field.setNumber(threshold , "DistMax", r) +# If +several +fields +are +defined: +minimum = gmsh.model.mesh.field.add("Min") +gmsh.model.mesh.field.setNumbers (minimum , " FieldsList ", [threshold ]) # add +other +fields +in the +list if needed +gmsh.model.mesh.field. setAsBackgroundMesh (minimum) +Finally, the options of the mesher are defined and the mesh is created. +gmsh.model.occ. synchronize () +gmsh.option.setNumber("General.Terminal" ,1) +38 + +gmsh.option.setNumber("Mesh.Optimize", True) +gmsh.option.setNumber("Mesh. OptimizeNetgen ", True) +gmsh.model.occ. synchronize () +# gmsh.option.setNumber (" Mesh. MshFileVersion ", 2.0) +gmsh.option.setNumber("Mesh. MeshSizeExtendFromBoundary ", 0) +gmsh.option.setNumber("Mesh. MeshSizeFromPoints ", 0) +gmsh.option.setNumber("Mesh. MeshSizeFromCurvature ", 0) +# +gmsh.model.mesh.generate(gdim) +gmsh.write("Mesh.msh") +gmsh.finalize () +Appendix B.2. Local refinement within FEniCSx +Using GMSH API, an exact circular interface is generated. +However, +a similar mesh could have been obtained within FEniCSx through the ap- +proximation of the circular interface around the indenter by local refining. +Here-after is proposed a minimal code for local refinement inside the circular +area. +First, the required libraries are imported and a box mesh is created. +## Librairies +import +dolfinx +import +numpy as np +from +dolfinx.mesh +import +create_box , CellType , refine , locate_entities , +meshtags +from +dolfinx.io +import +XDMFFile +from +mpi4py +import +MPI +# +## Box +# Dimensions +of the +sample +[Length , Width , Height] = [6e-4, 2.5e-4, 4e -5] +# Discretization +[nx ,ny ,nz] = [30 ,15 ,8] +mesh = create_box(MPI.COMM_WORLD ,np.array ([[0.0 ,0.0 ,0.0] ,[ Length , Width , +Height ]]) , [nx ,ny ,nz], cell_type=CellType. tetrahedron ) +Then a locator is introduced to identify all the edges (fdim = 1) which +are part of the region we aim to refine. +def +test_on_boundary (x): +return (np.sqrt(np.power(x[0] -3e-4 ,2)+np.power(x[1] ,2)) <=1.5e -4) +# +refine_boudaries = [(11 , +lambda x: test_on_boundary (x))] +Finally, a loop is performed to compute several times the refinement +(np.arange(N)), using the existing refine() function. +for _ in np.arange (2): +# Refinement +refine_indices , refine_markers = [], [] +fdim = mesh.topology.dim -2 +for (marker , locator) in +refine_boudaries : +39 + +facets = locate_entities (mesh , fdim , locator) +refine_indices .append(facets) +refine_markers .append(np.full_like(facets , marker)) +refine_indices = np.hstack( refine_indices ).astype(np.int32) +refine_markers = np.hstack( refine_markers ).astype(np.int32) +# indices +in +meshtag +must be sorted +sorted_facets_refine = np.argsort( refine_indices ) +refine_tag = meshtags(mesh , fdim , refine_indices [ sorted_facets_refine ], +refine_markers [ sorted_facets_refine ]) +mesh.topology. create_entities (fdim) +mesh = refine(mesh , refine_indices [ sorted_facets_refine ]) +The facets are tagged to apply boundary conditions and the mesh is +written as a .xdmf file. +def +Omega_top(x): +return +np. logical_and ((x[2] == Height), (np.sqrt(np.power(x[0] -3e-4 ,2)+ +np.power(x[1] ,2)) <=1.5e -4)) +# +def +Omega_loading (x): +return +np. logical_and ((x[2] == Height), (np.sqrt(np.power(x[0] -3e-4 ,2)+ +np.power(x[1] ,2)) >=1.2e -4)) +# +# Create +the +facet +tags (identify +the +boundaries) +# 1 = loading , 2 = top +minus +loading , 3 = bottom , 4 = left , 5 = right , 6 = +Front , 7 = back +boundaries = [(1, lambda x: Omega_loading (x)), +(2, lambda x: Omega_top(x)), +(3, lambda x: np.isclose(x[2], 0.0)), +(4, lambda x: np.isclose(x[0], 0.0)), +(5, lambda x: np.isclose(x[0], Length)), +(6, lambda x: np.isclose(x[1], 0.0)), +(7, lambda x: np.isclose(x[1], Width))] +# Mark +them +facet_indices , facet_markers = [], [] +fdim = mesh.topology.dim - 1 +for (marker , locator) in +boundaries: +facets = locate_entities (mesh , fdim , locator) +facet_indices .append(facets) +facet_markers .append(np.full_like(facets , marker)) +facet_indices = np.hstack( facet_indices ).astype(np.int32) +facet_markers = np.hstack( facet_markers ).astype(np.int32) +sorted_facets = np.argsort( facet_indices ) +facet_tag = meshtags(mesh , fdim , facet_indices [ sorted_facets ], facet_markers +[ sorted_facets ]) +facet_tag.name = "facets" +# Write +XDMF +mesh.topology. create_connectivity (mesh.topology.dim -1, mesh.topology.dim) +with +XDMFFile(mesh.comm , "facet_tags.xdmf", "w") as +xdmftag: +xdmftag.write_mesh(mesh) +xdmftag. write_meshtags (facet_tag) +xdmftag.close () +Figure B.6 gives the comparison of the mesh obtained using GMSH and +the one using local refinement. +40 + +(a) +(b) +Figure B.6: GMSH (a) and FEniCSx (b) generated meshes. +Appendix B.3. Import an external mesh (XDMF or MSH) +Once the mesh is generated as a tagged .msh or .xdmf file, one can con- +sider directly read them to compile the domain and read the markers using: +from +dolfinx.io.gmshio +import +read_from_msh +from +dolfinx.io +import +XDMFFile +# set +value to 0 if .xdmf , set it to 1 if .msh +mesher = 1 +# +if mesher +== 0: +# ######################### +## +Read +XDMF +mesh +## +# ######################### +filename = "filename.xdmf" +with +XDMFFile(MPI.COMM_WORLD , filename , "r") as file: +mesh = file.read_mesh () +mesh.topology. create_connectivity (mesh.topology.dim -1, mesh.topology +.dim) +facet_tag = file. read_meshtags (mesh , "tag.name") +# +elif +mesher == 1: +# ######################### +## Read +gmsh +mesh +## +# ######################### +mesh , cell_tag , facet_tag = read_from_msh ("filename.msh", MPI.COMM_WORLD +, 0, gdim =3) +# +else: +print(’The +mesh +type is +wrongly +defined. mesher +should +equal 0 for +xdmf +and 1 for msh +files.’) +exit () +41 + +Appendix C. Evaluate the function at a physical point +One strength of using FEniCSx is its ability to evaluate the solution at +given points, summing the contribution of the neighbor cells of the mesh 2. +The following code allowed to compute the figures presented for the results +of the sections 3 and ref 4. First, one need to define the points where to +evaluate the solution. +import +numpy as np +num_points = 11 +y_check = np.linspace (0,Height , num_points ) +points_for_time = np.array ([[ Width /2, 0., 0.], [Width /2, Height /2, 0.]]) +points_for_space = np.zeros (( num_points ,3)) +for ii in range(num_points): +points_for_space [ii ,0]= Width /2 +points_for_space [ii ,1]= y_check[ii] +points = np. concatenate (( points_for_time , points_for_space )) +The following step is to identify the cells contributing to the points. +from +dolfinx.geometry +import +BoundingBoxTree , compute_collisions , +compute_colliding_cells +tree = BoundingBoxTree (mesh , mesh.geometry.dim) +cell_candidates = compute_collisions (tree , points) +colliding_cells = compute_colliding_cells (mesh , cell_candidates , points) +# Here is an +example +to +select +cells +contributing +to the +first +and +second +points. +cells_y_0 = colliding_cells .links (0) +cells_y_H_over_2 = colliding_cells .links (1) +Knowing the shape of the functions to evaluate, lists are created and will +be updated during the resolution procedure. Regarding parallel computation, +these lists are only created on the first kernel. +from +mpi4py +import +MPI +if MPI.COMM_WORLD.rank == 0: +pressure_y_0 = np.zeros(num_steps , dtype=PETSc. ScalarType) +pressure_y_Height_over_2 = np.zeros(num_steps , dtype=PETSc. ScalarType ) +pressure_space0 = np.zeros(num_points , dtype=PETSc. ScalarType ) +pressure_space1 = np.zeros(num_points , dtype=PETSc. ScalarType ) +pressure_space2 = np.zeros(num_points , dtype=PETSc. ScalarType ) +A function is created to evaluate a function given the mesh, the function, +the contributing cells to the point and the list with its index to store the +evaluated value in. +def +evaluate_point (mesh , function , contributing_cells , point , output_list , +index): +from +mpi4py +import +MPI +2see https://jorgensd.github.io/dolfinx-tutorial/chapter2/ns_code2.html? +highlight=eval +42 + +function_eval = None +if len( contributing_cells ) > 0: +function_eval = function.eval(point , contributing_cells [:1]) +function_eval = mesh.comm.gather(function_eval , root =0) +# Choose +first +pressure +that is found +from +the +different +processors +if MPI.COMM_WORLD.rank == 0: +for +element +in +function_eval : +if +element +is not +None: +output_list[index ]= element [0] +break +pass +Finally, the problem is solved for each time steps. +The functions are +evaluated for all kernels and gathered on the first one where the first pressure +found by the different processors will be uploaded in the here-above lists. +# time +steps to +evaluate +the +pressure +in space: +n0 , n1 , n2 = 200 ,400 ,800 +# +t = 0 +L2_p = np.zeros(num_steps , dtype=PETSc. ScalarType ) +for n in range(num_steps): +t += dt +try: +num_its , converged = solver.solve(X0) +except: +if MPI.COMM_WORLD.rank == 0: +print(" ************* ") +print("Solver +failed") +print(" ************* ") +pass +X0.x. scatter_forward () +# Update +Value +Xn.x.array [:] = X0.x.array +Xn.x. scatter_forward () +__u , __p = X0.split () +# +# Export +the +results +__u.name = " Displacement " +__p.name = "Pressure" +xdmf. write_function (__u ,t) +xdmf. write_function (__p ,t) +# +# Compute +L2 norm +for +pressure +error_L2p += L2_error_p (mesh ,pressure_element ,__p) +L2_p[n] = error_L2p +# +# Solve +tracking +if MPI.COMM_WORLD.rank == 0: +print(f"Time +step {n}/{ num_steps}, Load {T.value}, L2 -error p { +error_L2p :.2e}") +# Evaluate +the +functions +# in time +if n == n0: +for ii in range(num_points ): +evaluate_point (mesh , __p , colliding_cells .links(ii +2) , points[ii ++2], +pressure_space0 , ii) +43 + +t0 = t +elif n==n1: +evaluate_point (mesh , __p , colliding_cells .links(ii +2) , points[ii ++2], +pressure_space1 , ii) +t1 = t +elif n==n2: +evaluate_point (mesh , __p , colliding_cells .links(ii +2) , points[ii ++2], +pressure_space2 , ii) +t2 = t +# +xdmf.close () +References +[1] S. 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Mech. 67 (2000) 17–21. +49 + diff --git a/KdFIT4oBgHgl3EQfaSsg/content/tmp_files/load_file.txt b/KdFIT4oBgHgl3EQfaSsg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f718f8c5c161889b7baf9212bd29f0bdac7777c2 --- /dev/null +++ b/KdFIT4oBgHgl3EQfaSsg/content/tmp_files/load_file.txt @@ -0,0 +1,1859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf,len=1858 +page_content='Highlights Multi-compartment poroelastic models of perfused biological soft tissues: implementation in FEniCSx T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Lavigne, St´ephane Urcun, Pierre-Yves Rohan, Giuseppe Scium`e, Davide Baroli, St´ephane P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Bordas Implementation of a poro-elastic formulation within FEniCSx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Single and double compartment columns are modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Elastic and Hyper-elastic solid scaffold are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Fast and accurate computation is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='11256v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='TO] 26 Jan 2023 Multi-compartment poroelastic models of perfused biological soft tissues: implementation in FEniCSx T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Lavignea,b,c, St´ephane Urcuna, Pierre-Yves Rohanb, Giuseppe Scium`ec, Davide Barolid,, St´ephane P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Bordasa,e,f aInstitute of Computational Engineering, Department of Engineering, University of Luxembourg, 6, avenue de la Fonte, Esch-sur-Alzette, L-4364, Luxembourg bArts et Metiers Institute of Technology, IBHGC, 151 bd de l’hopital, Paris, 75013, France cArts et Metiers Institute of Technology, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' of Bordeaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Bordeaux INP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' INRAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' I2M Bordeaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Avenue d’Aquitaine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Pessac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 33607,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' France dUniversit`a della Svizzera Italiana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Euler Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Lugano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Swiss eClyde Visiting Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Salt Lake City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' United States fDepartment of Medical Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' China Medical University Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' China Medical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Taichung,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Taiwan Abstract Soft biological tissues demonstrate strong time-dependent and strain-rate me- chanical behavior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' arising from their intrinsic viscoelasticity and fluid-solid interactions especially at sufficiently large time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The time-dependent mechanical properties of soft tissues influence their physiological functions and are linked to several pathological processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Poroelastic modelling rep- resents a promising approach because it allows to integrate multiscale/mul- tiphysics data to probe biologically relevant phenomena at a smaller scale and embed the relevant mechanisms at the larger scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The implementation of multiphasic flow poroelastic models howver is a complex undertaking, re- quiring extensive knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The FEniCSx Project provides a novel tool for the automated solution of partial differential equations by the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' This paper aims to provide the required tools to model the mixed formulation of poro-elasticity, from the theory to the implementation, within FEniCSx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Several benchmark cases are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' A column under confined compression conditions is compared to the Terzaghi analytical solution, us- ing the L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' An implementation of poro-hyper-elasticity is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' A bi-compartment column is compared to previously published results af a Cast3m implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' For all cases, accurate results are obtained in terms of a normalized Root Mean Square Error (RMSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Furthermore, the FEn- iCSx computation is found three times faster than the legacy FEniCS one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The benefits of parallel computation are also highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Keywords: Mixed Space, Poro-elasticity, Bi-compartment, FEniCSx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Introduction Numerous problems of relevance in biomechanics, have at their core, the presence of a deformable solid matrix which experiences flow-induced strain : skin (brain (Budday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [1], Hosseini-Farid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [2], Franceschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [3], Urcun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [4]), muscle tissue (Lavigne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [5]), tumour (Scium`e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [6], Scium`e [7], Oftadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [8]), artciular cartilage (Ateshian [9]) and lum- bar intervertebral discs (Argoubi and Shirazi-Adl [10]) just to name a few).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The time-dependent mechanical properties of soft tissues influence their phys- iological functions and are linked to several pathological processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Although a fluid-structure interaction (FSI) problem, the number and range of fluid flows is generally so vast that the direct approach of a defined boundary be- tween fluid and solid is impossible to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In these cases, homogenisation and statistical treatment of the material-fluid system is possibly the only way forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' A prominent technique of this type is that of poroelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Extensive studies have shown that poroelastic models can accuratley re- produce the time-dependent behavior of soft tissues under different loading conditions (Gimnich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [11], Argoubi and Shirazi-Adl [10], Peyrounette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [12], Siddique et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [13], Hosseini-Farid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [2], Franceschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [3], Lavigne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Compared to a visco-(hyper)-elastic formulation (Van Loocke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [14], Simms et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [15], Wheatley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [16], Vaidya and Wheat- ley [17]), the poroelastic properties are independant of the sample size (Urcun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Furthermore, a poroelastic approach can integrate multiscale/mul- tiphysics data to probe biologically relevant phenomena at a smaller scale and embed the relevant mechanisms at the larger scale (in particular, bio- chemistry of oxygen and of inflammatory signalling pathways), allowing the interpretation of the different time characteristics (Urcun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [18], Scium`e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [6], Scium`e [7], Gray and Miller [19], Mascheroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Email address: davide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='baroli@usi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='ch (Davide Baroli ) Preprint submitted to Elsevier January 27, 2023 In most commercially available FE software packages used in research in biomecahnics (ABAQUS, ANSYS, RADIOSS, etc), pre-programmed mate- rial models for soft biological tissues are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The disadvantage of these pre-programmed models is that they are presented to the user as a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Therefore, many researchers turn to implementing their own material formu- lations through user subroutines (the reader is refered, for example, to the excellent tutorial of Fehervary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [21] on the implementation of a nonlinear hyperelastic material model using user subroutines in ABAQUS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' This task, however, is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' When documentation is available, these only provide expressions, without any derivations, lack details and background informa- tion, making the implementation complex and error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In addition, in case of a custom formulation or the introduction of bio-chemical equations for example, specific computational skills are required making the task even more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In the end, the use of commercially available FE software packages limit the straightforward reproducibility of the research by other teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The interest for open-source tools has skyrocketed to increase the impact of the studies within the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' For Finite Element modeling, the FEn- iCS project (Alnæs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [22]) is an OpenAccess software which has proven its efficiency in biomechanics (Mazier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Based on a Python/C++ coding interface and the Unified Form Language, it allows to easily solve a defined variational form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Furthermore, its compatibility to open-source meshers like GMSH make its use appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The project has already shown is capacity solving large deformation problems (Mazier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [24]) and mixed formulations (Urcun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [18, 4], Bulle [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Previous work provided the implementation of poro-mechanics within the FEniCS project (Haagenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [26], Joodat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' However, the FEniCS project is legacy and has been replaced by the FEniCSx project in August 2022 (Alnæs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [28], Scroggs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [29, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The aim of this paper is to propose a step-by-step explanation on how to implement several poro-mechanical models in FEniCSx with a special at- tention to parallel computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' First, an instantaneous uni-axial confined compression of a porous elastic medium is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' This example corre- sponds to an avascular tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Then, the same single-compartment model is computed for a hyper-elastic solid scaffold followed by a 3D confined bi- compartment modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Confined compression of a column: geometrical definition The time-dependant response of soft tissues are often assessed based on confined compression creep and stress relaxation test data (Budday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [1], Hosseini-Farid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [2], Franceschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [3], Urcun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In this section, therefore, all the benchmark examples focus on uni-axial confined compression of a column sample as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Both 2D and 3D geometries are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The column is described by its width (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1*h) and height (h) in 2D and its length in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' h p0 us · x = 0 us · y = 0 pl = 0 (pb = 0) y x Figure 1: Load (red), Boundary conditions (blue) and mesh (gray) of the uni-axial confined compression of a porous 2D column of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' All the proposed codes were computed within a docker image of FEniCSx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The dolfinx version used in this paper is v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' FEniCSx is a proficient plat- form for parallel computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' All described codes here-under are compatible with multi-kernel computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The corresponding terminal command is: mpirun n python3 Where is the number of threads to use and <filename> is the python code of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Within the FEniCSx software, the domain (geometry) is discretized to match with the Finite Element (FE) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The space is thus divided in 4 nx × ny = 2 × 40 elements in 2D and nx × ny × nz = 2 × 40 × 2 elements in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The choice of the number of elements is further discussed section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In this article, the meshes are directly created within the FEniCSx environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' However, as a strong compatibility exists with the GMSH api (Geuzaine and Remacle [31]), it is recommended to use GMSH for this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' An example of the use of GMSH API for a more complex geometry is given section Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' It is worth noting that we identify all the boundaries of interest at this step for the future declaration of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 2D mesh Working in the python environment requires to import the working li- braries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' To create a 2D mesh, the first step is to import the following li- braries: import dolfinx import numpy as np from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh import create_rectangle , CellType , locate_entities , meshtags from mpi4py import MPI Then, the domain of resolution (mesh) is computed with: Width , Height = 1e-5, 1e-4 #[m] nx , ny = 2, 40 #[ ] mesh = create_rectangle (MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD , np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array ([[0 ,0] ,[ Width , Height ]]) , [nx ,ny], cell_type=CellType.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' quadrilateral ) Once the mesh object has been created, its boundaries are identified us- ing couples of (marker, locator) to tag with a marker value the elements of dimension fdim fulfilling the locator requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' For the 2D mesh, the (marker, locator) couples are given by # identifiers: 1 , 2, 3, 4 = bottom , right , top , left boundaries = [(1, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[1], 0)), (2, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[0], Width)), (3, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[1], Height)), (4, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[0], 0))] Finally the entities are marked by: facet_indices , facet_markers = [], [] # dimension of the elements we are looking for fdim = mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim - 1 for (marker , locator) in boundaries: facets = locate_entities (mesh , fdim , locator) facet_indices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(facets) facet_markers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='full_like(facets , marker)) facet_indices = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( facet_indices ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) facet_markers = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( facet_markers ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) sorted_facets = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='argsort( facet_indices ) 5 # the meshtags () function requires sorted facet_indices facet_tag = meshtags(mesh , fdim , facet_indices [ sorted_facets ], facet_markers [ sorted_facets ]) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3D mesh The method for a 3D mesh is similar to the 2D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' First the libraries are imported and the geometry is created using a 3D function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The (marker, locator) tuples are completed to describe all the boundaries of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The same tagging routine is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ## libraries import dolfinx import numpy from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh import create_box , CellType , locate_entities , meshtags from mpi4py import MPI ## Mesh generation Length , Height , Width = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1 #[m] nx , ny , nz = 2, 40, 2 mesh = create_box(MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD , numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array ([[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0] ,[ Length , Height , Width ]]) , [nx , ny , nz], cell_type=CellType.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hexahedron ) ## Define the boundaries of the domain: # 1, 2, 3, 4, 5, 6 = bottom , right , top , left , back , front boundaries = [(1, lambda x: numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[1], 0)), (2, lambda x: numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[0], Length)), (3, lambda x: numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[1], Height)), (4, lambda x: numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[0], 0)), (5, lambda x: numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[2], Width)), (6, lambda x: numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[2], 0))] facet_indices , facet_markers = [], [] fdim = mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim - 1 for (marker , locator) in boundaries: facets = locate_entities (mesh , fdim , locator) facet_indices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(facets) facet_markers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='full_like(facets , marker)) facet_indices = numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( facet_indices ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) facet_markers = numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( facet_markers ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) sorted_facets = numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='argsort( facet_indices ) facet_tag = meshtags(mesh , fdim , facet_indices [ sorted_facets ], facet_markers [ sorted_facets ]) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Single-compartment porous medium We propose to reproduce the instantaneous uni-axial confined compres- sion at the top surface of a single-compartment porous column of height h, Figure 1, described by a 2D elastic or a 3D hyper-elastic solid scaffold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Re- garding the 2D elastic case, the column has a height of h = 100µm, the instantaneous load p0 has a magnitude of 100Pa and is applied during 6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Regarding the 3D hyper-elastic case, the column has a height of h = 1m, the 6 instantaneous load p0 has a magnitude of p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3MPa and is applied during 100000 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The mechanical parameters are respectively given Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' To assess the reliability of our results, we compare our computed solutions to the Terzaghi’s analytical solution and to the results of Selvadurai and Suvorov [32], for the elastic and hyper-elastic scaffolds respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Parameter Symbol Value Unit Young modulus E 5000 Pa Poisson ratio ν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4 Intrinsic permeability kε 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='8 × 10−15 m2 Biot coefficient β 1 Density of phase α ρα kg m−3 IF viscosity µl 1 × 10−3 Pa s Porosity εl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 Solid grain Bulk modulus Ks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' × 1010 Pa Fluid Bulk modulus Kl 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 × 109 Pa Table 1: Elastic mechanical parameters to compare with the Terzaghi solution Parameter Symbol Value Unit Young modulus E 600000 Pa Poisson ratio ν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3 Bulk modulus K 500000 Pa Intrinsic permeability kε 3 × 10−14 m2 IF viscosity µl 1 × 10−3 Pa s Porosity εl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 Solid grain Bulk modulus Ks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' × 1010 Pa Fluid Bulk modulus Kl 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 × 109 or 5 × 105 Pa Biot coefficient β 1 − K Ks ≈ 1 Table 2: Hyper-elastic mechanical parameters from Selvadurai and Suvorov [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In the absence of information on the porosity, solid grain bulk modulus and fluid bulk modulus, the parameter are arbitrarily chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Terzaghi’s Analytical solution The Terzaghi consolidation problem is often used for benchmarking porous media mechanics, as an analytical solution of this problem exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' An im- plementation of this experiment was proposed by Haagenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' [26], 7 within the legacy FEniCS project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The Terzaghi problem consists in an uni-directional confined compression experiment of a column (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Assuming small and uni-directional strains, incompressible homogeneous phases and constant mechanical properties, the analytical expression of the pore pressure is given in terms of series in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' pl = 4p0 π +∞ � k=1 (−1)k−1 2k − 1 cos � (2k − 1)π 2 y h � exp � −(2k − 1)2π2 4 cvt h2 � (1) cv = kε µl(Sβ + β2 M ) (2) M = 3Ks(1 − ν) (1 + ν) (3) Sβ = β − εl 0 Ks + εl 0 Kl (4) Where p0=timposed ·n is the full applied load,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' y is the altitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' h is the initial height of the sample ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' cv is the consolidation coefficient defined by (Equation 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' M the longitudinal modulus (Equation 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Sβ the inverse of the Biot Modulus (Equation 4) and εl 0 is the initial porosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Governing equations Let one consider a bi-phasic structure composed of a solid scaffold filled with interstitial fluid (IF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The medium is assumed fully saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In this section, to set up the governing equations, we make the hypothesis of a Biot coefficient equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The following convention is assumed: •s denotes the solid phase and •l denotes the fluid phase (IF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The primary variables of the problem are the pressure applied in the pores of the porous medium, namely pl, and the displacement of the solid scaffold, namely us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (Equation 5) constrains the different volume fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The volume fraction of the phase α is defined by (Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' εl is called the porosity of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' εs + εl = 1 (5) εα = Volumeα Volumetotal (6) 8 Assuming that there is no inter-phase mass transport, the continuity equations (mass conservation) of the liquid and solid phases are respectively given by Equation 7 and Equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ∂ ∂t(ρlεl) + ∇ · (ρlεlvl) = 0 (7) ∂ ∂t(ρs(1 − εl)) + ∇ · (ρs(1 − εl)vs) = 0 (8) Regarding the distributivity of the divergence term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' with a scalar and V vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ∇ · (aV) = a∇ · (V) + ∇a · V (9) Applied to 7 and Equation 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' and considering the definition of the material derivative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Ds Dtf = ∂f ∂t + ∇f · vs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' the continuity equations are given by: Ds Dt(ρs(1 − εl)) + ρs(1 − εl)∇ · vs = 0 (10) Ds Dt(ρlεl) + ∇ · (ρlεl(vl − vs)) + ρlεl∇ · vs = 0 (11) For the fluid phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' the Darcy’s law (Equation 12) is used to evaluate the fluid flow in the porous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' εl(vl − vs) = −kε µl (∇pl − ρlg) (12) Where kε is the intrinsic permeability (m2), µl is the dynamic viscosity (Pa s) and g the gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Introducing the state law 1 ρα Dsρα Dt = 1 Kα Dpα Dt , Kα being the bulk modulus of the phase alpha, the Darcy’s law and summing 10 and Equation 11, we obtain: � εl Kl + 1 − εl Ks � Dspl Dt + ∇ · vs − ∇ · �kε µl ∇pl � = 0 (13) Where S = � εl Kl + 1−εl Ks � is called the storativity coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 9 Once the continuity equations are settled, one can define the quasi-static momentum balance of the porous medium, Equation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ∇ · ttot = 0 (14) Where ttot is the total Cauchy stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' We introduce an effective stress tensor denoted teff, responsible for all deformation of the solid scaffold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Then, ttot can be expressed as: ttot = teff − βplId (15) Where Id is the identity matrix and β is the Biot coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Finally, the governing equations of this single compartment porous medium are: � εl Kl + 1 − εl Ks � Dspl Dt + ∇ · vs − ∇ · �kε µl ∇pl � = 0 on Ω (16) ∇ · ttot = 0 on Ω (17) Three boundaries are defined: the first one, Γu has imposed displacement (Equation 18), the second one Γs has imposed external forces (Equation 19) and Γp is submitted to an imposed pressure (fluid leakage condition (Equation 20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' We obtain: teff = timposed on Γs (18) us = uimposed on Γu (19) p = 0 on Γp (20) According to Figure 1, Γp = Γs is the top surface and Γu covers the lateral and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Effective stress Two type of solid constitutive laws are considered: an elastic scaffold and a hyper-elastic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Linear elasticity In case of a Elastic scaffold, the effective stress tensor is defined as follows: ϵ(u) = 1 2(∇u + ∇uT) (21) teff = 2µϵ(us) + λtr(ϵ(us))Id (22) Where Id is the identity matrix and (λ, µ) the Lame coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Hyper-elasticity In case of an hyper-elastic scaffold, other quantities are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Let one introduce the deformation gradient F: F = Id + ∇us (23) Then, J is the determinant of F: J = det(F) (24) According to the classic formulation of a finite element procedure, we introduce C the right Cauchy-Green stress tensor and its first invariant I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' By definition: C = FTF (25) I1 = Tr(C) (26) The theory of hyper-elasticity defines a potential of elastic energy W(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The generalized Neo-Hookean potential (Equation 27) introduced by Treloar [33], implemented in Abaqus and used by Selvadurai and Suvorov [32] is evaluated in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' W(F) = µ 2 (J−2/3I1 − tr(Id)) + �λ 2 + µ 3 � ∗ (J − 1)2 (27) However, other potential were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' It was shown that the hyper- elastic potential can be expressed as the combination of a isochoric com- ponent and a volumetric component (Marino [34], Horgan and Saccomandi 11 [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' We define the lame coefficients by µ = E 2(1−ν) and λ = Eν (1+ν)(1−2ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' For a Neo-Hookean material, we further have: W(F) = ˜W(I1, J) + U(J) (28) Where ˜W(I1, J) is the isochoric part and U(J) the volumetric one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The study of Selvadurai and Suvorov [32] presented a compressible case (ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3) reaching high deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Therefore, a compressible formulation of the Neo- Hookean strain-energy potential from Pence and Gou [36], Horgan and Sac- comandi [35] is also computed for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Therefore, the implemented isochoric part of the strain energy potential is: ˜W1(I1, J) = µ 2 (I1 − tr(Id) − 2 log[J]) (29) Two different volumetric parts (U1 and U2) which were proposed in Doll and Schweizerhof [37] are implemented, U1(J) = λ 2 log[J]2 (30) U2(J) = λ 2(J − 1)2 (31) Finally, from the potential (Equation 28 or 27) derives the first Piola- Kirchhoff stress tensor as the effective stress such that: teff = ∂W ∂F (32) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Variational formulation For the computation of the Finite Element (FE) model, the variational form of Equation 16 and Equation 17 is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Let one consider (q,v) the test functions defined in the mixed space L2 0(Ω) × [H1(Ω)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' With a first order approximation in time, Equation 16 gives: S dt � Ω (pl − pl n)qdΩ + 1 dt � Ω ∇ · (us − us n)qdΩ +kε µl � Ω ∇pl∇qdΩ = 0, ∀ q ∈ L2 0(Ω) (33) 12 Similarly, by integrating by part Equation 17, and including the Neumann boundary conditions, we get: � Ω teff : ∇vdΩ − � Ω βpl∇ · vdΩ − � Γs timposed · n · vdΓs = 0, ∀ v ∈ [H1(Ω)]2 (34) The first order approximation in time impose to define the initial condi- tions which are fixed according to Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Parameter Symbol Value Unit Displacement us 0 m Displacement at previous step us n 0 m IF pressure pl timposed · n Pa IF pressure at previous step pl n 0 Pa Table 3: Initial conditions for the single compartment model Finally, the problem to solve is: Find (pl, us) ∈ L2 0(Ω) × [H1(Ω)]2 such that Equation 33 and Equation 34 are verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 2D linear elastic solid scaffold 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' FEniCSx implementation This section aims to provide a possible implementation of a 2D elastic problem and its comparison with the Terzaghi analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Conversely to the former FEniCS project, Dolfinx is based on a more explicit use of the libraries and requires to import them in the FEniCSx environment separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Therefore, each function used in the following implementation of the problem needs to be imported as a first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' import numpy as np from dolfinx import nls from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='fem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='petsc import NonlinearProblem from ufl import VectorElement , FiniteElement , MixedElement , TestFunctions , TrialFunction from ufl import Measure , FacetNormal from ufl import nabla_div , dx , dot , inner , grad , derivative , split from petsc4py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='PETSc import ScalarType from mpi4py import MPI from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='fem import (Constant , dirichletbc , Function , FunctionSpace , locate_dofs_topological ) from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='io import XDMFFile 13 Then, the time parametrization is introduced, the load value T such that timposed = p0 · n with n the outward normal to the mesh, and the material parameters which are defined as ufl constants over the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ## Time parametrization t = 0 # Start time Tf = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' # End time num_steps = 1000 # Number of time steps dt = (Tf -t)/num_steps # Time step size ## Material parameters E = Constant(mesh , ScalarType (5000)) nu = Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4)) lambda_m = Constant(mesh , ScalarType(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value*nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value /((1+ nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value) (1 -2*nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value)))) mu = Constant(mesh , ScalarType (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value /(2*(1+ nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value)))) rhos = Constant(mesh , ScalarType (1)) kepsilon = Constant(mesh , ScalarType (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='8e -15)) mul = Constant(mesh , ScalarType (1e -2)) rhol = Constant(mesh , ScalarType (1)) beta = Constant(mesh , ScalarType (1)) epsilonl = Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2)) Kf = Constant(mesh , ScalarType (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 e9)) Ks = Constant(mesh , ScalarType (1 e10)) S = (epsilonl/Kf)+(1- epsilonl)/Ks ## Mechanical loading pinit = 100 #[Pa] T = Constant(mesh ,ScalarType (-pinit)) The surface element for integration based on the tags and the normals of the mesh are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' # Create the surfacic element ds = Measure("ds", domain=mesh , subdomain_data =facet_tag) # compute the mesh normals to express t^imposed = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='normal normal = FacetNormal (mesh) Two type of elements are defined for displacement and pressure, then combined to obtain the mixed space (MS) of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' displacement_element = VectorElement ("CG", mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='ufl_cell (), 2) pressure_element = FiniteElement ("CG", mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='ufl_cell (), 1) MS = FunctionSpace (mesh , MixedElement ([ displacement_element , pressure_element ])) The space of resolution being defined, we can introduce the Dirichlet boundary conditions according to Equation 19, Equation 20 and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' # 1 = bottom: uy=0, 2 = right: ux=0, 3= top: pl=0 drainage , 4= left: ux=0 bcs = [] fdim = mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim - 1 # uy=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (1) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1))) # ux=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (2) 14 dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0))) # ux=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (4) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0))) # leakage p=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (3) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1))) The problem depends on the time Equation 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Initial conditions in dis- placement and pressure are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Therefore, we defined X0 the unknown function and Xn the solution at the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Giving the collapse() function, we identify the initial displacement function Un and its mapping within the Xn solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Then, its values are set to 0 and reassigned in Xn using the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='scatter forward() allows to update the values of Xn in case of parallel computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The same method is used to set up the initial pressure field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' To fit with the studied problems, the load is instantaneously applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Therefore, the initial pore pressure of the sample is assumed equal to p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' # X0 , Xn: Solution and previous functions of space X0 = Function(MS) Xn = Function(MS) # Initial values # Solid Displacement Un_ , Un_to_MS = MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='collapse () FUn_ = Function(Un_) with FUn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='localForm () as initial_local : initial_local .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='set(ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0)) # Assign in Xn and broadcast to all the threads Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[Un_to_MS] = FUn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # IF Pressure Pn_ , Pn_to_MS = MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='collapse () FPn_ = Function(Pn_) with FPn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='localForm () as initial_local : initial_local .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='set(ScalarType(pinit)) # Assign in Xn and broadcast to all the threads Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[Pn_to_MS] = FPn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () The deformation and effective stress given Equation 21 and Equation 22 are defined by the following function: def teff_Elastic (u,lambda_m ,mu): from ufl import sym , grad , nabla_div , Identity ## Deformation epsilon = sym(grad(u)) ## Stress return lambda_m * nabla_div(u) * Identity(u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' geometric_dimension ()) + 2* mu*epsilon 15 Finally, splitting the two functions X0, Xn, and introducing the test func- tions, the weak form is implemented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='p =split(X0) u_n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='p_n=split(Xn) # Set up the test functions v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='q = TestFunctions (MS) # Equation 33 F = (1/dt)*nabla_div(u-u_n)*q*dx + (kepsilon/mul)*dot(grad(p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='grad(q))*dx + ( S/dt )*(p-p_n)*q*dx # Equation 34 F += inner(grad(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='teff(u))*dx - beta * p * nabla_div(v)*dx - T*inner(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' normal)*ds (3) Introducing the trial function of the mixed space dX0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' we define the non- linear problem based on the variational form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' the unknown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' the boundary conditions and the Jacobian: dX0 = TrialFunction (MS) Js = derivative(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' X0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' dX0) Problem = NonlinearProblem (F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' X0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' bcs = bcs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' J = Js) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Solving and results To solve the non-linear problem defined here-above, a Newton solver is tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' solver = nls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='petsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' NewtonSolver (mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm , Problem) # Absolute tolerance solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='atol = 5e -10 # relative tolerance solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rtol = 1e -11 solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' convergence_criterion = " incremental " The parameters were set according to Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' During the resolution, we computed for each step the error in L2-norm in pressure defined Equation 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' These formulations are easily evaluated within the FEniCSx environment by defining the following functions: E(pl) = �� Ω(pl − pex)2dx �� Ω(pex)2dx (35) Where pex is the exact solution, computed from the Terzaghi’s analytical formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' def terzaghi_p(x): kmax =1e3 p0 ,L=pinit ,Height cv = kepsilon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value/mul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value *( lambda_m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value +2* mu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value) pression =0 16 for k in range (1,int(kmax)): pression +=p0 *4/ np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pi*( -1) **(k -1) /(2*k -1)*np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='cos ((2*k -1) *0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5* np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pi*(x [1]/L))*np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='exp ( -(2*k-1) **2*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='25* np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pi **2* cv*t/L**2) pl=pression return pl def L2_error_p(mesh ,pressure_element ,__p): V2 = FunctionSpace (mesh , pressure_element ) pex = Function(V2) pex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' interpolate (terzaghi_p ) L2_errorp , L2_normp = form(inner(__p - pex , __p - pex) * dx), form(inner (pex , pex) * dx) error_localp = assemble_scalar (L2_errorp)/ assemble_scalar (L2_normp) error_L2p = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sqrt(mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='allreduce(error_localp , op=MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='SUM)) return error_L2p To get a code suitable for parallel computation, the solutions needed to be gathered on a same processor using the MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='allreduce() function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Once the error functions were defined, the problem is solved within the time loop: # Create an output xdmf file to store the values xdmf = XDMFFile(mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm , ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='/ terzaghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='xdmf", "w") xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='write_mesh(mesh) # Solve the problem and evaluate values of interest t = 0 L2_p = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros(num_steps , dtype=PETSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ScalarType ) for n in range(num_steps): t += dt num_its , converged = solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='solve(X0) X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # Update Value Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array [:] = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () __u , __p = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='split () # Export the results __u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name = " Displacement " __p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name = "Pressure" xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_function (__u ,t) xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_function (__p ,t) # Compute L2 norm for pressure error_L2p = L2_error_p (mesh ,pressure_element ,__p) L2_p[n] = error_L2p # Solve tracking if mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rank == 0: print(f"Time step {n}, Number of iterations {num_its}, Load {T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value }, L2 -error p {error_L2p :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2e}") xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='close () The results obtained for pressure and displacements are provided Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The code to evaluate the pressure at given points is provided Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The curves show the efficiency of the simulation to reproduce the an- alytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The accuracy of the simulation was also supported by the estimation of the error based on the L2-norm of the pressure equal to 17 (a) (b) Figure 2: Comparison of the computed pore pressure against the analytical solution: in (a) time and (b) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The pressure was well recovered based on the evaluation of the L2-norm error (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='57 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='46) × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='57 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='46) × 10−3 which is deemed satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The same problem was solved using the legacy FEniCS version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The proposed FEniCSx implementa- tion was faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' It was computed in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='48 seconds compared to the previously 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='82 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' To show the efficiency of the parallel computation, the 3D case Appendix A is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' For a given spatio-temporal discretisation, a larger com- putational time of 1 hour 4 minutes 29 seconds is needed using FEniCSx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' To reduce the time, the code naturally supports parallel computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The same code was run for several number of threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Computed on 2 threads, the code required 53 minutes 27 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' For 4 threads, the running time was further reduced to 46 minutes 27 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Finally, using 8 threads, the computation time was reduced up to 28 minutes 9 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Finally, a convergence analysis on the meshing of the column was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The L2 error metric was used and its evolution for a nx × ny discretized mesh is given Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' As we could have expected from the 1D behavior of a confined compression Terzaghi case, the error is almost independent from the nx choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Figure 3(a) shows that a ny ≥ 10 gives better estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' According to Figure 3(b), a balance between precision and computation time must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The more elements, the higher the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' To ensure obtaining a reliable solution, a mesh of nx × ny = 2 × 40 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 18 1e2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 Analytic y=0 Analytic y=h/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='8 FEniCSx y=0 Pressure (Pa) FEniCSxy=h/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 0 1 2 3 4 5 6 time (s)1e-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 Analytic FEniCSx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='8 (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6 Height ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4 t=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='8s t=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4s t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 0 10 20 30 40 50 60 70 Pressure(Pa)(a) (b) Figure 3: Convergence analysis for a nx × ny discretized mesh: L-2 norm (a) and Compu- tation time (b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3D hyper-elastic scaffold 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' FEniCSx implementation The implementation method of the 3D case is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' However, spe- cial attention must be placed on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Indeed, moving from 2D to 3D introduces two more boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Therefore, the Dirichlet boundary conditions definition is completed with: # uz=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (5) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2))) # uz=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (6) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2))) The effective stress tensor is also different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' As an example, the stress tensor resulting from the potential W(F) = ˜W1(I1, J) + U1(J) is defined in FEniCSx by: def teff(u,lambda_m ,mu): from ufl import variable , Identity , grad , det , tr , ln , diff ## Deformation gradient F = variable(Identity(len(u)) + grad(u)) J = variable(det(F)) ## Right Cauchy -Green tensor C = variable(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='T * F) ##Invariants of deformation tensors Ic = variable(tr(C)) ## Potential W = (mu / 2) * (Ic - 3) - mu * ln(J) + (lambda_m / 2) * (ln(J))**2 return diff(W, F) 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='075 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='050 rr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='025 0 10 4 6 hy 20 8 xu 30 10Computation time (s) 40 30 30 20 20 30 20 2 10 4 6 nx 8 10 0All other developed potential are available in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Results The same solver options as for the 2D case were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' To limit the computation time, the time step was made variable: dt=500 for t ∈ [0, 20000], dt=1000 for t ∈ [20000, 60000] and dt=10000 for t ∈ [60000, 100000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' A total of 84 time steps was then considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The parameters were set according to Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The results for the previ- ously defined strain-energy potential are given Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Each finite element problem was computed in 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3 seconds on 8 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Independently from the choice of the potential, the consolidated pressure was retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' On the contrary, the resulting displacement depends on the chosen potential but a same order of magnitude is found for all the cases and describe well the observations proposed in Selvadurai and Suvorov [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In the absence of information about the porosity or the fluid bulk mod- ulus in the referent study, two fluid bulk modulus were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In case where the fluid bulk modulus is made close to the water one (Kf = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2×109), the hyper-elastic material well recovers the expected values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' However, mis- matches appear for a linear scaffold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' This can result from the use of a elastic law for large deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In case of a lower value of the fluid bulk modulus Kf = 5×105 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=', this can correspond to a non-constant value of the perme- ability and the porosity), the elastic behavior was recovered but differences on the hyper-elastic formulation were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' We believe that these differences result from a permeability depending on the stress state of the column which has not been developed in the referent paper (’Initial values of the permeability and viscosity are the same for all three materials.’ from Selvadurai and Suvorov [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 20 (a) (b) (c) (d) Figure 4: Kf = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 × 109: (a) Displacement of the to surface points and (b) pressure at the bottom of the column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Kf = 5 × 105: (c) Displacement of the to surface points and (d) pressure at the bottom of the column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The computed Linear Elastic (LE) and Neo-Hookean (NH) for both volumetric functions and the found calibrated parameters are super-imposed with the expected values from Selvadurai and Suvorov [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Confined bi-compartment porous-elastic medium Sections 3 proposed a poro-mechanical modeling of a single-compartment porous medium (suitable for an avascularised tissue for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In case of in vivo modeling, at least one more fluid phase is required: the blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' A 3D confined compression example of a column of height 100 µm is proposed, based on the here-after variational formulation and Scium`e [7] study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The load is applied as a sinusoidal ramp up to the magnitude of 100 Pa during 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Then, the load is sustained for 125 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' For more complex geometries, a gmsh example of a rectangle geometry indented by a cylindrical beam on its top surface and the corresponding local 21 le- Displacement (m) 2 3 0 20000 40000 60000 80000 100000 time (s)1e5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 LE (l1-3-2log())+log()2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 (l1-32log() +( -1)2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 (-2/3/1 -3) + (+)* (- 1)2 Linear Elastic [29] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 Neo-Hooke [29] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 0 20000 40000 60000 80000 100000 time (s)le- Displacement (m) 2 3 0 20000 40000 60000 80000 100000 time (s)1e5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 (Pa) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 Pressure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 0 20000 40000 60000 80000 100000 time (s)refinement are proposed Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Parameter Symbol Value Unit Young modulus E 5000 Pa Poisson ratio ν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 IF viscosity µl 1 Pa s Intrinsic permeability kε 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' × 10−14 m2 Biot coefficient β 1 Density of phase α ρα kg m−3 Porosity εl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 Vessel Bulk modulus Kν 1 × 103 Pa vessel Intrinsic permeability kε b 2 × 10−16 or 4 × 10−16 m2 Blood viscosity µb 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 × 10−3 Pa s Initial vascular porosity εb 0 0% or 2% or 4% Vascular porosity εb Equation 48 Table 4: Mechanical parameters for the bi-compartment model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Governing Equations Let one consider a vascular multi-compartment structure composed of a solid scaffold filled with interstitial fluid (IF) and blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The medium is assumed fully saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The following convention is assumed: •s denotes the solid phase, •l denotes the interstitial fluid phase (IF) and •b denotes the vascular part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The primary variables of the problem are the pressure applied in the pores of the extra-vascular part of the porous medium, namely pl, the blood pressure, namely pb, and the displacement of the solid scaffold, namely us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (Equation 36) links the different volume fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The volume fraction of the phase α is defined by (Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' εl is called the extra-vascular porosity of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' εs + εl + εb = 1 (36) Assuming that there is no inter-phase mass transport (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' the IF and the blood are assumed pure phases), the continuity equations (mass conservation) of the solid, the IF and the blood phases are respectively given by Equation 37, 38, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 22 ∂ ∂t(ρs(1 − εl − εb)) + ∇ · (ρs(1 − εl − εb)vs) = 0 (37) ∂ ∂t(ρlεl) + ∇ · (ρlεlvl) = 0 (38) ∂ ∂t(ρbεb) + ∇ · (ρbεbvb) = 0 (39) According to section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2, and dividing each equation by the corresponding density, the continuity equations can be re-expressed as: Ds Dt(1 − εl − εb) + (1 − εl − εb)∇ · vs = 0 (40) Dsεl Dt + ∇ · (εl(vl − vs)) + εl∇ · vs = 0 (41) Dsεb Dt + ∇ · (εb(vb − vs)) + εb∇ · vs = 0 (42) For the fluid phase, Darcy’s law (Equation 43, 44) is used to evaluate the fluid flow in the porous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' εl(vl − vs) = −kε µl (∇pl − ρlg) (43) εb(vb − vs) = −kb µb(∇pb − ρbg) (44) Where kε, kb are the intrinsic permeabilities (m2), µl, µb are the dynamic viscosities (Pa s), pl, pb the pressures and g the gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Equation 39 gives the following relationship: Dsεl Dt = −Dsεb Dt + (1 − εl − εb)∇ · vs (45) Considering Equations 43, 45, Equation 41 becomes: −Dsεb Dt − ∇ · (kε µl ∇pl) + (1 − εb)∇ · vs = 0 (46) 23 Then, reading Equation 44, Equation 42 gives: Dsεb Dt − ∇ · (kb µb∇pb) + εb∇ · vs = 0 (47) Considering a vascular tissue, we assume that the blood vessels are mostly surrounded by IF so they have weak direct interaction with the solid scaf- fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Furthermore, the vessels are assumed compressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Therefore, a state equation for the volume fraction of blood is introduced Equation 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' εb = εb 0 · � 1 − pl − pb Kν � (48) Where εb 0 denotes the blood volume fraction when pl = pb, Kν is the vessel compressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' It follows that Equations 46, 47 can be re-written as: − εb 0 Kν �Dspl Dt − Dspb Dt � − ∇ · (kε µl ∇pl) + (1 − εb)∇ · vs = 0 (49) εb 0 Kν �Dspl Dt − Dspb Dt � − ∇ · (kb µb∇pb) + εb∇ · vs = 0 (50) Once the continuity equations are settled, one can define the quasi-static momentum balance of the porous medium, Equation 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ∇ · ttot = 0 (51) Where ttot is the total Cauchy stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' We introduce an effective stress tensor denoted teff, responsible for all deformation of the solid scaffold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Then, ttot can be expressed as: ttot = teff − (1 − ζ)plId − ζpbId (52) ϵ(u) = 1 2(∇u + ∇uT) (53) teff = 2µϵ(us) + λtr(ϵ(us))Id (54) ζ = εb 0 � 1 − 2pl − pb Kν � (55) 24 Four boundaries are defined: the first one, Γu has imposed displacement (Equation 56), the second one Γs has imposed external forces (Equation 57) and Ωp has imposed pressure (fluid leakage condition (Equation 58, 59)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' We obtain: teff = timposed on Γs (56) us = uimposed on Γu (57) pl = 0 on Γp (58) pb = 0 on Γp (59) The initial conditions are given Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Parameter Symbol Value Unit Displacement us 0 m Displacement at previous step us n 0 m IF pressure pl 0 Pa IF pressure at previous step pl n 0 Pa Blood pressure pb 0 Pa Blood pressure at previous time step pb 0 Pa Vascular porosity εb εb 0 Table 5: Initial conditions for the bi-compartment model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Variational Form For the computation of the FE model, the variational form of Equation 49- 51 must be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Let one consider (ql,qb,v) the test functions defined in the mixed space L2 0(Ω) × L2 0(Ω) × [H1(Ω)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' With a first order approximation in time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Equation 49,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 50 gives: 25 − εb 0 Kν 1 dt � Ω (pb − pb n − pl + pl n)qldΩ + 1 − εb dt � Ω ∇ · (us − us n)qldΩ +kε µl � Ω ∇pl∇qldΩ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ∀ ql ∈ L2 0(Ω) (60) εb Kν 1 dt � Ω (pb − pb n − pl + pl n)qbdΩ + εb dt � Ω ∇ · (us − us n)qbdΩ +kb µb � Ω ∇pb∇qbdΩ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ∀ qb ∈ L2 0(Ω) (61) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' by integrating by part Equation 51,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' and including the Neumann boundary conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' we get: � Ω teff : ∇vdΩ − � Ω (1 − ζ)pl∇ · vdΩ − � Ω ζpb∇ · vdΩ − � Γs timposed · vdΓs = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ∀ v ∈ [H1(Ω)]3 (62) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' FEniCSx Implementation This section provides the code of a multi-compartment 3D column in confined compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' In order to evaluate the FEniCSx implementation, this case is similar to the Cast3m solution proposed in Scium`e [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3 cases are studied: avascular tissue, vascular porosity of 2% and vascular porosity of 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The load is applied as a sine ramp during 5 seconds and then sustained during 125 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The time discretisation is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' t, t_ramp , t_sust = 0, 5, 125 # Start time Tf = t_ramp+t_sust # End time num_steps = 1301 # Number of time steps dt = (Tf -t)/num_steps # Time step size We then introduce the material parameters according to Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The three cases of vascularisation and Equation 55 are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' E = Constant(mesh , ScalarType (5000)) nu = Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2)) kepsilon_l = Constant(mesh , ScalarType (1e -14)) mu_l = Constant(mesh , ScalarType (1)) 26 lambda_m = Constant(mesh , ScalarType(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value*nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value /((1+ nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value) (1 -2*nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value)))) mu = Constant(mesh , ScalarType (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value /(2*(1+ nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value)))) Knu = Constant(mesh , ScalarType (1000)) # compressibility of the vessels mu_b = Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='004)) #dynamic mu_l of the blood case =1 if case ==0: epsilon_b_0=Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='00)) #initial vascular porosity k_b=Constant(mesh , ScalarType (2e -16)) #intrinsic permeability of vessels def zeta(pl ,pb): return Constant(mesh ,ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=')) elif case ==1: epsilon_b_0=Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='02)) #initial vascular porosity k_b=Constant(mesh , ScalarType (2e -16)) #intrinsic permeability of vessels def zeta(pl ,pb): return epsilon_b_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value *(1 -2*(pl -pb)/Knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value) elif case ==2: epsilon_b_0 = Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='04)) #initial vascular porosity k_b = Constant(mesh , ScalarType (4e -16)) #intrinsic permeability of vessels def zeta(pl ,pb): return epsilon_b_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value *(1 -2*(pl -pb)/Knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value) Then, the integration space, boundary and initial conditions are set up for the displacement, the IF pressure and the blood pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ## Mechanical loading (Terzaghi) pinit = 200 #[Pa] T = Constant(mesh ,ScalarType (-pinit)) ## Define Mixed Space (R2 ,R, R) -> (u,pl , pb) element = VectorElement ("CG", mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='ufl_cell (), 2) pressure_element = FiniteElement ("CG", mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='ufl_cell (), 1) MS = FunctionSpace (mesh , MixedElement ([ element , pressure_element , pressure_element ])) # Create the solution and initial spaces X0 = Function(MS) Xn = Function(MS) # Create the surfacic element ds = Measure("ds", domain=mesh , subdomain_data =facet_tag) # compute the normals normal = FacetNormal (mesh) # Define the Dirichlet conditions bcs = [] # uy=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (1) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1))) # ux=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (2) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0))) # ux=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (4) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0) , fdim , facets) 27 bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0))) # uz=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (5) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2))) # uz=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (6) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2))) # leakage pl=pb=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (3) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1))) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2))) # Set Initial values # Displacement Un_ , Un_to_MS = MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='collapse () FUn_ = Function(Un_) with FUn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='localForm () as initial_local : initial_local .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='set(ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0)) # Update Xn for all threads Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[Un_to_MS] = FUn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # IF Pressure Pn_ , Pn_to_MS = MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='collapse () FPn_ = Function(Pn_) with FPn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='localForm () as initial_local : initial_local .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='set(ScalarType (0)) # Update Xn for all threads Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[Pn_to_MS] = FPn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # Blood Pressure Pbn_ , Pbn_to_MS = MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='collapse () FPbn_ = Function(Pbn_) with FPbn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='localForm () as initial_local : initial_local .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='set(ScalarType (0)) # Update Xn for all threads Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[Pbn_to_MS] = FPbn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () Internal variables are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The vessels are compressible so we include the evolution of the vascular porosity as a function representing Equation 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' # Internal variables: vascular porosity Poro_space = FunctionSpace (mesh , pressure_element ) poro_b = Function(Poro_space ) # vascular porosity # Initialize with poro_b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='localForm () as initial_local : initial_local .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='set(ScalarType( epsilon_b_0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value)) # Update poro_b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () poro_b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name="poro_b" A xdmf file is opened to store the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' xdmf = XDMFFile(mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm , "terzaghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='xdmf", "w") xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='write_mesh(mesh) 28 The test functions as well as the variational form are introduced according to Equations 60, 61, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' pl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' pb = split(X0) u_n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' pl_n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' pb_n = split(Xn) v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ql ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' qb = TestFunctions (MS) dx = Measure("dx",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' metadata ={" quadrature_degree ": 4}) F = (1- poro_b)*(1/ dt)*nabla_div(u-u_n)*ql*dx + ( kepsilon_l /( mu_l) )*dot( grad(pl),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='grad(ql) )*dx - ( epsilon_b_0 /Knu)*( (1/ dt)*(pb -pb_n -pl+pl_n) )* ql*dx F += poro_b *(1/ dt)*nabla_div(u-u_n)*qb*dx + ( k_b /( mu_b) )*dot( grad(pb),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' grad(qb) )*dx + ( epsilon_b_0 /Knu)*( (1/ dt)*(pb -pb_n -pl+pl_n) )*qb*dx F += inner(grad(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='teff(u))*dx - (1-zeta(pl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pb))*pl*nabla_div(v)*dx - zeta( pl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pb)*pb*nabla_div(v)*dx - T*inner(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='normal)*ds (3) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' the problem to be solved is defined and a Newton method is used for each time step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' the vascular porosity is updated and the results are stored in the xdmf file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' dX0 = TrialFunction (MS) J = derivative(F, X0 , dX0) Problem = NonlinearProblem (F, X0 , bcs = bcs , J = J) solver = nls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='petsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' NewtonSolver (mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm , Problem) # Set Newton solver options solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='atol = 5e -10 solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rtol = 1e -11 solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' convergence_criterion = " incremental " t = 0 for n in range(num_steps): t += dt if t < t_ramp: f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 * (1 - np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='cos(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pi*t/t_ramp)) else: f1 = 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value = 200*f1 num_its , converged = solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='solve(X0) X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # Update Value Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array [:] = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # Update porosity poro_b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array [:] = epsilon_b_0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value *(1 -(1/ Knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value)*(X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[ Pn_to_MS]-X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[Pbn_to_MS ])) poro_b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # Save data __u , __pl , __pb = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='split () __u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name = " Displacement " __pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name = "Pressure IF" __pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name = "Pressure blood" xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_function (__u ,t) xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_function (__pl ,t) xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_function (__pb ,t) xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_function (poro_b ,t) xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='close () 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Results (a) (b) (c) Figure 5: Comparison of the results obtained using FEniCSx against Scium`e [7] results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' All results were shifted to obtain similar figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The solid, doted and dashed lines respectively represent the 0%, 2% 4% initial vascular porosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (a) Evolution of the pressure at the bottom points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (b) Displacement of the top points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (c) Vascular porosity at the bottom points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The behaviour was well retrieved for all the cases with a NRMSE lower than 10% for all variables according to Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The evolution of the vascular and interstitial pressures at the bottom points and the vertical displacement at the top points are provided Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Each solution was obtained in 6 ± 2 minutes on 8 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The overall behavior of the interstitial fluid pressure, the blood pressure and the solid displacement were retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' To quantitatively assess the reliability of our implemented model, The normalized root mean square error (NRMSE, Equa- tion 63) was computed for each case with the results obtained with Cast3m in Scium`e [7], Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=" 30 200 Load 0% 150 p'atthe bottom points 100 50 pbatthe bottom points 0 10 5 0 5 10 15 20 25 30 time (s)1e-6 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 Vertical Displacementoftoppoints 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 (w) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 0% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 10 0 10 20 30 time (s)1e-2 4 2% 3 4% 1 Vascular porosity at the bottom points 0 0 20 40 60 80 100 120 time (s)Load Sciume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (2021), g = 0 Sciume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (2021), g = 2 Sciume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' (2021), eg = 4NRMSE(x, xref) = � 1 N � i∈[1,N](x − xref)2 mean(xref) (63) Parameter 0% 2% 4% pl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1 % uy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='7 % pb 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='7 % 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='8 % εb 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6 % Table 6: NRMSE computed for each studied variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The NRMSE was found lower than 10% for all unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The differences are assumed to result from the method of resolution which differs between Cast3m and FEniCSx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Indeed, the Cast3m procedure relies on a staggered solver whereas our results were obtained using a monolithic solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The order of magnitudes of the NRMSE made us however consider our solution as trustworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Conclusion The objective of this paper was to propose a step-by-step explanation on how to implement several poro-mechanical models in FEniCSx with a special attention to parallel computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Several benchmark cases for a mixed formulation were evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' First, a confined column was simulated under compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Accurate results according to the L2-norm were found compared to the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Furthermore, the code was computed 3 times faster than in the legacy FEniCS environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Then, a possible im- plementation of an hyper-elastic formulation was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The model was validated using Selvadurai and Suvorov [32] values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Finally, a confined bi- compartment sample was simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The results were compared to Scium`e [7] data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Small differences were observed due to the choice of the solver (staggered or monolithic) but remained acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The authors hope that this paper will contribute to facilitate the use of poroelasticity in the biome- chanical engineering community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' This article and its supplementary material constitute a starting point to implement their own material models at a pre- ferred level of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Supplementary material The python codes corresponding to the workflows and the docker file of this article are made available for 2D and 3D cases on the following link: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='com/Th0masLavigne/Dolfinx_Porous_Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Declaration of Competing Interest Authors have no conflicts of interest to report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Acknowledgment This research was funded in whole, or in part, by the Luxembourg Na- tional Research Fund (FNR),grant reference No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 17013182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' For the purpose of open access, the author has applied a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0) license to any Author Accepted Manuscript ver- sion arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The present project is also supported by the National Research Fund, Luxembourg, under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' C20/MS/14782078/QuaC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 32 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 3D Terzaghi example Here-after is proposed a minimal working code corresponding to the 2D case included within the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' import numpy as np import csv from petsc4py import PETSc import dolfinx from dolfinx import nls from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='io import XDMFFile from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh import CellType , create_box , locate_entities_boundary , locate_entities , meshtags from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='fem import (Constant , dirichletbc , Function , FunctionSpace , locate_dofs_topological , form , assemble_scalar ) from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='fem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='petsc import NonlinearProblem from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='geometry import BoundingBoxTree , compute_collisions , compute_colliding_cells from petsc4py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='PETSc import ScalarType from mpi4py import MPI from ufl import (FacetNormal , Identity , Measure , TestFunctions , TrialFunction , VectorElement , FiniteElement , dot , dx , inner , grad , nabla_div , div , sym , MixedElement , derivative , split) # def epsilon(u): return sym(grad(u)) # def teff(u): return lambda_m * nabla_div(u) * Identity(u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' geometric_dimension ()) + 2* mu*epsilon(u) # kmax =1e3 def terzaghi_p(x): p0 ,L=pinit ,Height cv = permeability .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value/viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value *( lambda_m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value +2* mu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value) pression =0 for k in range (1,int(kmax)): pression +=p0 *4/ np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pi*( -1) **(k -1) /(2*k -1)*np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='cos ((2*k -1) *0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5* np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pi*(x [1]/L))*np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='exp ( -(2*k-1) **2*0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='25* np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pi **2* cv*t/L**2) pl=pression return pl # def L2_error_p(mesh ,pressure_element ,__p): V2 = FunctionSpace (mesh , pressure_element ) pex = Function(V2) pex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' interpolate (terzaghi_p ) L2_errorp , L2_normp = form(inner(__p - pex , __p - pex) * dx), form(inner (pex , pex) * dx) error_localp = assemble_scalar (L2_errorp)/ assemble_scalar (L2_normp) error_L2p = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sqrt(mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='allreduce(error_localp , op=MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='SUM)) return error_L2p # ## Create the domain / mesh Height = 1e-4 #[m] Width = 1e-5 #[m] Length = 1e-5 #[m] 33 mesh = create_box(MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD , np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array ([[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0] ,[ Length , Width , Height ]]) , [8, 8, 20], cell_type=CellType.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' tetrahedron ) # ## Define the boundaries: # 1 = bottom , 2 = right , 3=top , 4=left , 5=back , 6= front boundaries = [(1, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[2], 0)), (2, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[0], Length)), (3, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[2], Height)), (4, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[0], 0)), (5, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[1], Width)), (6, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[1], 0))] # facet_indices , facet_markers = [], [] fdim = mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim - 1 for (marker , locator) in boundaries: facets = locate_entities (mesh , fdim , locator) facet_indices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(facets) facet_markers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='full_like(facets , marker)) facet_indices = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( facet_indices ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) facet_markers = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( facet_markers ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) sorted_facets = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='argsort( facet_indices ) facet_tag = meshtags(mesh , fdim , facet_indices [ sorted_facets ], facet_markers [ sorted_facets ]) # ## Time parametrization t = 0 # Start time Tf = 6 # End time num_steps = 1000 # Number of time steps dt = (Tf -t)/num_steps # Time step size # ## Material parameters E = Constant(mesh , ScalarType (5000)) nu = Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='4)) lambda_m = Constant(mesh , ScalarType(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value*nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value /((1+ nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value) (1 -2*nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value)))) mu = Constant(mesh , ScalarType (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value /(2*(1+ nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value)))) rhos = Constant(mesh , ScalarType (1)) permeability = Constant(mesh , ScalarType (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='8e -15)) viscosity = Constant(mesh , ScalarType (1e -2)) rhol = Constant(mesh , ScalarType (1)) beta = Constant(mesh , ScalarType (1)) porosity = Constant(mesh , ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2)) Kf = Constant(mesh , ScalarType (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2 e9)) Ks = Constant(mesh , ScalarType (1 e10)) S = (porosity/Kf)+(1- porosity)/Ks # ## Mechanical loading pinit = 100 #[Pa] T = Constant(mesh ,ScalarType (-pinit)) # # Create the surfacic element ds = Measure("ds", domain=mesh , subdomain_data =facet_tag) normal = FacetNormal (mesh) # # Define Mixed Space (R2 ,R) -> (u,p) displacement_element = VectorElement ("CG", mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='ufl_cell (), 2) pressure_element = FiniteElement ("CG", mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='ufl_cell (), 1) 34 MS = FunctionSpace (mesh , MixedElement ([ displacement_element , pressure_element ])) # # Define the Dirichlet condition # 1 = bottom: uy=0, 2 = right: ux=0, 3= top: pl=0 drainage , 4= left: ux=0 bcs = [] # uz=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (1) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (2))) # ux=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (2) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0))) # ux=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (4) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0))) # uy=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (5) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1))) # uy=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (6) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1))) # drainage p=0 facets = facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='find (3) dofs = locate_dofs_topological (MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1) , fdim , facets) bcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(dirichletbc(ScalarType (0) , dofs , MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1))) # # Create the initial and solution spaces X0 = Function(MS) Xn = Function(MS) # # Initial values # Un_ , Un_to_MS = MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='collapse () FUn_ = Function(Un_) with FUn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='localForm () as initial_local : initial_local .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='set(ScalarType (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0)) # # Update Xn Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[Un_to_MS] = FUn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # Pn_ , Pn_to_MS = MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sub (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='collapse () FPn_ = Function(Pn_) with FPn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='localForm () as initial_local : initial_local .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='set(ScalarType(pinit)) # # Update Xn Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array[Pn_to_MS] = FPn_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # # Variational form # Identify the unknowns from the function 35 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='p =split(X0) u_n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='p_n=split(Xn) # Set up the test functions v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='q = TestFunctions (MS) # Equation 17 F = (1/dt)*nabla_div(u-u_n)*q*dx + ( permeability /viscosity)*dot(grad(p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' grad(q))*dx + ( S/dt )*(p-p_n)*q*dx # Equation 18 F += inner(grad(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='teff(u))*dx - beta * p * nabla_div(v)*dx - T*inner(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' normal)*ds (3) # Non linear problem definition dX0 = TrialFunction (MS) J = derivative(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' X0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' dX0) Problem = NonlinearProblem (F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' X0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' bcs = bcs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' J = J) # set up the non -linear solver solver = nls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='petsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' NewtonSolver (mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm , Problem) # Absolute tolerance solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='atol = 5e -10 # relative tolerance solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rtol = 1e -11 solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' convergence_criterion = " incremental " # t = 0 L2_p = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros(num_steps , dtype=PETSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ScalarType ) for n in range(num_steps): t += dt num_its , converged = solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='solve(X0) X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # Update Value Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array [:] = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () __u , __p = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='split () # Compute L2 norm for pressure error_L2p = L2_error_p (mesh ,pressure_element ,__p) L2_p[n] = error_L2p # Solve tracking if mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rank == 0: print(f"Time step {n}, Number of iterations {num_its}, Load {T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value }, L2 -error p {error_L2p :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2e}") if mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rank == 0: print(f"L2 error p, min {np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='min(L2_p):.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2e}, mean {np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mean(L2_p):.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2e}, max {np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='max(L2_p):.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2e}, std {np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='std(L2_p):.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2e}") Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Local refinement A 3D geometry can be meshed using the GMSH API of python (Geuzaine and Remacle [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' This allows to represent complex geometries including circle arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' An optimized and locally refined mesh can be therefore obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' This example uses the method proposed in the FEniCS project tutorial 1 pro- 1see https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='fenicsproject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='org/dolfinx/main/python/demos/demo_gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' html 36 vided by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Dokken and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' An alternative procedure in the FEniCSx environment with local refinement is then proposed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Meshing using GMSH API First, the environment is initialized and the physical variables required for the box/cylinder creation are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' import gmsh import numpy as np # gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='initialize () # # box parameters [Length , Width , Height] = [6e-4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5e-4, 4e -5] # cylinder parameters xc ,yc ,zc ,dx ,dy ,dz , r = 6e-4/2 , 0, 0, 0, 0, 4e-5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5e-4 # expected dimension of the mesh gdim = 3 The geometries are created using built-in functions of GMSH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' potential duplicates are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' # create the geometry box = gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='addBox (0, 0, 0, Length , Width , Height) cylinder = gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='addCylinder (xc ,yc ,zc ,dx ,dy ,dz , r,tag =1000 , angle=np .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='pi) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' synchronize () # Remove duplicate entities and synchronize gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' removeAllDuplicates () gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' synchronize () Physical groups are defined: the volumes for the 3D meshing and the surfaces for tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Surface groups were identified based on the coordinates of the center of mass of each surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' surfaces , volumes = [gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' getEntities (d) for d in [ gdim -1, gdim ]] print(volumes) # Volumes gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (volumes [0][0] , [volumes [0][1]] , 1) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (volumes [0][0] , 1, ’Half_Cylinder ’) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (volumes [1][0] , [volumes [1][1]] , 1) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (volumes [1][0] , 1, ’Box ’) # 1 = loading , 2 = top minus loading , 3 = bottom , 4 = left , 5 = right , 6 = Front , 7 = back bottom_marker , front_marker , back_marker , left_marker , right_marker , top_marker , indenter_marker = 3, 6, 7, 4, 5, 2, 1 bottom , front , back , left , right , top , indenter = [] ,[] ,[] ,[] ,[] ,[] ,[] boundaries = gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' getBoundary (volumes , oriented=False) for boundary in boundaries: center_of_mass = gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' getCenterOfMass (boundary [0], boundary [1]) if np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose( center_of_mass [1], Width): back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(boundary [1]) elif np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose( center_of_mass [1], 0): 37 front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(boundary [1]) elif np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose( center_of_mass [0], 0): left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(boundary [1]) elif np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose( center_of_mass [0], Length): right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(boundary [1]) elif np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose( center_of_mass [2], 0): bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(boundary [1]) elif np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose( center_of_mass [2], Height) and center_of_mass [1]> Width /3: top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(boundary [1]) else: indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(boundary [1]) # mark the surfaces gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (boundaries [0][0] , bottom , bottom_marker ) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (boundaries [0][0] , bottom_marker , ’bottom ’) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (boundaries [0][0] , front , front_marker ) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (boundaries [0][0] , front_marker , ’front ’) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (boundaries [0][0] , back , back_marker ) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (boundaries [0][0] , back_marker , ’back ’) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (boundaries [0][0] , left , left_marker ) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (boundaries [0][0] , left_marker , ’left ’) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (boundaries [0][0] , right , right_marker ) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (boundaries [0][0] , right_marker , ’right ’) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (boundaries [0][0] , top , top_marker ) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (boundaries [0][0] , top_marker , ’top ’) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' addPhysicalGroup (boundaries [0][0] , indenter , indenter_marker ) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setPhysicalName (boundaries [0][0] , indenter_marker , ’indenter ’) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' synchronize () # Write a geo file for verification in the GMSH GUI gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='write(’Geom_2reelle_8EP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' geo_unrolled ’) Then, a threshold function is defined over a distance field to mesh the circular area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' This allows for creating an adaptive mesh: coarse far from the circular area, refine close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' indenter_interface = surfaces [0][1] distance = gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='add("Distance") gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumbers (distance , "FacesList", [ indenter_interface ]) # A threshold function is defined: resolution = r/10 threshold = gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='add("Threshold") gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber(threshold , "IField", distance) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber(threshold , "LcMin", resolution ) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber(threshold , "LcMax", 5* resolution) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber(threshold , "DistMin", 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6*r) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber(threshold , "DistMax", r) # If several fields are defined: minimum = gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='add("Min") gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumbers (minimum , " FieldsList ", [threshold ]) # add other fields in the list if needed gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' setAsBackgroundMesh (minimum) Finally, the options of the mesher are defined and the mesh is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' synchronize () gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber("General.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='Terminal" ,1) 38 gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber("Mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='Optimize", True) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber("Mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' OptimizeNetgen ", True) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' synchronize () # gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber (" Mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' MshFileVersion ", 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber("Mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' MeshSizeExtendFromBoundary ", 0) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber("Mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' MeshSizeFromPoints ", 0) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='setNumber("Mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' MeshSizeFromCurvature ", 0) # gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='generate(gdim) gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='write("Mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='msh") gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='finalize () Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Local refinement within FEniCSx Using GMSH API, an exact circular interface is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' However, a similar mesh could have been obtained within FEniCSx through the ap- proximation of the circular interface around the indenter by local refining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Here-after is proposed a minimal code for local refinement inside the circular area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' First, the required libraries are imported and a box mesh is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ## Librairies import dolfinx import numpy as np from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='mesh import create_box , CellType , refine , locate_entities , meshtags from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='io import XDMFFile from mpi4py import MPI # ## Box # Dimensions of the sample [Length , Width , Height] = [6e-4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5e-4, 4e -5] # Discretization [nx ,ny ,nz] = [30 ,15 ,8] mesh = create_box(MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD ,np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array ([[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0 ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0] ,[ Length , Width , Height ]]) , [nx ,ny ,nz], cell_type=CellType.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' tetrahedron ) Then a locator is introduced to identify all the edges (fdim = 1) which are part of the region we aim to refine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' def test_on_boundary (x): return (np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='power(x[0] -3e-4 ,2)+np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='power(x[1] ,2)) <=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5e -4) # refine_boudaries = [(11 , lambda x: test_on_boundary (x))] Finally, a loop is performed to compute several times the refinement (np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='arange(N)), using the existing refine() function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' for _ in np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='arange (2): # Refinement refine_indices , refine_markers = [], [] fdim = mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim -2 for (marker , locator) in refine_boudaries : 39 facets = locate_entities (mesh , fdim , locator) refine_indices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(facets) refine_markers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='full_like(facets , marker)) refine_indices = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( refine_indices ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) refine_markers = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( refine_markers ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) # indices in meshtag must be sorted sorted_facets_refine = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='argsort( refine_indices ) refine_tag = meshtags(mesh , fdim , refine_indices [ sorted_facets_refine ], refine_markers [ sorted_facets_refine ]) mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' create_entities (fdim) mesh = refine(mesh , refine_indices [ sorted_facets_refine ]) The facets are tagged to apply boundary conditions and the mesh is written as a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='xdmf file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' def Omega_top(x): return np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' logical_and ((x[2] == Height), (np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='power(x[0] -3e-4 ,2)+ np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='power(x[1] ,2)) <=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='5e -4)) # def Omega_loading (x): return np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' logical_and ((x[2] == Height), (np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='power(x[0] -3e-4 ,2)+ np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='power(x[1] ,2)) >=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2e -4)) # # Create the facet tags (identify the boundaries) # 1 = loading , 2 = top minus loading , 3 = bottom , 4 = left , 5 = right , 6 = Front , 7 = back boundaries = [(1, lambda x: Omega_loading (x)), (2, lambda x: Omega_top(x)), (3, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[2], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0)), (4, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[0], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0)), (5, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[0], Length)), (6, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[1], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='0)), (7, lambda x: np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='isclose(x[1], Width))] # Mark them facet_indices , facet_markers = [], [] fdim = mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim - 1 for (marker , locator) in boundaries: facets = locate_entities (mesh , fdim , locator) facet_indices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(facets) facet_markers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='append(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='full_like(facets , marker)) facet_indices = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( facet_indices ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) facet_markers = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='hstack( facet_markers ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='astype(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='int32) sorted_facets = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='argsort( facet_indices ) facet_tag = meshtags(mesh , fdim , facet_indices [ sorted_facets ], facet_markers [ sorted_facets ]) facet_tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name = "facets" # Write XDMF mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' create_connectivity (mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim -1, mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim) with XDMFFile(mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm , "facet_tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='xdmf", "w") as xdmftag: xdmftag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='write_mesh(mesh) xdmftag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_meshtags (facet_tag) xdmftag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='close () Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6 gives the comparison of the mesh obtained using GMSH and the one using local refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' 40 (a) (b) Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='6: GMSH (a) and FEniCSx (b) generated meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Import an external mesh (XDMF or MSH) Once the mesh is generated as a tagged .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='msh or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='xdmf file, one can con- sider directly read them to compile the domain and read the markers using: from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='gmshio import read_from_msh from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='io import XDMFFile # set value to 0 if .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='xdmf , set it to 1 if .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='msh mesher = 1 # if mesher == 0: # ######################### ## Read XDMF mesh ## # ######################### filename = "filename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='xdmf" with XDMFFile(MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD , filename , "r") as file: mesh = file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='read_mesh () mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' create_connectivity (mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim -1, mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='topology .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim) facet_tag = file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' read_meshtags (mesh , "tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name") # elif mesher == 1: # ######################### ## Read gmsh mesh ## # ######################### mesh , cell_tag , facet_tag = read_from_msh ("filename.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='msh", MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD , 0, gdim =3) # else: print(’The mesh type is wrongly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' mesher should equal 0 for xdmf and 1 for msh files.’) exit () 41 Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Evaluate the function at a physical point One strength of using FEniCSx is its ability to evaluate the solution at given points, summing the contribution of the neighbor cells of the mesh 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The following code allowed to compute the figures presented for the results of the sections 3 and ref 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' First, one need to define the points where to evaluate the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' import numpy as np num_points = 11 y_check = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='linspace (0,Height , num_points ) points_for_time = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array ([[ Width /2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ], [Width /2, Height /2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=']]) points_for_space = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros (( num_points ,3)) for ii in range(num_points): points_for_space [ii ,0]= Width /2 points_for_space [ii ,1]= y_check[ii] points = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' concatenate (( points_for_time , points_for_space )) The following step is to identify the cells contributing to the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' from dolfinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='geometry import BoundingBoxTree , compute_collisions , compute_colliding_cells tree = BoundingBoxTree (mesh , mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='dim) cell_candidates = compute_collisions (tree , points) colliding_cells = compute_colliding_cells (mesh , cell_candidates , points) # Here is an example to select cells contributing to the first and second points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' cells_y_0 = colliding_cells .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='links (0) cells_y_H_over_2 = colliding_cells .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='links (1) Knowing the shape of the functions to evaluate, lists are created and will be updated during the resolution procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Regarding parallel computation, these lists are only created on the first kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' from mpi4py import MPI if MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rank == 0: pressure_y_0 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros(num_steps , dtype=PETSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ScalarType) pressure_y_Height_over_2 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros(num_steps , dtype=PETSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ScalarType ) pressure_space0 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros(num_points , dtype=PETSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ScalarType ) pressure_space1 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros(num_points , dtype=PETSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ScalarType ) pressure_space2 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros(num_points , dtype=PETSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ScalarType ) A function is created to evaluate a function given the mesh, the function, the contributing cells to the point and the list with its index to store the evaluated value in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' def evaluate_point (mesh , function , contributing_cells , point , output_list , index): from mpi4py import MPI 2see https://jorgensd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='io/dolfinx-tutorial/chapter2/ns_code2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='html?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' highlight=eval 42 function_eval = None if len( contributing_cells ) > 0: function_eval = function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='eval(point , contributing_cells [:1]) function_eval = mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='gather(function_eval , root =0) # Choose first pressure that is found from the different processors if MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rank == 0: for element in function_eval : if element is not None: output_list[index ]= element [0] break pass Finally, the problem is solved for each time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' The functions are evaluated for all kernels and gathered on the first one where the first pressure found by the different processors will be uploaded in the here-above lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' # time steps to evaluate the pressure in space: n0 , n1 , n2 = 200 ,400 ,800 # t = 0 L2_p = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='zeros(num_steps , dtype=PETSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' ScalarType ) for n in range(num_steps): t += dt try: num_its , converged = solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='solve(X0) except: if MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rank == 0: print(" ************* ") print("Solver failed") print(" ************* ") pass X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () # Update Value Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array [:] = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='array Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' scatter_forward () __u , __p = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='split () # # Export the results __u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name = " Displacement " __p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='name = "Pressure" xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_function (__u ,t) xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' write_function (__p ,t) # # Compute L2 norm for pressure error_L2p = L2_error_p (mesh ,pressure_element ,__p) L2_p[n] = error_L2p # # Solve tracking if MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='COMM_WORLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='rank == 0: print(f"Time step {n}/{ num_steps}, Load {T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='value}, L2 -error p { error_L2p :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='2e}") # Evaluate the functions # in time if n == n0: for ii in range(num_points ): evaluate_point (mesh , __p , colliding_cells .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='links(ii +2) , points[ii +2], pressure_space0 , ii) 43 t0 = t elif n==n1: evaluate_point (mesh , __p , colliding_cells .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='links(ii +2) , points[ii +2], pressure_space1 , ii) t1 = t elif n==n2: evaluate_point (mesh , __p , colliding_cells .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='links(ii +2) , points[ii +2], pressure_space2 , ii) t2 = t # xdmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content='close () References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Budday, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' Ovaert, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFIT4oBgHgl3EQfaSsg/content/2301.11256v1.pdf'} +page_content=' A.' metadata={'source': 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of blockchain technol- +ogy, there has recently been a renewed interest in the design, +implementation and evaluation of decentralized systems. Most +of these systems are intended to be deployed at scale and in +heterogeneous environments with real users and unpredictable +workloads. Nevertheless, most research in this field evaluates +such systems in controlled environments that poorly reflect +the complex conditions of real-world environments. In this +work, we argue that deployment is crucial to understanding +decentralized mechanisms in a real-world environment and an +enabler to building more robust and sustainable systems. We +highlight the merits of deployment by comparing this approach +with other experimental setups and show how our lab applied +a deployment-first methodology. We then outline how we use +Tribler, our peer-to-peer file-sharing application, to deploy and +monitor decentralized mechanisms at scale. We illustrate the +application of our methodology by describing a deployment +trial in experimental tokenomics. Finally, we summarize four +lessons learned from multiple deployment trials where we +applied our methodology. +Index Terms—Decentralized Systems, Research Methodol- +ogy, Experimental Setups, System Failures. +I. INTRODUCTION +The scale and complexity of distributed systems have +increased tremendously since the field’s inception. We live +in an era of complex ultra-large-scale networks characterized +by the number of participating nodes and by high het- +erogeneity, flexibility, non-trivial social dependencies, and +emergent properties [3], [11], [25]. Ensuring the proper +functioning, deployment, monitoring and maintenance of +such systems requires system designers to obtain insights +into their performance and correctness. Given the scale +and unpredictability of such systems, obtaining engineering +insights presents unique challenges for researchers and de- +velopers. +Decentralized blockchain applications embody the charac- +teristics of complex ultra-large-scale networks [2], [24]. At +the same time, the costs of failures in these applications are +very high due to built-in financial mechanisms [29]. Though +realistic experimental setups such as testnets aim to address +these challenges to a degree, properly testing and evaluating +such systems remains a challenging endeavour [5]. +The mainstream paradigm in distributed systems research +is a top-down design that focuses on predicting performance, +failures and limitations through experimentation in con- +trolled environments. These experiments usually are carried +out as simulations or emulations informing researchers’ +design choices [4]. An empirical study of failures can +provide invaluable insights into different types of distributed +systems [12], [13], [26]. Detection of events that make +a system fail to operate according to its specifications - +detection of failures - is often a critical task in distributed +systems given complex interdependencies [17]. +However, empirical experimentation, guided by workloads +extracted from a deployed system, is uncommon in academic +research [4]. Mature research methodologies with an em- +phasis on deployment are still missing even in empirically- +driven fields of distributed systems research, for example, +blockchain applications [18]. +We address this gap by presenting our deployment-first +methodology for designing and evaluating decentralized +systems.1 We specifically focus on findings that can be +obtained from the study of failures after the deployment. Our +methodology is based on nearly two decades of experience +developing the Tribler software [8], [22], [28], serving +as infrastructure for deploying and evaluating decentral- +ized mechanisms at scale. We demonstrate that while the +deployment-first research methodology can be demanding +in terms of time investment, additional insights obtained +are worth this time investment. A nuanced evaluation of +trade-offs between different experimental setups is also +instrumental in designing research methodologies. +In summary, this work makes the following contributions: +1) We compare different experimental setups and high- +light the merits of deployment as a critical step in the +research methodology when building and evaluating +distributed systems (Section II). +2) We formulate our deployment-first research method- +ology to evaluate decentralized mechanisms at scale +(Section III). We also describe how we use Tribler, +our decentralized file-sharing application and research +vehicle for conducting large-scale deployment trials. +3) We illustrate how we applied our research methodol- +ogy to obtain unique insights in a complex use case on +tokenomics in decentralized networks (Section III-B). +4) Based on our experiences with our methodology, we +summarize four lessons we learned (Section III-C). +II. THE STATUS QUO OF DECENTRALIZED SYSTEMS +EXPERIMENTS +This work argues that deployment experiments are es- +sential to build robust decentralized mechanisms. We first +1The scope of this paper is primarily limited to decentralized systems. +However, some of these insights could be relevant to a wider field of +distributed systems. +arXiv:2301.04508v1 [cs.DC] 11 Jan 2023 + +1. Mechanism +Design +2. Software +Implementation +3. Experiments +(in controlled +environment) +5. Deployment +and +Monitoring +4. refine +Most research +ends at this step +Fig. 1. The standard research methodology to design, build, evaluate and deploy decentralized systems. +describe the standard methodology to research distributed +systems and then outline the merits of deployment as part +of the research methodology. +A. Standard Research Methodology +Figure 1 shows the standard research methodology to +design, build, evaluate, and deploy decentralized systems. +This research methodology is based on reports in the +fields’ literature [4], [14], [15], on discussions with other +researchers, and our own experiences. The following five +steps describe this methodology: +1) Mechanism Design. A researcher starts by designing +a particular mechanism. Research with an exclusive +focus on understanding theoretical models is usually +limited to this step [4]. +2) Software Implementation. The researcher then works +on a software implementation of the designed mech- +anism. Assuming that the original design was well- +executed, the implementation phase should not result +in significant changes to the original models. As such, +we left out this feedback loop from Figure 1. +3) Experiments. With the implementation, the researcher +conducts experiments in an environment controlled by +the analyst, e.g., on a local computer or a compute +cluster. These experiments usually aim to verify the +implementation’s correctness and quantify system met- +rics, e.g., scalability and fault tolerance. +4) Refining The Design Using Experimental Results. +Based on the experimental results obtained in the +previous step, the researcher updates the mechanism +design, updates the accompanying implementation and +re-runs experiments. For instance, an experiment re- +vealing that a particular design has low scalability +(e.g., in the number of participants the system can +support) might require the researcher to identify bot- +tlenecks in the mechanism and resolve them. +5) Deployment and Monitoring. The researcher can +deploy the algorithm in a real-world setting after the +experiments are finished. The deployed software will +likely be continuously monitored to detect failures or +anomalies. +Most academic research does not further test and evaluate +their mechanisms using deployment and ends at step (4) +[4]. This is not unexpected since local experiments usually +suffice to prove the mechanism’s trade-offs, correctness +or performance to a scientific community. As such, the +time investment and resource costs do not justify the need +for deployment.2 We argue, however, that trade-offs and +limitations associated with the usage of experimental setups +to evaluate decentralized systems are more nuanced. +B. The Merits of Deployment +We start by comparing the trade-offs between different +experimental setups used to evaluate decentralized systems. +Based on our literature research, we choose to compare the +following four experimental setups: +1) A simulation is a model of an application tested on a +model of an environment; +2) An emulation is an application that runs in an envi- +ronment where some parts of it are modelled; +3) A testnet is an application that is deployed on multiple +machines run by researchers or volunteer testers; +4) A real-world deployment is an application that end +users run on their machines. +We compare in Table I eight different properties of these +experimental setups and briefly discuss them below. +Costs of experiments. It is not always feasible to rep- +resent the costs of experiments in commensurable scales. +Most often, the costs of experiments can be described by +the monetary costs of purchasing necessary computation +resources. Both in simulation and emulation, these costs are +relatively manageable. However, in the case of a deployed +system, experimentation costs can be much higher or lower, +depending on the application context. Experiments on a +deployed system that require changes in system parameters +or functionality can cause failures and loss of users or market +share [5]. However, experiments on a deployed system can +have meager costs in some situations, e.g. if volunteer users +provide their resources.3 In testnets, if most of the resources +are provided by volunteer users, the experiment costs can be +low for researchers. However, attracting a sufficient amount +of users for a testnet can also require some initial investments +and upfront costs to get traction, as can be the case with +marketing costs for blockchain testnets [10]. +Scalability. of experiments is strongly correlated with +the costs of experiments. In the case of simulations, it is +relatively cheap to scale up the simulation size. In emulation, +the upper bound for scalability is typically limited by the +available computational resources of a testbed. In the case +2In contrast to that, the deployment of novel system designs is a +common practice in the blockchain industry [18]. However, the quality +of experimental evaluations is lacking compared to academic research, as +industry whitepapers often present biased and inflated results [20]. +3See https://github.com/ethereum/ropsten/blob/master/revival.md. + +Property +Simulation +Emulation +Testnet +Real-world Deployment +Cost of experiment +Medium +Medium +Low/High +Low/High +Scalability +Medium/High +Resource constrained +Resource constrained +Resource constrained +Environmental realism +Low +Low/Medium +High +Very High +Failures discoverability +Impossible +Low +Medium +High +Reproducibility +High +Medium +Low +Low +Control +High +High +Medium +Low +Speed of change +Fast +Fast +Medium +Slow +Debugability +High +Medium +Medium +Low +TABLE I +A comparison between four different experimental setups: simulation, emulation, testnet, and real-world deployment. +of a testnet, the scalability is limited to the computational +resources available to the researcher. In a real-world deploy- +ment, the scale of the experiment is usually limited by the +number of end-users and the resources they contribute. +Environmental realism. This is one of the two key +features that set testnets and deployments apart from other +experimental setups. Environmental realism is comprised of +three different parameters: (1) client heterogeneity, e.g., dif- +ferences in hardware capabilities; (2) variability in external +parameters such as network conditions; (3) the effect of +user behaviour. A real-world deployment captures all these +three parameters. The key difference with the testnet is the +effect of user behaviour; e.g., a testnet can be exclusive to +expert users, reducing realism. User incentives in testnets +are also sometimes simplified, e.g., in blockchain testnets, +there usually are no financial incentives. Both in simulation +and emulation, all three parameters of environmental realism +are lower compared to other experimental setups. External +parameters such as network conditions can be more realistic +with emulation. Realistic client heterogeneity is challenging +to represent realistically in an emulation conducted with het- +erogeneous hardware. One observation is that environmental +realism can be improved for simulation and emulation if +these setups are designed with values known from mea- +surements of deployed systems. Two key factors limit such +measurements: first, the measured system should have a very +similar application context; Second, it is not certain if the +results of the measurements still hold as the system evolves. +There has been some work that aims to bring environmen- +tal realism to simulation or emulation setups. Sarzyniec et al. +present Distem, a virtualisation platform to enable resource +heterogeneity in a homogeneous compute cluster [23]. While +this is a step to make experiments more accurate, the failure +model and user-generated workloads are not carried over +from the real-world environment. Recent work on scalability +experiments with BFT consensus protocols proposes a simu- +lator [1]. Understanding the realism gap between simulators +and real-world environments is a key part of this work. +Discoverability of new failures. This is another key +property that distinguishes controlled and uncontrolled ex- +periment setups. Certain types of failures can only be dis- +covered in a real-world environment, particularly emergent +and user-caused failures [13]. Testnets can reveal certain +types of emergent and partial failures [5]. Relatively fewer +novel types of failures can be revealed by experiments using +emulation setups. In principle, simulations do not allow for +discovering new types of failures. +Reproducibility. This parameter is tied to the availability +of the same setup to different researchers. Simulation, at +least in theory, allows for the highest level of reproducibil- +ity, given that all artefacts can easily be published. With +emulations, the evaluated software can be made available, +but access to an identical experimental testbed is not always +available. It could be argued that while reproducibility is +low for both testnets and deployed systems, a testnet allows +for a relatively easier replication of experimental conditions, +which is almost impossible with deployed systems. +Control. Simulation and emulation are the experimental +setups that give researchers the most control over the flow +of their experiments. With testnets and real-world deployed +systems, researchers usually have little to no control over +the system while it is running since there is a dependency +on the volunteers or end users that are running the software. +Speed of change. The speed at which an experiment +can be modified is a distinguishing factor between different +experimental setups. This speed of change is usually the +lowest in a deployed real-world system, given the delays +caused by the propagation of software updates. Testnets +can allow for somewhat quicker deployment cycles. In both +cases, deployment and the collection of results can be time- +consuming. Simulation and emulation setups have relatively +low external constraints and quickly be changed. +Debugability. Discovering, analyzing and reproducing +software bugs in deployed systems require dedicated infras- +tructure and can be a time-consuming process. In compari- +son, testnets and emulation provide more debugability since +the researcher has a higher level of control. Simulation can +provide the highest discoverability of bugs as long as the +scale of the simulation is not too large. +This analysis shows that we can not have a com- +prehensive evaluation without deployment. We need +to account for environmental realism and failures, +which are detectable only in a real-world scenario. + +1. Mechanism +Design +2. Software +Implementation +3. Experiments +(in controlled +environment) +5. Deployment +and +Monitoring +4. refine +6. refine +6. refine +Fig. 2. Our deployment-first research methodology. The key difference with Figure 1 is that we directly apply new insights or real-world workload traces +to the mechanism design and experiments, respectively (step 6). +III. TRIBLER: DEPLOYING AND MONITORING +DECENTRALIZED MECHANISMS AT SCALE +Tribler is our lab’s peer-to-peer file-sharing software and +research vehicle to deploy and evaluate decentralized mech- +anisms at scale [8]. We have used the Tribler software for al- +most two decades to obtain unique insights into the complex +interactions and dynamics in live peer-to-peer networks [22]. +Over 30 PhD researchers and BSc/MSc students have used +our software to evaluate their mechanisms. Tribler also has +a stable user base that enables longitudinal deployment +experiments. Over 1.8 million users have downloaded the +Tribler software, and at the time of writing, Tribler has +40’000 unique monthly users.4 +Tribler was initially designed as a file-sharing application +that allows users to download torrent files anonymously +using a custom onion-routing protocol [16]. Tribler uses +the IPv8 networking library that supports authenticated +messaging and enables the construction and maintenance +of decentralized overlays. Over the years, however, Tribler +has evolved from a BitTorrent download client to a versatile +application with features such as keyword search, bundling +torrents into channels, and reputation mechanisms to address +free-riding behaviour [19]. +A. Deployment-First Research Methodology +Tribler is a vital part of our labs’ research methodology +since it enables us to deploy and evaluate decentralized +mechanisms at scale. Tribler also allowed us to try out a +different research methodology with an increased focus on +deployment. Originating from our experiences with Tribler +and deployment efforts, we now present our deployment-first +methodology of decentralized systems design and refinement +based on a continuous experimentation cycle. Figure 2 +visualizes an updated approach to the traditional research +methodology shown in Figure 1. We argue that in continuous +experimentation, the system’s deployment stage does not +take place after experiments (step 3 in Figure 2). Instead, +we treat deployment as a critical next step in our research +methodology that happens after experiments. Potential find- +ings from deployment studies include discovering new types +of failures that do not occur in a controlled environment +and novel insights or performance issues caused by the +unpredictability of real-world environments. These insights +4See https://release.tribler.org +feed directly into the refinement of the design and exper- +iments in the following two ways (step 6 in Figure 2). +First, we leverage our new insights to update and improve +the decentralized mechanism, similar to how the standard +research methodology uses experimental results for refine- +ment. Second, we use information obtained from deployment +to refine our experiments in a controlled environment. This +can be done, for example, by replaying a workload trace +obtained from the live network during in-house experiments +to evaluate mechanisms under a more realistic workload. A +key focus during deployment is on monitoring the mecha- +nism to detect failure or anomalies. This is further discussed +in Section III-C. +Our methodology does not substitute the need for exper- +iments in controlled environments. On the opposite, data +obtained from a deployment trial, such as network character- +istics, the performance of clients, and user behaviour, should +be used to address the limitations of other experimental +setups, such as simulation and emulation. Therefore, this +data increases the realism of local experiments and helps in +further validating mechanisms before deployment. +B. Motivating Use-Case: Experimental Tokenomics +We now describe how we have applied our deployment- +first methodology during a recent deployment trial. This +trial uses tokenomics to address free-riding behaviour while +downloading content with Tribler. +Mechanism Design and Objectives. A fundamental issue +in peer-to-peer networks is free-riding behaviour, where one +peer takes more resources from the community than it con- +tributes [9]. In Tribler, this manifests as a user downloading +more data from others than contributing back (seeding). +Earlier work established that free-riding behaviour in Tribler +is typical, resulting in fewer uploaders and degradation of +download speed [19]. Since our anonymous downloading +mechanism increases resource usage even further, addressing +free-riding behaviour became an important issue as the +Tribler network grew. +Our solution to free-riding combines three complemen- +tary mechanisms, each designed, evaluated and deployed in +Tribler using our deployment-first methodology (also see +Figure 3). The first mechanism is a lightweight, decen- +tralized ledger named TrustChain, which stores all pair- +wise bandwidth transfers between users in the network in +the form of records [21]. TrustChain is designed explicitly + +1. Accounting +Mechanism +2. Reputation +Mechanism +3. Resource Allocation +Mechanism +Fig. 3. +Three complementary mechanisms we used to address free- +riding behaviour in Tribler [7]. We evaluated each mechanism using our +deployment-first methodology. +for lightweight accounting in decentralized networks and +is highly scalable in the number of participants because it +avoids a global consensus mechanism. Users share these +records with other users using a simple gossiping mech- +anism. Our second mechanism is a reputation mechanism +that, based on received records, computes a trustworthiness +score for other users. The third mechanism is a resource +allocation mechanism that determines for each user to which +other users it will upload data. The combined working +of these mechanisms allows users to identify free-riders +themselves and consequentially refuse them services while +giving honest users preferential treatment. Additional details +and experimental results can be found in our other work [7]. +Applying our Deployment-First Methodology. We de- +ployed each of the three mechanisms and went through +various deployment cycles to improve and fine-tune them. +As a first step, we designed TrustChain, implemented it +and conducted correctness and validation experiments on our +compute cluster (steps 1-3 in Figure 2). We then integrated +TrustChain into the Tribler software, implemented a crawler +to gather created records, and published a new software +release. Due to the lack of real-world traces and insights, +we could not adequately set some parameters, for example, +the interval at which TrustChain records are shared with +other users. Only after a few deployment cycles did we have +insights on setting such parameters. +We +continuously +monitored +the +created +TrustChain +records in the Tribler network, and we were able to detect +various failures and design shortcomings that were not +discovered during our local experiments. For example, our +deployment revealed that our initial design of TrustChain +was falling short because a user can only be engaged in +recording one transaction at the same time. This shortcoming +significantly limited the speed at which records could be cre- +ated and is an essential limitation since the Tribler software +frequently communicates with other users simultaneously. It +bootstrapped a redesign of the format of TrustChain records +with support for concurrent transactions (see [7]). At the +same time, we used the collected TrustChain records to +start designing our reputation mechanism (see [21]). We +also discovered various bugs in the deployment stage, for +example, one bug was related to database corruption that +occasionally occurs on a particular version of Windows. +C. Lessons Learned +We have conducted multiple deployment trials with Tri- +bler. Due to space constraints, we cannot discuss all in- +sights obtained when applying our deployment-first research +methodology. However, we will summarize four lessons we +learned when working on the previously described use case +and our other deployment trials. +Lesson I: Plan for Mechanism Upgrades and Maintain- +ing Backwards Compatibility. Tribler consists of various +mechanisms that we continuously monitor and improve. +Upgrading these mechanisms sometimes required us to +make changes that break compatibility with prior versions. +This compatibility break results in the fragmentation of the +network since users with different versions of a particular +mechanism can no longer communicate with each other. +Additionally, such breaking changes often require software +logic that updates locally stored data (e.g., in a database) to +be compatible with the new mechanism. +We aim to minimize the number of breaking mechanism +changes to avoid too much fragmentation of our network and +to ensure sufficient usage of newly deployed mechanisms. +This aim is also motivated by our observation that users are +relatively slow in updating their Tribler software when a new +release is published, especially if the benefits of the soft- +ware update are unclear.5 During our deployment trials, we +learned that we should plan for mechanism upgrades already +while designing a particular mechanism. We note that this +problem is not exclusive to Tribler since many blockchain +systems occasionally have to upgrade their network protocol +by releasing a new software version or forking the network, +e.g., to fix security issues or improve performance [27]. +Lesson II: The Importance of Monitoring. Contin- +uously monitoring the behaviour of new mechanisms is +critical to detect failures and anomalies in deployment [28]. +We engineer a crawler during every deployment trial and +provision it when a new Tribler release is published. This +crawler joins a particular overlay network as a peer, sends +data queries to other Tribler instances and persists the +retrieved information in a local database. This practice +is comparable to collecting, analyzing and visualizing the +transactions made in blockchain networks. +We have deployed multiple crawlers to gather data from +our live network. For example, alongside the TrustChain +ledger, we also deployed a crawler that collects records cre- +ated by users. However, due to churn, the crawler sometimes +is unable to collect particular data points. Because a user +could have gone offline before the crawler sent a request, +our datasets did not always contain all the data points we +required. Despite this, the data collected during deployment +revealed a large-scale outage due to a software bug since the +number of created TrustChain records dropped significantly. +These experiences taught us that monitoring infrastructure +is crucial to planning a deployment. +Lesson +III: +Document +all +Design +Decisions +and +Changes. +Successfully +applying +our +deployment-first +methodology requires adequate planning and introduces +unique challenges for developers and researchers. In early +deployment trials, we could have documented our design +and deployment decisions better and, therefore, would +have avoided repeating prior mistakes. Over the years, we +5See https://release.tribler.org. + +adopted the open science approach [6] to publicly record all +our source code, design decisions and meeting minutes. We +also carefully report our observations from the deployment +environment and document failures to avoid repeating +particular mistakes in future iterations of a mechanism. This +open science approach is now an essential aspect of our +Tribler development cycle and research methodology. Open +science also helps other researchers understand and replicate +our prior results. It is also beneficial for users interested +in understanding how the Tribler software behaves, what +data is being collected, and what mechanisms are being +executed on their devices. All this information is publicly +available on our GitHub repository.6 +Lesson IV: Do not Deploy Too Much at Once. A +common mistake we made during early deployment trials +was that we tended to include multiple new features or +mechanisms in a single release. Not only did this prolong +the time between software releases, but it also increased +the risk of breaking the Tribler software when there was +a defect in one of the newly-deployed mechanisms. It also +made it impossible to isolate the effects of specific changes. +To avoid these risks, we currently aim to include at most one +new feature per release and aim for short release cycles. For +example, we shipped each of the mechanisms described in +the use case in Section III-B with separate releases with a +few months between them. +We also learned that mechanism design is an incremental +process that requires multiple iterations to grow and become +fruitful. For example, when designing a socio-economic +mechanism, it is often impossible to adequately parameterize +the mechanism since the dynamics of the deployment envi- +ronment are not known apriori by the researcher. Only in +response to data collected from a real-world environment +the mechanism can be made robust and optimized for a +particular application domain. +IV. THE ROAD AHEAD +We have argued that the increasing complexity and de- +pendencies on decentralized systems such as blockchain +applications require more robust and mature experimentation +methodologies. Such methodologies are needed to identify +new types of failures in realistic environments. We argued +that deployment should be explicitly integrated as a key step +in the research methodology to improve the evaluation of +decentralized mechanisms. +We have presented our deployment-first approach that +goes beyond the standard research methodologies. We also +presented Tribler, our research vehicle for deploying decen- +tralized mechanisms. We showed how we use insights from +deployment trials to improve the design of decentralized +mechanisms and their experiments. We have shown that ex- +perimental setups based on deployment provide (1) insights +into new types of failures; and (2) a foundation for the +design of realistic experiments in controlled environments. +6See https://github.com/tribler/tribler/issues. +By describing a tokenomics use case, we demonstrated the +feasibility of our deployment-first approach in practice. +Our deployment-first approach is a continuously evolving +methodology. One possible extension is the addition of +infrastructure and approaches for A/B testing decentralized +mechanisms. This approach would serve different algorithms +and parameters to distinct subsets of users. +REFERENCES +[1] Christian Berger et al. 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Cryptology +ePrint Archive, 2022. + diff --git a/UtE3T4oBgHgl3EQfagrv/content/tmp_files/load_file.txt b/UtE3T4oBgHgl3EQfagrv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3029a00574361fd5247fb33392f847c2dde9cc05 --- /dev/null +++ b/UtE3T4oBgHgl3EQfagrv/content/tmp_files/load_file.txt @@ -0,0 +1,448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf,len=447 +page_content='A Deployment-First Methodology to Mechanism Design and Refinement in Distributed Systems Martijn de Vos, Georgy Ishmaev, Johan Pouwelse, Stefanie Roos Delft University of Technology, The Netherlands Abstract—Catalyzed by the popularity of blockchain technol- ogy, there has recently been a renewed interest in the design, implementation and evaluation of decentralized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Most of these systems are intended to be deployed at scale and in heterogeneous environments with real users and unpredictable workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Nevertheless, most research in this field evaluates such systems in controlled environments that poorly reflect the complex conditions of real-world environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In this work, we argue that deployment is crucial to understanding decentralized mechanisms in a real-world environment and an enabler to building more robust and sustainable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We highlight the merits of deployment by comparing this approach with other experimental setups and show how our lab applied a deployment-first methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We then outline how we use Tribler, our peer-to-peer file-sharing application, to deploy and monitor decentralized mechanisms at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We illustrate the application of our methodology by describing a deployment trial in experimental tokenomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Finally, we summarize four lessons learned from multiple deployment trials where we applied our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Index Terms—Decentralized Systems, Research Methodol- ogy, Experimental Setups, System Failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' INTRODUCTION The scale and complexity of distributed systems have increased tremendously since the field’s inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We live in an era of complex ultra-large-scale networks characterized by the number of participating nodes and by high het- erogeneity, flexibility, non-trivial social dependencies, and emergent properties [3], [11], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Ensuring the proper functioning, deployment, monitoring and maintenance of such systems requires system designers to obtain insights into their performance and correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Given the scale and unpredictability of such systems, obtaining engineering insights presents unique challenges for researchers and de- velopers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Decentralized blockchain applications embody the charac- teristics of complex ultra-large-scale networks [2], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' At the same time, the costs of failures in these applications are very high due to built-in financial mechanisms [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Though realistic experimental setups such as testnets aim to address these challenges to a degree, properly testing and evaluating such systems remains a challenging endeavour [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The mainstream paradigm in distributed systems research is a top-down design that focuses on predicting performance, failures and limitations through experimentation in con- trolled environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' These experiments usually are carried out as simulations or emulations informing researchers’ design choices [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' An empirical study of failures can provide invaluable insights into different types of distributed systems [12], [13], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Detection of events that make a system fail to operate according to its specifications - detection of failures - is often a critical task in distributed systems given complex interdependencies [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' However, empirical experimentation, guided by workloads extracted from a deployed system, is uncommon in academic research [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Mature research methodologies with an em- phasis on deployment are still missing even in empirically- driven fields of distributed systems research, for example, blockchain applications [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We address this gap by presenting our deployment-first methodology for designing and evaluating decentralized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='1 We specifically focus on findings that can be obtained from the study of failures after the deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Our methodology is based on nearly two decades of experience developing the Tribler software [8], [22], [28], serving as infrastructure for deploying and evaluating decentral- ized mechanisms at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We demonstrate that while the deployment-first research methodology can be demanding in terms of time investment, additional insights obtained are worth this time investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A nuanced evaluation of trade-offs between different experimental setups is also instrumental in designing research methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In summary, this work makes the following contributions: 1) We compare different experimental setups and high- light the merits of deployment as a critical step in the research methodology when building and evaluating distributed systems (Section II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 2) We formulate our deployment-first research method- ology to evaluate decentralized mechanisms at scale (Section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We also describe how we use Tribler, our decentralized file-sharing application and research vehicle for conducting large-scale deployment trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 3) We illustrate how we applied our research methodol- ogy to obtain unique insights in a complex use case on tokenomics in decentralized networks (Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 4) Based on our experiences with our methodology, we summarize four lessons we learned (Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' THE STATUS QUO OF DECENTRALIZED SYSTEMS EXPERIMENTS This work argues that deployment experiments are es- sential to build robust decentralized mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We first 1The scope of this paper is primarily limited to decentralized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' However, some of these insights could be relevant to a wider field of distributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='04508v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='DC] 11 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Mechanism Design 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Software Implementation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Experiments (in controlled environment) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Deployment and Monitoring 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' refine Most research ends at this step Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The standard research methodology to design, build, evaluate and deploy decentralized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' describe the standard methodology to research distributed systems and then outline the merits of deployment as part of the research methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Standard Research Methodology Figure 1 shows the standard research methodology to design, build, evaluate, and deploy decentralized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This research methodology is based on reports in the fields’ literature [4], [14], [15], on discussions with other researchers, and our own experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The following five steps describe this methodology: 1) Mechanism Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A researcher starts by designing a particular mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Research with an exclusive focus on understanding theoretical models is usually limited to this step [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 2) Software Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The researcher then works on a software implementation of the designed mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Assuming that the original design was well- executed, the implementation phase should not result in significant changes to the original models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' As such, we left out this feedback loop from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 3) Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' With the implementation, the researcher conducts experiments in an environment controlled by the analyst, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=', on a local computer or a compute cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' These experiments usually aim to verify the implementation’s correctness and quantify system met- rics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=', scalability and fault tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 4) Refining The Design Using Experimental Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Based on the experimental results obtained in the previous step, the researcher updates the mechanism design, updates the accompanying implementation and re-runs experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' For instance, an experiment re- vealing that a particular design has low scalability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=', in the number of participants the system can support) might require the researcher to identify bot- tlenecks in the mechanism and resolve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 5) Deployment and Monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The researcher can deploy the algorithm in a real-world setting after the experiments are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The deployed software will likely be continuously monitored to detect failures or anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Most academic research does not further test and evaluate their mechanisms using deployment and ends at step (4) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This is not unexpected since local experiments usually suffice to prove the mechanism’s trade-offs, correctness or performance to a scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' As such, the time investment and resource costs do not justify the need for deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='2 We argue, however, that trade-offs and limitations associated with the usage of experimental setups to evaluate decentralized systems are more nuanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The Merits of Deployment We start by comparing the trade-offs between different experimental setups used to evaluate decentralized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Based on our literature research, we choose to compare the following four experimental setups: 1) A simulation is a model of an application tested on a model of an environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 2) An emulation is an application that runs in an envi- ronment where some parts of it are modelled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 3) A testnet is an application that is deployed on multiple machines run by researchers or volunteer testers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 4) A real-world deployment is an application that end users run on their machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We compare in Table I eight different properties of these experimental setups and briefly discuss them below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Costs of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' It is not always feasible to rep- resent the costs of experiments in commensurable scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Most often, the costs of experiments can be described by the monetary costs of purchasing necessary computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Both in simulation and emulation, these costs are relatively manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' However, in the case of a deployed system, experimentation costs can be much higher or lower, depending on the application context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Experiments on a deployed system that require changes in system parameters or functionality can cause failures and loss of users or market share [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' However, experiments on a deployed system can have meager costs in some situations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' if volunteer users provide their resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='3 In testnets, if most of the resources are provided by volunteer users, the experiment costs can be low for researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' However, attracting a sufficient amount of users for a testnet can also require some initial investments and upfront costs to get traction, as can be the case with marketing costs for blockchain testnets [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' of experiments is strongly correlated with the costs of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In the case of simulations, it is relatively cheap to scale up the simulation size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In emulation, the upper bound for scalability is typically limited by the available computational resources of a testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In the case 2In contrast to that, the deployment of novel system designs is a common practice in the blockchain industry [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' However, the quality of experimental evaluations is lacking compared to academic research, as industry whitepapers often present biased and inflated results [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 3See https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='com/ethereum/ropsten/blob/master/revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Property ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Emulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Testnet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Real-world Deployment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Cost of experiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low/High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low/High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Scalability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium/High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Resource constrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Resource constrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Resource constrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Environmental realism ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low/Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Very High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Failures discoverability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Impossible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Reproducibility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Speed of change ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Fast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Fast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Slow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Debugability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='TABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='A comparison between four different experimental setups: simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' emulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' testnet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' and real-world deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' of a testnet, the scalability is limited to the computational resources available to the researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In a real-world deploy- ment, the scale of the experiment is usually limited by the number of end-users and the resources they contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Environmental realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This is one of the two key features that set testnets and deployments apart from other experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Environmental realism is comprised of three different parameters: (1) client heterogeneity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=', dif- ferences in hardware capabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' (2) variability in external parameters such as network conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' (3) the effect of user behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A real-world deployment captures all these three parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The key difference with the testnet is the effect of user behaviour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=', a testnet can be exclusive to expert users, reducing realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' User incentives in testnets are also sometimes simplified, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=', in blockchain testnets, there usually are no financial incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Both in simulation and emulation, all three parameters of environmental realism are lower compared to other experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' External parameters such as network conditions can be more realistic with emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Realistic client heterogeneity is challenging to represent realistically in an emulation conducted with het- erogeneous hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' One observation is that environmental realism can be improved for simulation and emulation if these setups are designed with values known from mea- surements of deployed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Two key factors limit such measurements: first, the measured system should have a very similar application context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Second, it is not certain if the results of the measurements still hold as the system evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' There has been some work that aims to bring environmen- tal realism to simulation or emulation setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Sarzyniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' present Distem, a virtualisation platform to enable resource heterogeneity in a homogeneous compute cluster [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' While this is a step to make experiments more accurate, the failure model and user-generated workloads are not carried over from the real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Recent work on scalability experiments with BFT consensus protocols proposes a simu- lator [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Understanding the realism gap between simulators and real-world environments is a key part of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Discoverability of new failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This is another key property that distinguishes controlled and uncontrolled ex- periment setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Certain types of failures can only be dis- covered in a real-world environment, particularly emergent and user-caused failures [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Testnets can reveal certain types of emergent and partial failures [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Relatively fewer novel types of failures can be revealed by experiments using emulation setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In principle, simulations do not allow for discovering new types of failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This parameter is tied to the availability of the same setup to different researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Simulation, at least in theory, allows for the highest level of reproducibil- ity, given that all artefacts can easily be published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' With emulations, the evaluated software can be made available, but access to an identical experimental testbed is not always available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' It could be argued that while reproducibility is low for both testnets and deployed systems, a testnet allows for a relatively easier replication of experimental conditions, which is almost impossible with deployed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Simulation and emulation are the experimental setups that give researchers the most control over the flow of their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' With testnets and real-world deployed systems, researchers usually have little to no control over the system while it is running since there is a dependency on the volunteers or end users that are running the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Speed of change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The speed at which an experiment can be modified is a distinguishing factor between different experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This speed of change is usually the lowest in a deployed real-world system, given the delays caused by the propagation of software updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Testnets can allow for somewhat quicker deployment cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In both cases, deployment and the collection of results can be time- consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Simulation and emulation setups have relatively low external constraints and quickly be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Debugability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Discovering, analyzing and reproducing software bugs in deployed systems require dedicated infras- tructure and can be a time-consuming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In compari- son, testnets and emulation provide more debugability since the researcher has a higher level of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Simulation can provide the highest discoverability of bugs as long as the scale of the simulation is not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This analysis shows that we can not have a com- prehensive evaluation without deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We need to account for environmental realism and failures, which are detectable only in a real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Mechanism Design 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Software Implementation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Experiments (in controlled environment) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Deployment and Monitoring 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' refine 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' refine 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' refine Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Our deployment-first research methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The key difference with Figure 1 is that we directly apply new insights or real-world workload traces to the mechanism design and experiments, respectively (step 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' TRIBLER: DEPLOYING AND MONITORING DECENTRALIZED MECHANISMS AT SCALE Tribler is our lab’s peer-to-peer file-sharing software and research vehicle to deploy and evaluate decentralized mech- anisms at scale [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We have used the Tribler software for al- most two decades to obtain unique insights into the complex interactions and dynamics in live peer-to-peer networks [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Over 30 PhD researchers and BSc/MSc students have used our software to evaluate their mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Tribler also has a stable user base that enables longitudinal deployment experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='8 million users have downloaded the Tribler software, and at the time of writing, Tribler has 40’000 unique monthly users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='4 Tribler was initially designed as a file-sharing application that allows users to download torrent files anonymously using a custom onion-routing protocol [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Tribler uses the IPv8 networking library that supports authenticated messaging and enables the construction and maintenance of decentralized overlays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Over the years, however, Tribler has evolved from a BitTorrent download client to a versatile application with features such as keyword search, bundling torrents into channels, and reputation mechanisms to address free-riding behaviour [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Deployment-First Research Methodology Tribler is a vital part of our labs’ research methodology since it enables us to deploy and evaluate decentralized mechanisms at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Tribler also allowed us to try out a different research methodology with an increased focus on deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Originating from our experiences with Tribler and deployment efforts, we now present our deployment-first methodology of decentralized systems design and refinement based on a continuous experimentation cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Figure 2 visualizes an updated approach to the traditional research methodology shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We argue that in continuous experimentation, the system’s deployment stage does not take place after experiments (step 3 in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Instead, we treat deployment as a critical next step in our research methodology that happens after experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Potential find- ings from deployment studies include discovering new types of failures that do not occur in a controlled environment and novel insights or performance issues caused by the unpredictability of real-world environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' These insights 4See https://release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='tribler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='org feed directly into the refinement of the design and exper- iments in the following two ways (step 6 in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' First, we leverage our new insights to update and improve the decentralized mechanism, similar to how the standard research methodology uses experimental results for refine- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Second, we use information obtained from deployment to refine our experiments in a controlled environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This can be done, for example, by replaying a workload trace obtained from the live network during in-house experiments to evaluate mechanisms under a more realistic workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A key focus during deployment is on monitoring the mecha- nism to detect failure or anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This is further discussed in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Our methodology does not substitute the need for exper- iments in controlled environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' On the opposite, data obtained from a deployment trial, such as network character- istics, the performance of clients, and user behaviour, should be used to address the limitations of other experimental setups, such as simulation and emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Therefore, this data increases the realism of local experiments and helps in further validating mechanisms before deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Motivating Use-Case: Experimental Tokenomics We now describe how we have applied our deployment- first methodology during a recent deployment trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This trial uses tokenomics to address free-riding behaviour while downloading content with Tribler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Mechanism Design and Objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A fundamental issue in peer-to-peer networks is free-riding behaviour, where one peer takes more resources from the community than it con- tributes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In Tribler, this manifests as a user downloading more data from others than contributing back (seeding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Earlier work established that free-riding behaviour in Tribler is typical, resulting in fewer uploaders and degradation of download speed [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Since our anonymous downloading mechanism increases resource usage even further, addressing free-riding behaviour became an important issue as the Tribler network grew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Our solution to free-riding combines three complemen- tary mechanisms, each designed, evaluated and deployed in Tribler using our deployment-first methodology (also see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The first mechanism is a lightweight, decen- tralized ledger named TrustChain, which stores all pair- wise bandwidth transfers between users in the network in the form of records [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' TrustChain is designed explicitly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Accounting Mechanism 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Reputation Mechanism 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Resource Allocation Mechanism Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Three complementary mechanisms we used to address free- riding behaviour in Tribler [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We evaluated each mechanism using our deployment-first methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' for lightweight accounting in decentralized networks and is highly scalable in the number of participants because it avoids a global consensus mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Users share these records with other users using a simple gossiping mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Our second mechanism is a reputation mechanism that, based on received records, computes a trustworthiness score for other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The third mechanism is a resource allocation mechanism that determines for each user to which other users it will upload data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' The combined working of these mechanisms allows users to identify free-riders themselves and consequentially refuse them services while giving honest users preferential treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Additional details and experimental results can be found in our other work [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Applying our Deployment-First Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We de- ployed each of the three mechanisms and went through various deployment cycles to improve and fine-tune them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' As a first step, we designed TrustChain, implemented it and conducted correctness and validation experiments on our compute cluster (steps 1-3 in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We then integrated TrustChain into the Tribler software, implemented a crawler to gather created records, and published a new software release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Due to the lack of real-world traces and insights, we could not adequately set some parameters, for example, the interval at which TrustChain records are shared with other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Only after a few deployment cycles did we have insights on setting such parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We continuously monitored the created TrustChain records in the Tribler network, and we were able to detect various failures and design shortcomings that were not discovered during our local experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' For example, our deployment revealed that our initial design of TrustChain was falling short because a user can only be engaged in recording one transaction at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This shortcoming significantly limited the speed at which records could be cre- ated and is an essential limitation since the Tribler software frequently communicates with other users simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' It bootstrapped a redesign of the format of TrustChain records with support for concurrent transactions (see [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' At the same time, we used the collected TrustChain records to start designing our reputation mechanism (see [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We also discovered various bugs in the deployment stage, for example, one bug was related to database corruption that occasionally occurs on a particular version of Windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Lessons Learned We have conducted multiple deployment trials with Tri- bler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Due to space constraints, we cannot discuss all in- sights obtained when applying our deployment-first research methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' However, we will summarize four lessons we learned when working on the previously described use case and our other deployment trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Lesson I: Plan for Mechanism Upgrades and Maintain- ing Backwards Compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Tribler consists of various mechanisms that we continuously monitor and improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Upgrading these mechanisms sometimes required us to make changes that break compatibility with prior versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This compatibility break results in the fragmentation of the network since users with different versions of a particular mechanism can no longer communicate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Additionally, such breaking changes often require software logic that updates locally stored data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=', in a database) to be compatible with the new mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We aim to minimize the number of breaking mechanism changes to avoid too much fragmentation of our network and to ensure sufficient usage of newly deployed mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This aim is also motivated by our observation that users are relatively slow in updating their Tribler software when a new release is published, especially if the benefits of the soft- ware update are unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='5 During our deployment trials, we learned that we should plan for mechanism upgrades already while designing a particular mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We note that this problem is not exclusive to Tribler since many blockchain systems occasionally have to upgrade their network protocol by releasing a new software version or forking the network, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=', to fix security issues or improve performance [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Lesson II: The Importance of Monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Contin- uously monitoring the behaviour of new mechanisms is critical to detect failures and anomalies in deployment [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We engineer a crawler during every deployment trial and provision it when a new Tribler release is published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This crawler joins a particular overlay network as a peer, sends data queries to other Tribler instances and persists the retrieved information in a local database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This practice is comparable to collecting, analyzing and visualizing the transactions made in blockchain networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We have deployed multiple crawlers to gather data from our live network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' For example, alongside the TrustChain ledger, we also deployed a crawler that collects records cre- ated by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' However, due to churn, the crawler sometimes is unable to collect particular data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Because a user could have gone offline before the crawler sent a request, our datasets did not always contain all the data points we required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Despite this, the data collected during deployment revealed a large-scale outage due to a software bug since the number of created TrustChain records dropped significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' These experiences taught us that monitoring infrastructure is crucial to planning a deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Lesson III: Document all Design Decisions and Changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Successfully applying our deployment-first methodology requires adequate planning and introduces unique challenges for developers and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In early deployment trials, we could have documented our design and deployment decisions better and, therefore, would have avoided repeating prior mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Over the years, we 5See https://release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='tribler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' adopted the open science approach [6] to publicly record all our source code, design decisions and meeting minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We also carefully report our observations from the deployment environment and document failures to avoid repeating particular mistakes in future iterations of a mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This open science approach is now an essential aspect of our Tribler development cycle and research methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Open science also helps other researchers understand and replicate our prior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' It is also beneficial for users interested in understanding how the Tribler software behaves, what data is being collected, and what mechanisms are being executed on their devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' All this information is publicly available on our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='6 Lesson IV: Do not Deploy Too Much at Once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A common mistake we made during early deployment trials was that we tended to include multiple new features or mechanisms in a single release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Not only did this prolong the time between software releases, but it also increased the risk of breaking the Tribler software when there was a defect in one of the newly-deployed mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' It also made it impossible to isolate the effects of specific changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' To avoid these risks, we currently aim to include at most one new feature per release and aim for short release cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' For example, we shipped each of the mechanisms described in the use case in Section III-B with separate releases with a few months between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We also learned that mechanism design is an incremental process that requires multiple iterations to grow and become fruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' For example, when designing a socio-economic mechanism, it is often impossible to adequately parameterize the mechanism since the dynamics of the deployment envi- ronment are not known apriori by the researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Only in response to data collected from a real-world environment the mechanism can be made robust and optimized for a particular application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' THE ROAD AHEAD We have argued that the increasing complexity and de- pendencies on decentralized systems such as blockchain applications require more robust and mature experimentation methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Such methodologies are needed to identify new types of failures in realistic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We argued that deployment should be explicitly integrated as a key step in the research methodology to improve the evaluation of decentralized mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We have presented our deployment-first approach that goes beyond the standard research methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We also presented Tribler, our research vehicle for deploying decen- tralized mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We showed how we use insights from deployment trials to improve the design of decentralized mechanisms and their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' We have shown that ex- perimental setups based on deployment provide (1) insights into new types of failures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' and (2) a foundation for the design of realistic experiments in controlled environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' 6See https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content='com/tribler/tribler/issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' By describing a tokenomics use case, we demonstrated the feasibility of our deployment-first approach in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Our deployment-first approach is a continuously evolving methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' One possible extension is the addition of infrastructure and approaches for A/B testing decentralized mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' This approach would serve different algorithms and parameters to distinct subsets of users.' metadata={'source': 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and scalability of blockchain systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In DAPPS, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [21] Pim Otte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Trustchain: A sybil-resistant scalable blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' FGCS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [22] Johan Pouwelse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Tribler: a social-based peer-to-peer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Concurrency and computation: Practice and experience, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [23] Luc Sarzyniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Design and evaluation of a virtual experimental environment for distributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In PDP, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [24] Claudio J Tessone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Stochastic modelling of blockchain consen- sus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' arXiv, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [25] Alessandro Vespignani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Predicting the Behavior of Techno-Social Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Science, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [26] Guosai Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' What can we learn from four years of data center hardware failures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In DSN, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [27] Alexei Zamyatin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' A wild velvet fork appears!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' inclusive blockchain protocol changes in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In Financial Cryptography, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [28] Niels Zeilemaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Tribler: P2p media search and sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' In ACM Multimedia, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' [29] Liyi Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Sok: Decentralized finance (defi) attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} +page_content=' Cryptology ePrint Archive, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQfagrv/content/2301.04508v1.pdf'} diff --git a/V9E3T4oBgHgl3EQf0gvl/content/tmp_files/2301.04739v1.pdf.txt b/V9E3T4oBgHgl3EQf0gvl/content/tmp_files/2301.04739v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2116f79d2c4f3e6f0fb932524715992d56ccb7e5 --- /dev/null +++ b/V9E3T4oBgHgl3EQf0gvl/content/tmp_files/2301.04739v1.pdf.txt @@ -0,0 +1,666 @@ +Making sense of noise: introducing students to stochastic processes in +order to better understand biological behaviors +Michael W. Klymkowsky +Molecular, Cellular & Developmental Biology, University of Colorado Boulder, Boulder CO. +80309. (klym@colorado.edu : ORCID: 0000-0001-5816-9771) +Abstract: Biological systems are +characterized by the ubiquitous roles of +weak, that is, non-covalent molecular +interactions, small, often very small, numbers +of specific molecules per cell, and Brownian +motion. These combine to produce stochastic +behaviors at all levels from the molecular and +cellular to the behavioral. That said, students +are rarely introduced to the ubiquitous role of +stochastic processes in biological systems, +and how they produce unpredictable +behaviors. Here I present the case that they +need to be and provide some suggestions as +to how it might be approached. +Background: Three recent events combined to spur this reflection on stochasticity in +biological systems, how it is taught, and why it matters. The first was an article describing an +approach to introducing students to homeostatic processes in the context of the bacterial lac +operon (Booth et al., 2022), an adaptive gene regulatory system controlled in part by +stochastic events. The second was in-class student responses to the question, why do +interacting molecules “come back apart” (dissociate). Finally, there is the increasing attention +paid to what are presented as deterministic genetic factors, as illustrated by Kathryn Harden's +“The Genetic Lottery: Why DNA matters for social equality” (Harden, 2021). Previous work +suggests that students, and perhaps some instructors (see Garvin-Doxas and Klymkowsky, +2008; Klymkowsky et al., 2016; 2010), find the ubiquity, functional roles, and implications of +stochastic, that is inherently unpredictable processes, difficult to recognize and apply. Given +their practical and philosophical implications, it seems essential to introduce students to +stochasticity early in their educational journeys.   +What is stochasticity and why is it important for understanding biological systems? +Stochasticity results when intrinsically unpredictable events, e.g. molecular collisions, impact +the behavior of a system. There are a number of drivers of stochastic behaviors. Perhaps the +most obvious, and certainly the most ubiquitous in biological systems is thermal motion. The +Making sense of noise +Wednesday, January 11, 2023 + of +1 +13 +Image generated using the Java Genetic Drift applet +(http://darwin.eeb.uconn.edu/simulations/jdk1.0/ +drift.html) in 2014 - the applet no longer appears to be +available + +population size = 50, 100 generations +0 +100 +Generationsmany molecules within a solution (or a cell) are moving, they have kinetic energy – the energy +of motion and mass. The exact momentum of each molecule cannot, however, be accurately +and completely characterized without perturbing the system (echos of Heisenberg). Given the +impossibility of completely characterizing the system, we are left uncertain as to the state of +the system’s components, who is bound to whom, going forward.  +Through collisions energy is exchanged between molecules.  A number of chemical +processes are driven by the energy delivered through such collisions. Think about a typical +chemical reaction. In the course of the reaction, atoms are rearranged – bonds are broken (a +process that requires energy) and bonds are formed (a process that releases energy). Many +(most) of the chemical reactions that occur in biological systems require catalysts to bring their +required activation energies into the range available within the cell.   +1 +What makes the impact of thermal motion even more critical for biological systems is +that many (most) regulatory interactions and macromolecular complexes, the molecular +machines discussed by Alberts (1998) are based on relatively weak, non-covalent surface- +surface interactions between or within molecules. Such interactions are central to most +regulatory processes, from the activation of signaling pathways to the control of gene +expression. The specificity and stability of these non-covalent interactions, which include those +involved in determining the three-dimensional structure of macromolecules, are directly +impacted by thermal motion, and so by temperature – one reason controlling body temperature +is important.  +So why are these interactions stochastic and why does it matter?  A signature property +of a stochastic process is that while it may be predictable when large numbers of atoms, +molecules, or interactions are involved, the behaviors of individual atoms, molecules, and +interactions are not. A classic example, arising from factors intrinsic to the atom, is the decay of +radioactive isotopes. While the half-life of a large enough population of a radioactive isotope is +well defined, when any particular atom will decay is, in current theory, unknowable, a concept +difficult for students (see Hull and Hopf, 2020). This is the reason we cannot accurately predict +whether Schrȍdinger’s cat is alive or dead. The same behavior applies to the binding of a +2 +regulatory protein to a specific site on a DNA molecule and its subsequent dissociation: +predictable in large populations, not-predictable for individual molecules. The situation is +exacerbated by the fact that biological systems are composed of cells and cells are, typically, +small, and so contain relatively few molecules of each type (Milo and Phillips, 2015). There are +typically one or two copies of each gene in a cell, and these may be different from one another +(when heterozygous). The expression of any one gene depends upon the binding of specific +proteins, transcription factors, that act to activate or repress gene expression. In contrast to a +number of other cellular proteins, “as a rule of thumb, the concentrations of such transcription + For this discussion we ignore entropy, a factor that figures in whether a particular reaction in favorable or +1 +unfavorable, that is whether, and the extent to which it occurs. + Four common misconceptions about quantum physics [link] +2 +Making sense of noise +Wednesday, January 11, 2023 + of +2 +13 + +factors are in the nM range, corresponding to only 1-1000 copies per cell in bacteria or 103-106 +in mammalian cells” (Milo and Phillips, 2015). Moreover, while DNA binding proteins bind to +specific DNA sequences with high affinity, they also bind to DNA “non-specifically” in a largely +sequence independent manner with low affinity. Given that there are many more non-specific +(non-functional) binding sites in the DNA than functional ones, the effective concentration of a +particular transcription factor can be significantly lower than its total cellular concentration +would suggest. For example, in the case of the lac repressor of the bacterium Escherichia coli +(discussed further below), there are estimated to be ~10 molecules of the tetrameric lac +repressor per cell, but “non-specific affinity to the DNA causes >90% of LacI copies to be +bound to the DNA at locations that are not the cognate promoter site” (Milo and Phillips, 2015); +at most only a few molecules are free in the cytoplasm and available to bind to specific +regulatory sites.  Such low affinity binding to DNA allows proteins to undergo one-dimensional +diffusion, a process that can greatly speed up the time it takes for a DNA binding protein to +“find” high affinity binding sites (Stanford et al., 2000; von Hippel and Berg, 1989). Most +transcription factors bind in a functionally significant manner to hundreds to thousands of gene +regulatory sites per cell, often with distinct binding affinities. The effective binding affinity can +also be influenced by positive and negative interactions with other transcription and accessory +factors, chromatin structure, and DNA modifications. Functional complexes can take time to +assemble, and once assembled can initiate multiple rounds of polymerase binding and +activation, leading to a stochastic phenomena known as transcriptional bursting. An analogous +process occurs with RNA-dependent polypeptide synthesis (translation). The result, +particularly for genes expressed at lower levels, is that stochastic (unpredictable) bursts of +transcription/translation can lead to functionally significant changes in protein levels (Raj et al., +2010; Raj and van Oudenaarden, 2008). +There are many examples of stochastic +behaviors in biological systems. Originally noted by +Novick and Weiner (1957) in their studies of the lac +operon, it was clear that gene expression occurred +in an all or none manner. This effect was revealed +in a particularly compelling study by Elowitz et al +(2002) who used lac operon promoter elements to +drive expression of transgenes encoding cyan and +yellow fluorescent proteins (on a single plasmid) in +E. coli (→).  The observed behaviors were +dramatic; genetically identical cells were found to +express, stochastically, one, the other, both, or +neither transgene. The stochastic expression of +genes and downstream effects appear to be the +source of much of the variance found in organisms +Making sense of noise +Wednesday, January 11, 2023 + of +3 +13 +Figure adapted from Elowitz et al 2002 + +oriC +1.51Mb +1.46 Mb +E.Coli +intC<>YFP +galK<>CFP +4.6Mbwith the same genotype in the same environmental conditions (Honegger and de Bivort, 2018). +Beyond gene expression, the unpredictable effects of stochastic processes can be seen +at all levels of biological organization, from the biased random walk behaviors that underlie +various forms of chemotaxis (e.g. Spudich and Koshland, 1976) and the search behaviors in C. +elegans (Roberts et al., 2016) and other animals (Smouse et al., 2010), the noisiness in the +opening of individual neuronal voltage-gated ion channels (Braun, 2021; Neher and Sakmann, +1976), and various processes within the immune system (Hodgkin et al., 2014), to variations in +the behavior of individual organisms (e.g. the leafhopper example cited by Honegger and de +Bivort, 2018). Stochastic events are involved in a range of “social” processes in bacteria +(Bassler and Losick, 2006). Their impact serves as a form of “bet-hedging” in populations that +generate phenotypic variation in a homogeneous environment (see Symmons and Raj, 2016). +Stochastic events can regulate the efficiency of replication-associated error-prone mutation +repair (Uphoff et al., 2016) leading to increased variation in a population, particularly in +response to environmental stresses. Stochastic “choices” made by cells can be seen as +questions asked of the environment, the system’s response provides information that informs +subsequent regulatory decisions (see Lyon, 2015) and the selective pressures on individuals in +a population (Jablonka and Lamb, 2005). Together stochastic processes introduce a non- +deterministic (i.e. unpredictable) element into higher order behaviors (Murakami et al., 2017; +Roberts et al., 2016). +Controlling stochasticity: While stochasticity can be useful, it also needs to be controlled. +Not surprisingly then there are a number of strategies for “noise-suppression”, ranging from +altering regulatory factor concentrations, the formation of covalent disulfide bonds between or +within polypeptides, and regulating the activity of repair systems associated with DNA +replication, polypeptide folding, and protein assembly via molecular chaperones and targeted +degradation. For example, the identification of “cellular competition” effects has revealed that +“eccentric cells” (sometimes, and perhaps unfortunately referred to as of “losers”) can be +induced to undergo apoptosis (die) or migration in response to their “normal” neighbors +(Akieda et al., 2019; Di Gregorio et al., 2016; Ellis et al., 2019; Hashimoto and Sasaki, 2020; +Lima et al., 2021). +Student understanding of stochastic processes: There is ample evidence that students +(and perhaps some instructors as well) are confused by or uncertain about the role of thermal +motion, that is the transfer of kinetic energy via collisions, and the resulting stochastic +behaviors in biological systems. As an example, Champagne-Queloz et al (2016; 2017) found +that few students, even after instruction through molecular biology courses, recognize that +collisions with other molecules were  responsible for the disassembly of molecular complexes. +In fact, many adopt a more “deterministic” model for molecular disassembly after instruction +(see part A panel figure on next page). In earlier studies, we found evidence for a similar +confusion among instructors (part B of figure on the next page)(Klymkowsky et al., 2010).  +Making sense of noise +Wednesday, January 11, 2023 + of +4 +13 + +Introducing stochasticity to students: Given that understanding stochastic (random) +processes can be difficult for many (e.g. Garvin-Doxas and Klymkowsky, 2008; Taleb, 2005), +the question facing course designers and instructors is when and how best to help students +develop an appreciation for the ubiquity, specific roles, and implications of stochasticity- +dependent processes at all levels in biological systems. I would suggest that  introducing +students to the dynamics of non-covalent molecular interactions, prevalent in biological +systems in the context of stochastic interactions (i.e. kinetic theory) rather than a ∆G-based +approach may be useful. We can use the probability of garnering the energy needed to disrupt +an interaction to present concepts of binding specificity (selectivity) and stability. Developing an +understanding of the formation and  +disassembly of molecular interactions +builds on the same logic that Albert +Einstein and Ludwig Böltzman used to +demonstrate the existence of atoms and +molecules and the reversibility of +molecular reactions (Bernstein, 2006). +Moreover, as noted by Samoilov et al (2006) "stochastic mechanisms open novel classes of +regulatory, signaling, and organizational choices that can serve as efficient and effective +biological solutions to problems that are more complex, less robust, or otherwise suboptimal to +deal with in the context of purely deterministic systems." +The selectivity (specificity) and stability of molecular interactions can be understood +from an energetic perspective – comparing the enthalpic and entropic differences between +bound and unbound states. What is often missing from such discussions, aside from the fact of +Making sense of noise +Wednesday, January 11, 2023 + of +5 +13 + +B +90 +Once two molecule bind to one another, how could they +come back apart again? +introductory courses +Theywould havetobind to +yetanothermolecule- + correct responses +Thecomplexwillneedto +bedegraded +68 +Collionswithothermolecules +couldknockthemapart +(correct) +advanced courses +45 +A chemical reaction must +changethestructure +teachers +ofoneofthemolecules +(mostpopular) +23 +noanswe +% +A pre- and post-instruction analysis indicates that "Random +molecular collisions are not recognized as the major source +of breaking molecular interactions. The best answer (2) +0 +reflects the fact that molecules interacting and dissociated +dissociation question +from one another in response to the transfer of energy, +Comparison of responses by different grouls +typically collisions with other molecules, sufficient to +to this questions (BCl Q18)(modified from +overcome their interaction energy." (from +Kiymkowsky et al., 2010). +Champagne-Queloz 2016 use with permission)."What made the greatest impression upon the student, +however, was less the technical construction of mechanics +or the solution of complicated problems than the +achievement of mechanics in areas which apparently had +nothing to do with mechanics ... above all the kinetic theory +of gases:" - A. Einstein (Bernstein)their inherent complexity, particularly in terms of calculating changes in entropy and exactly +what is meant by energy (Cooper and Klymkowsky, 2013) is that many students enter biology +classes without a robust understanding of enthalpy, entropy, or free energy (Carson and +Watson, 2002).  Presenting students with a molecular  collision, kinetic theory-based +mechanism for the dissociation of molecular interactions, may help them better understand +(and apply) both the dynamics and specificity of molecular interactions. We can gage the +strength of an interaction (the sum of the forces stabilizing an interaction) based on the amount +of energy (derived from collisions with other molecules) needed to disrupt it.  The implication of +student responses to relevant Biology Concepts Instrument (BCI) questions and beSocratic +activities (data not shown), as well as a number of studies in chemistry, is that few students +consider the kinetic/vibrational energy delivered through collisions with other molecules (a +function of temperature), as key to explaining why interactions break (see Carson and Watson, +2002 and references therein).  Although this +paper is 20 years old, there is little or no +evidence that the situation has improved. +Moreover, there is evidence that the +conventional focus on mathematics-centered, +free energy calculations in the absence of +conceptual understanding may serve as an +unnecessary barrier to the inclusion of a more socioeconomically diverse, and under-served +populations of students (Ralph et al., 2022; Stowe and Cooper, 2019). +The lac operon as a context for introducing stochasticity: Studies of the E. coli  lac operon +hold an iconic place in the history of molecular biology and are often found in introductory +courses, although typically presented in a deterministic context. The mutational analysis of the +lac operon helped define key elements involved in gene regulation (Jacob and Monod, 1961; +Monod et al., 1963). Booth et al (2022) used the lac operon as the context for their “modeling +and simulation lesson”, Advanced Concepts in Regulation of the Lac Operon. Given its +inherently stochastic regulation (Choi et al., 2008; Elowitz et al., 2002; Novick and Weiner, +1957; Vilar et al., 2003), the lac operon is a good place to start introducing students to +stochastic processes. In this light, it is worth noting that Booth et al describes the behavior of +the lac operon as “leaky”, which would seem to imply a low, but continuous level of expression, +much as a leaky faucet continues to drip. As this is a peer-reviewed lesson, it seems likely that +it reflects widely held mis-understandings of how stochastic processes are introduced to, and +understood by students and instructors. +E. coli cells respond to the presence of lactose in growth media in a biphasic manner, +termed diauxie, due to “the inhibitory action of certain sugars, such as glucose, on adaptive +enzymes (meaning an enzyme that appears only in the presence of its substrate)” (Blaiseau +and Holmes, 2021). When these (preferred) sugars are depleted from the media, growth +Making sense of noise +Wednesday, January 11, 2023 + of +6 +13 + +" The essence of thermodynamics, however, is a study of +interactions. Terms like entropy and Gibbs free energy +cannot be understood as isolated entities that can be +transformed into one another. If students are to use these +concepts to make predictions about whether reactions can +occur, they need to understand them in the context of +chemical' processes'" - Carson & Watson, 2002slows. If lactose is present, however, growth will resume following a delay associated with the +expression of the proteins encoded by the operon that enables the cell to import and +metabolize lactose. Although the term homeostatic is used repeatedly by Booth et al, the lac +operon is part of an adaptive, rather than a homeostatic, system. In the absence of glucose, +cyclic AMP (cAMP) levels in the cell rise. cAMP binds to and activates the catabolite activator +protein (CAP), encoded for by the crp gene. Activation of CAP leads to the altered expression +of a number of target genes, whose products are involved in adaption to the stress associated +with the absence of common and preferred metabolites. cAMP-activated CAP acts as both a +transcriptional repressor and activator, “and has been shown to regulate hundreds of genes in +the E. coli genome, earning it the status of “global” or “master” regulator” (Frendorf et al., +2019). It is involved in the adaptation to environmental factors, rather than maintaining the cell +in a particular state (homeostasis).  +The lac operon is a classic polycistronic bacterial gene, encoding three distinct +polypeptides: lacZ (β-galactosidase), lacY (β-galactoside permease), and lacA (galactoside +acetyltransferase). When glucose or other preferred energy sources are present, expression of +the lac operon is blocked by the inactivity of CAP. The CAP protein is a homodimer and its +binding to DNA is regulated by the binding of the allosteric effector cAMP.  cAMP is generated +from ATP by the enzyme adenylate cyclase, encoded by the cya gene. In the absence of +glucose the enyzme encoded by the crr gene is phosphorylated and acts to activate adenylate +cyclase (Krin et al., 2002).  As cAMP levels increase, cAMP binds to the CAP protein, leading +to a dramatic change in its structure (↑), such that the protein’s  DNA binding domain becomes +available to interact with promoter sequences (figure from Sharma et al., 2009).  +Binding of activated (cAMP-bound) CAP is not, +by itself sufficient to activate expression of the lac +operon because of the presence of the constitutively +expressed lac repressor protein, encoded for by the +lacI gene. The active repressor is a tetramer, present +at very low levels (~10 molecules) per cell. The lac +operon contains three repressor (“operator”) binding +sites; the tetrameric repressor can bind two operator +sites simultaneously (figure → from Palanthandalam- +Madapusi and Goyal, 2011). In the absence of +Making sense of noise +Wednesday, January 11, 2023 + of +7 +13 + +DNA binding helices +cAMP +DNA +boundto +binding +interaction +DNA +recruit"RNA +polymerase + +synthesize +mRNA +Sharma et al., 2009 (frames from movie S1)Looped non-coding +DNA +Genes +Lacz +LacY +LacA +Lactose- +RepressorProtein +fromPalanthandalam-Madapusi&Goyal,2011lactose, but in the presence of cAMP-activated CAP, the operon is expressed in discrete +“bursts” (Novick and Weiner, 1957; Vilar et al., 2003). Choi et al (2008) found that these burst +come in two types, short and long, with the size of the burst referring to the number of mRNA +molecules synthesized (figure adapted from Choi et al ↓). The difference between burst sizes +arises from the length of time that the +operon’s repressor binding sites are +unoccupied by repressor. As noted +above, the tetravalent repressor +protein can bind to two operator sites +at the same time. When released from +one site, polymerase binding and +initiation produces a small number of +mRNA molecules. Persistent binding +to the second site means that the +repressor concentration remains locally high, favoring rapid rebinding to the operator and the +cessation of transcription (RNA synthesis). When the repressor releases from both operator +sites, a rarer event, it is free to diffuse away and interact (non-specifically, i.e. with low affinity) +with other DNA sites in the cell, leaving the lac operator sites unoccupied for a longer period of +time. The number of such non-specific binding sites greatly exceeds the number (three) of +specific binding sites in the operon. The result is the synthesis of a larger “burst” (number) of +mRNA molecules. The average length of time that the operator sites remain unoccupied is a +function of the small number of repressor molecules present and the repressor’s low but +measurable non-sequence specific binding to DNA.  +The expression of the lac operon leads to the appearance of β-galactosidase and β- +galactoside permease. An integral membrane protein, β-galactoside permease enables +extracellular lactose to enter the cell while cytoplasmic β-galactosidase catalyzes its +breakdown and the generation of allolactone, which binds to the lac repressor protein, +inhibiting its binding to operator sites, and so removing repression of transcription. In the +absence of lactose, there are few if any of the proteins (β-galactosidase and β-galactoside +permease) needed to activate the expression of the lac operon, so the obvious question is +how, when lactose does appear in the extracellular media, does the lac operon turn on? Booth +et al and the Wikipedia entry on the lac operon (accessed 29 June 2022) describe the turn on +of the lac operon as “leaky” (see above). The molecular modeling studies of Vilar et al and +Choi et al (which, together with Novick and Weiner, are not cited by Booth et al) indicate that +the system displays distinct threshold and maintenance concentrations of lactose needed for +stable lac gene expression. The term “threshold” does not occur in the Booth et al article. More +importantly, when cultures are examined at the single cell level, what is observed is not a +uniform increase in lac expression in all cells, as might be expected in the context of leaky +expression, but more sporadic (noisy) behaviors. Increasing numbers of cells are “full on” in +terms of lac operon expression over time when cultured in lactose concentrations above the +Making sense of noise +Wednesday, January 11, 2023 + of +8 +13 + +口 +口 +Lowintracellularinducerconcentration +Inducer +Lacl +Laco site +DNA +mRNA +Smallburst +Largeburst +口 +Time (min) +0 +10 +20 +30 +40 +50 +Atime-lapsesequenceinthepresenceof5omMTMG,onesuchcell switchesphenotypeoperon’s activation threshold. This illustrates the distinctly different implications of a leaky +versus a stochastic process in terms of their impacts on gene expression. While a leak is a +macroscopic metaphor that produces a +continuous, dependable, regular flow (drips), +the occurrence of “bursts” of gene expression +implies a stochastic (unpredictable) process +(← figure from Vilar et al).  +As the ubiquity and functionally +significant roles of stochastic processes in +biological systems becomes increasingly +apparent, e.g. in the prediction of phenotypes +from genotypes (Karavani et al., 2019; +Mostafavi et al., 2020), helping students +appreciate and understand the un-predictable, +that is stochastic, aspects of biological +systems becomes increasingly important. As an example, revealed dramatically through the +application of single cell RNA sequencing studies, variations in gene expression between cells +of the same "type" impacts organismic development and a range of behaviors. For example, in +diploid eukaryotic cells is now apparent that in many cells, and for many genes, only one of the +two alleles present is expressed; such “monoallelic” expression can impact a range of +processes (Gendrel et al., 2014). Given that stochastic processes are often not well conveyed +through conventional chemistry courses (Williams et al., 2015) or effectively integrated into, +and built upon in molecular (and other) biology curricula; presenting them explicitly in +introductory biology courses seems necessary and appropriate. +Summary:  There are a number of ways +that do not involve complex mathematical +(or chemical energy and entropy) +calculations through which introductory level +biology students can be introduced into the  +stochastic features of biological systems, +including the complex relationship between +genotype and phenotype.   These approaches can help students appreciate (and be +3 +immunized against) the increasingly popular (apparently) illusion of genetic determinism, as +illustrated by Harden’s (Harden, 2021) Genetic Lottery, reviewed by Feldman  & Riskin (2022) +and Coop & Przeworski (2022).  + In the context of genetics and evolutionary mechanisms, the process of genetic drift seems an appropriate +3 +context within which to introduce stochastic processes. Students consider behavior of systems as a function of +population size (see https://youtu.be/B5M_C8gBvYo). Recently we have discovered a new genetic drift applet +here. +Making sense of noise +Wednesday, January 11, 2023 + of +9 +13 + +"ln a now classic result (Gartner, 1990), 30 years +of inbreeding experiments on laboratory mice +and rats in shared environments eliminated only +20-30%ofobservedvarianceinanumberof +phenotypes.Theremaining70-80%was +referred to as the ‘intangible variance'." +- Honegger & de Bivort, 20181.0 +↑cell1 +totalpopulation +0.6 +Vilar et al +measurement +Lacz +cell3→ +0.4 +maximal +0.2 +cell2-→ +% +0.0 +0 +50 +100 +150 +200 +time (minutes) +additionof inducer(lactose)It may also help make sense of discussions of whether humans (and other organisms) +have “free will”.  Clearly the situation is complex. From a scientific perspective we are +analyzing systems without recourse to non-natural processes. At the same time, “Humans +typically experience freely selecting between alternative courses of action” (Maoz et al., 2019) +(Maoz et al., 2019a; see also Maoz et al., 2019b).  It seems possible that recognizing the +intrinsically unpredictable nature of many biological processes (including those of the central +nervous system) may lead us to conclude that whether or not free will exists is in fact a non- +scientific, unanswerable (and perhaps largely meaningless) question.  +Acknowledgment: Thanks to Melanie Cooper and Nick Galati for taking a look and +Chhavinder Singh for getting it started. 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Journal of Chemical +Education 92, 1979–1987. +Making sense of noise +Wednesday, January 11, 2023 + of +13 +13 + diff --git a/V9E3T4oBgHgl3EQf0gvl/content/tmp_files/load_file.txt b/V9E3T4oBgHgl3EQf0gvl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2f78fe756be8975036876f7950d9582a917fe5e --- /dev/null +++ b/V9E3T4oBgHgl3EQf0gvl/content/tmp_files/load_file.txt @@ -0,0 +1,728 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf,len=727 +page_content='Making sense of noise: introducing students to stochastic processes in order to better understand biological behaviors Michael W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Klymkowsky Molecular, Cellular & Developmental Biology, University of Colorado Boulder, Boulder CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' 80309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' (klym@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='edu : ORCID: 0000-0001-5816-9771) Abstract: Biological systems are characterized by the ubiquitous roles of weak, that is, non-covalent molecular interactions, small, often very small, numbers of specific molecules per cell, and Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' These combine to produce stochastic behaviors at all levels from the molecular and cellular to the behavioral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' That said, students are rarely introduced to the ubiquitous role of stochastic processes in biological systems, and how they produce unpredictable behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Here I present the case that they need to be and provide some suggestions as to how it might be approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Background: Three recent events combined to spur this reflection on stochasticity in biological systems, how it is taught, and why it matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The first was an article describing an approach to introducing students to homeostatic processes in the context of the bacterial lac operon (Booth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2022), an adaptive gene regulatory system controlled in part by stochastic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The second was in-class student responses to the question, why do interacting molecules “come back apart” (dissociate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=" Finally, there is the increasing attention paid to what are presented as deterministic genetic factors, as illustrated by Kathryn Harden's “The Genetic Lottery: Why DNA matters for social equality” (Harden, 2021)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Previous work suggests that students, and perhaps some instructors (see Garvin-Doxas and Klymkowsky, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Klymkowsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' 2010), find the ubiquity, functional roles, and implications of stochastic, that is inherently unpredictable processes, difficult to recognize and apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Given their practical and philosophical implications, it seems essential to introduce students to stochasticity early in their educational journeys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' What is stochasticity and why is it important for understanding biological systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Stochasticity results when intrinsically unpredictable events, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' molecular collisions, impact the behavior of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' There are a number of drivers of stochastic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Perhaps the most obvious, and certainly the most ubiquitous in biological systems is thermal motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The Making sense of noise Wednesday, January 11, 2023 of 1 13 Image generated using the Java Genetic Drift applet (http://darwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='eeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='uconn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='edu/simulations/jdk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='0/ drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='html) in 2014 - the applet no longer appears to be available population size = 50, 100 generations 0 100 Generationsmany molecules within a solution (or a cell) are moving, they have kinetic energy – the energy of motion and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The exact momentum of each molecule cannot, however, be accurately and completely characterized without perturbing the system (echos of Heisenberg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Given the impossibility of completely characterizing the system, we are left uncertain as to the state of the system’s components, who is bound to whom, going forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Through collisions energy is exchanged between molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' A number of chemical processes are driven by the energy delivered through such collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Think about a typical chemical reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In the course of the reaction, atoms are rearranged – bonds are broken (a process that requires energy) and bonds are formed (a process that releases energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Many (most) of the chemical reactions that occur in biological systems require catalysts to bring their required activation energies into the range available within the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' 1 What makes the impact of thermal motion even more critical for biological systems is that many (most) regulatory interactions and macromolecular complexes, the molecular machines discussed by Alberts (1998) are based on relatively weak, non-covalent surface- surface interactions between or within molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Such interactions are central to most regulatory processes, from the activation of signaling pathways to the control of gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The specificity and stability of these non-covalent interactions, which include those involved in determining the three-dimensional structure of macromolecules, are directly impacted by thermal motion, and so by temperature – one reason controlling body temperature is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' So why are these interactions stochastic and why does it matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' A signature property of a stochastic process is that while it may be predictable when large numbers of atoms, molecules, or interactions are involved, the behaviors of individual atoms, molecules, and interactions are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' A classic example, arising from factors intrinsic to the atom, is the decay of radioactive isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' While the half-life of a large enough population of a radioactive isotope is well defined, when any particular atom will decay is, in current theory, unknowable, a concept difficult for students (see Hull and Hopf, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' This is the reason we cannot accurately predict whether Schrȍdinger’s cat is alive or dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The same behavior applies to the binding of a 2 regulatory protein to a specific site on a DNA molecule and its subsequent dissociation: predictable in large populations, not-predictable for individual molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The situation is exacerbated by the fact that biological systems are composed of cells and cells are, typically, small, and so contain relatively few molecules of each type (Milo and Phillips, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' There are typically one or two copies of each gene in a cell, and these may be different from one another (when heterozygous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The expression of any one gene depends upon the binding of specific proteins, transcription factors, that act to activate or repress gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In contrast to a number of other cellular proteins, “as a rule of thumb, the concentrations of such transcription For this discussion we ignore entropy, a factor that figures in whether a particular reaction in favorable or 1 unfavorable, that is whether, and the extent to which it occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Four common misconceptions about quantum physics [link] 2 Making sense of noise Wednesday, January 11, 2023 of 2 13 factors are in the nM range, corresponding to only 1-1000 copies per cell in bacteria or 103-106 in mammalian cells” (Milo and Phillips, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Moreover, while DNA binding proteins bind to specific DNA sequences with high affinity, they also bind to DNA “non-specifically” in a largely sequence independent manner with low affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Given that there are many more non-specific (non-functional) binding sites in the DNA than functional ones, the effective concentration of a particular transcription factor can be significantly lower than its total cellular concentration would suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' For example, in the case of the lac repressor of the bacterium Escherichia coli (discussed further below), there are estimated to be ~10 molecules of the tetrameric lac repressor per cell, but “non-specific affinity to the DNA causes >90% of LacI copies to be bound to the DNA at locations that are not the cognate promoter site” (Milo and Phillips, 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' at most only a few molecules are free in the cytoplasm and available to bind to specific regulatory sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Such low affinity binding to DNA allows proteins to undergo one-dimensional diffusion, a process that can greatly speed up the time it takes for a DNA binding protein to “find” high affinity binding sites (Stanford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' von Hippel and Berg, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Most transcription factors bind in a functionally significant manner to hundreds to thousands of gene regulatory sites per cell, often with distinct binding affinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The effective binding affinity can also be influenced by positive and negative interactions with other transcription and accessory factors, chromatin structure, and DNA modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Functional complexes can take time to assemble, and once assembled can initiate multiple rounds of polymerase binding and activation, leading to a stochastic phenomena known as transcriptional bursting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' An analogous process occurs with RNA-dependent polypeptide synthesis (translation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The result, particularly for genes expressed at lower levels, is that stochastic (unpredictable) bursts of transcription/translation can lead to functionally significant changes in protein levels (Raj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Raj and van Oudenaarden, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' There are many examples of stochastic behaviors in biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Originally noted by Novick and Weiner (1957) in their studies of the lac operon, it was clear that gene expression occurred in an all or none manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' This effect was revealed in a particularly compelling study by Elowitz et al (2002) who used lac operon promoter elements to drive expression of transgenes encoding cyan and yellow fluorescent proteins (on a single plasmid) in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' coli (→).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The observed behaviors were dramatic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' genetically identical cells were found to express, stochastically, one, the other, both, or neither transgene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The stochastic expression of genes and downstream effects appear to be the source of much of the variance found in organisms Making sense of noise Wednesday, January 11, 2023 of 3 13 Figure adapted from Elowitz et al 2002 oriC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='51Mb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='46 Mb E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='Coli intC<>YFP galK<>CFP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='6Mbwith the same genotype in the same environmental conditions (Honegger and de Bivort, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Beyond gene expression, the unpredictable effects of stochastic processes can be seen at all levels of biological organization, from the biased random walk behaviors that underlie various forms of chemotaxis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Spudich and Koshland, 1976) and the search behaviors in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' elegans (Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2016) and other animals (Smouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2010), the noisiness in the opening of individual neuronal voltage-gated ion channels (Braun, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Neher and Sakmann, 1976), and various processes within the immune system (Hodgkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2014), to variations in the behavior of individual organisms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' the leafhopper example cited by Honegger and de Bivort, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Stochastic events are involved in a range of “social” processes in bacteria (Bassler and Losick, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Their impact serves as a form of “bet-hedging” in populations that generate phenotypic variation in a homogeneous environment (see Symmons and Raj, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Stochastic events can regulate the efficiency of replication-associated error-prone mutation repair (Uphoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2016) leading to increased variation in a population, particularly in response to environmental stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Stochastic “choices” made by cells can be seen as questions asked of the environment, the system’s response provides information that informs subsequent regulatory decisions (see Lyon, 2015) and the selective pressures on individuals in a population (Jablonka and Lamb, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Together stochastic processes introduce a non- deterministic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' unpredictable) element into higher order behaviors (Murakami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Controlling stochasticity: While stochasticity can be useful, it also needs to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Not surprisingly then there are a number of strategies for “noise-suppression”, ranging from altering regulatory factor concentrations, the formation of covalent disulfide bonds between or within polypeptides, and regulating the activity of repair systems associated with DNA replication, polypeptide folding, and protein assembly via molecular chaperones and targeted degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' For example, the identification of “cellular competition” effects has revealed that “eccentric cells” (sometimes, and perhaps unfortunately referred to as of “losers”) can be induced to undergo apoptosis (die) or migration in response to their “normal” neighbors (Akieda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Di Gregorio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Hashimoto and Sasaki, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Student understanding of stochastic processes: There is ample evidence that students (and perhaps some instructors as well) are confused by or uncertain about the role of thermal motion, that is the transfer of kinetic energy via collisions, and the resulting stochastic behaviors in biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' As an example, Champagne-Queloz et al (2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' 2017) found that few students, even after instruction through molecular biology courses, recognize that collisions with other molecules were responsible for the disassembly of molecular complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In fact, many adopt a more “deterministic” model for molecular disassembly after instruction (see part A panel figure on next page).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In earlier studies, we found evidence for a similar confusion among instructors (part B of figure on the next page)(Klymkowsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Making sense of noise Wednesday, January 11, 2023 of 4 13 Introducing stochasticity to students: Given that understanding stochastic (random) processes can be difficult for many (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Garvin-Doxas and Klymkowsky, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Taleb, 2005), the question facing course designers and instructors is when and how best to help students develop an appreciation for the ubiquity, specific roles, and implications of stochasticity- dependent processes at all levels in biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' I would suggest that introducing students to the dynamics of non-covalent molecular interactions, prevalent in biological systems in the context of stochastic interactions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' kinetic theory) rather than a ∆G-based approach may be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' We can use the probability of garnering the energy needed to disrupt an interaction to present concepts of binding specificity (selectivity) and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Developing an understanding of the formation and disassembly of molecular interactions builds on the same logic that Albert Einstein and Ludwig Böltzman used to demonstrate the existence of atoms and molecules and the reversibility of molecular reactions (Bernstein, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Moreover, as noted by Samoilov et al (2006) "stochastic mechanisms open novel classes of regulatory, signaling, and organizational choices that can serve as efficient and effective biological solutions to problems that are more complex, less robust, or otherwise suboptimal to deal with in the context of purely deterministic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='" The selectivity (specificity) and stability of molecular interactions can be understood from an energetic perspective – comparing the enthalpic and entropic differences between bound and unbound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' What is often missing from such discussions, aside from the fact of Making sense of noise Wednesday, January 11, 2023 of 5 13 B 90 Once two molecule bind to one another, how could they come back apart again?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' introductory courses Theywould havetobind to yetanothermolecule- correct responses Thecomplexwillneedto bedegraded 68 Collionswithothermolecules couldknockthemapart (correct) advanced courses 45 A chemical reaction must changethestructure teachers ofoneofthemolecules (mostpopular) 23 noanswe % A pre- and post-instruction analysis indicates that "Random molecular collisions are not recognized as the major source of breaking molecular interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The best answer (2) 0 reflects the fact that molecules interacting and dissociated dissociation question from one another in response to the transfer of energy, Comparison of responses by different grouls typically collisions with other molecules, sufficient to to this questions (BCl Q18)(modified from overcome their interaction energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='" (from Kiymkowsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Champagne-Queloz 2016 use with permission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' "What made the greatest impression upon the student, however, was less the technical construction of mechanics or the solution of complicated problems than the achievement of mechanics in areas which apparently had nothing to do with mechanics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' above all the kinetic theory of gases:" - A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Einstein (Bernstein)their inherent complexity, particularly in terms of calculating changes in entropy and exactly what is meant by energy (Cooper and Klymkowsky, 2013) is that many students enter biology classes without a robust understanding of enthalpy, entropy, or free energy (Carson and Watson, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Presenting students with a molecular collision, kinetic theory-based mechanism for the dissociation of molecular interactions, may help them better understand (and apply) both the dynamics and specificity of molecular interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' We can gage the strength of an interaction (the sum of the forces stabilizing an interaction) based on the amount of energy (derived from collisions with other molecules) needed to disrupt it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The implication of student responses to relevant Biology Concepts Instrument (BCI) questions and beSocratic activities (data not shown), as well as a number of studies in chemistry, is that few students consider the kinetic/vibrational energy delivered through collisions with other molecules (a function of temperature), as key to explaining why interactions break (see Carson and Watson, 2002 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Although this paper is 20 years old, there is little or no evidence that the situation has improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Moreover, there is evidence that the conventional focus on mathematics-centered, free energy calculations in the absence of conceptual understanding may serve as an unnecessary barrier to the inclusion of a more socioeconomically diverse, and under-served populations of students (Ralph et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Stowe and Cooper, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The lac operon as a context for introducing stochasticity: Studies of the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' coli lac operon hold an iconic place in the history of molecular biology and are often found in introductory courses, although typically presented in a deterministic context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The mutational analysis of the lac operon helped define key elements involved in gene regulation (Jacob and Monod, 1961;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Monod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Booth et al (2022) used the lac operon as the context for their “modeling and simulation lesson”, Advanced Concepts in Regulation of the Lac Operon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Given its inherently stochastic regulation (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Elowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Novick and Weiner, 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Vilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2003), the lac operon is a good place to start introducing students to stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In this light, it is worth noting that Booth et al describes the behavior of the lac operon as “leaky”, which would seem to imply a low, but continuous level of expression, much as a leaky faucet continues to drip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' As this is a peer-reviewed lesson, it seems likely that it reflects widely held mis-understandings of how stochastic processes are introduced to, and understood by students and instructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' coli cells respond to the presence of lactose in growth media in a biphasic manner, termed diauxie, due to “the inhibitory action of certain sugars, such as glucose, on adaptive enzymes (meaning an enzyme that appears only in the presence of its substrate)” (Blaiseau and Holmes, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' When these (preferred) sugars are depleted from the media, growth Making sense of noise Wednesday, January 11, 2023 of 6 13 " The essence of thermodynamics, however, is a study of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Terms like entropy and Gibbs free energy cannot be understood as isolated entities that can be transformed into one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' If students are to use these concepts to make predictions about whether reactions can occur, they need to understand them in the context of chemical\' processes\'" - Carson & Watson, 2002slows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' If lactose is present, however, growth will resume following a delay associated with the expression of the proteins encoded by the operon that enables the cell to import and metabolize lactose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Although the term homeostatic is used repeatedly by Booth et al, the lac operon is part of an adaptive, rather than a homeostatic, system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In the absence of glucose, cyclic AMP (cAMP) levels in the cell rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' cAMP binds to and activates the catabolite activator protein (CAP), encoded for by the crp gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Activation of CAP leads to the altered expression of a number of target genes, whose products are involved in adaption to the stress associated with the absence of common and preferred metabolites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' cAMP-activated CAP acts as both a transcriptional repressor and activator, “and has been shown to regulate hundreds of genes in the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' coli genome, earning it the status of “global” or “master” regulator” (Frendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' It is involved in the adaptation to environmental factors, rather than maintaining the cell in a particular state (homeostasis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The lac operon is a classic polycistronic bacterial gene, encoding three distinct polypeptides: lacZ (β-galactosidase), lacY (β-galactoside permease), and lacA (galactoside acetyltransferase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' When glucose or other preferred energy sources are present, expression of the lac operon is blocked by the inactivity of CAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The CAP protein is a homodimer and its binding to DNA is regulated by the binding of the allosteric effector cAMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' cAMP is generated from ATP by the enzyme adenylate cyclase, encoded by the cya gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In the absence of glucose the enyzme encoded by the crr gene is phosphorylated and acts to activate adenylate cyclase (Krin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' As cAMP levels increase, cAMP binds to the CAP protein, leading to a dramatic change in its structure (↑), such that the protein’s DNA binding domain becomes available to interact with promoter sequences (figure from Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Binding of activated (cAMP-bound) CAP is not, by itself sufficient to activate expression of the lac operon because of the presence of the constitutively expressed lac repressor protein, encoded for by the lacI gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The active repressor is a tetramer, present at very low levels (~10 molecules) per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The lac operon contains three repressor (“operator”) binding sites;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' the tetrameric repressor can bind two operator sites simultaneously (figure → from Palanthandalam- Madapusi and Goyal, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In the absence of Making sense of noise Wednesday, January 11, 2023 of 7 13 DNA binding helices cAMP DNA boundto binding interaction DNA recruit"RNA polymerase + synthesize mRNA Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2009 (frames from movie S1)Looped non-coding DNA Genes Lacz LacY LacA Lactose- RepressorProtein fromPalanthandalam-Madapusi&Goyal,2011lactose, but in the presence of cAMP-activated CAP, the operon is expressed in discrete “bursts” (Novick and Weiner, 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Vilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Choi et al (2008) found that these burst come in two types, short and long, with the size of the burst referring to the number of mRNA molecules synthesized (figure adapted from Choi et al ↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The difference between burst sizes arises from the length of time that the operon’s repressor binding sites are unoccupied by repressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' As noted above, the tetravalent repressor protein can bind to two operator sites at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' When released from one site, polymerase binding and initiation produces a small number of mRNA molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Persistent binding to the second site means that the repressor concentration remains locally high, favoring rapid rebinding to the operator and the cessation of transcription (RNA synthesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' When the repressor releases from both operator sites, a rarer event, it is free to diffuse away and interact (non-specifically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' with low affinity) with other DNA sites in the cell, leaving the lac operator sites unoccupied for a longer period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The number of such non-specific binding sites greatly exceeds the number (three) of specific binding sites in the operon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The result is the synthesis of a larger “burst” (number) of mRNA molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The average length of time that the operator sites remain unoccupied is a function of the small number of repressor molecules present and the repressor’s low but measurable non-sequence specific binding to DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The expression of the lac operon leads to the appearance of β-galactosidase and β- galactoside permease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' An integral membrane protein, β-galactoside permease enables extracellular lactose to enter the cell while cytoplasmic β-galactosidase catalyzes its breakdown and the generation of allolactone, which binds to the lac repressor protein, inhibiting its binding to operator sites, and so removing repression of transcription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In the absence of lactose, there are few if any of the proteins (β-galactosidase and β-galactoside permease) needed to activate the expression of the lac operon, so the obvious question is how, when lactose does appear in the extracellular media, does the lac operon turn on?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Booth et al and the Wikipedia entry on the lac operon (accessed 29 June 2022) describe the turn on of the lac operon as “leaky” (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The molecular modeling studies of Vilar et al and Choi et al (which, together with Novick and Weiner, are not cited by Booth et al) indicate that the system displays distinct threshold and maintenance concentrations of lactose needed for stable lac gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' The term “threshold” does not occur in the Booth et al article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' More importantly, when cultures are examined at the single cell level, what is observed is not a uniform increase in lac expression in all cells, as might be expected in the context of leaky expression, but more sporadic (noisy) behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Increasing numbers of cells are “full on” in terms of lac operon expression over time when cultured in lactose concentrations above the Making sense of noise Wednesday, January 11, 2023 of 8 13 口 口 Lowintracellularinducerconcentration Inducer Lacl Laco site DNA mRNA Smallburst Largeburst 口 Time (min) 0 10 20 30 40 50 Atime-lapsesequenceinthepresenceof5omMTMG,onesuchcell switchesphenotypeoperon’s activation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' This illustrates the distinctly different implications of a leaky versus a stochastic process in terms of their impacts on gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' While a leak is a macroscopic metaphor that produces a continuous, dependable, regular flow (drips), the occurrence of “bursts” of gene expression implies a stochastic (unpredictable) process (← figure from Vilar et al).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' As the ubiquity and functionally significant roles of stochastic processes in biological systems becomes increasingly apparent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' in the prediction of phenotypes from genotypes (Karavani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Mostafavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2020), helping students appreciate and understand the un-predictable, that is stochastic, aspects of biological systems becomes increasingly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' As an example, revealed dramatically through the application of single cell RNA sequencing studies, variations in gene expression between cells of the same "type" impacts organismic development and a range of behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' For example, in diploid eukaryotic cells is now apparent that in many cells, and for many genes, only one of the two alleles present is expressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' such “monoallelic” expression can impact a range of processes (Gendrel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Given that stochastic processes are often not well conveyed through conventional chemistry courses (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2015) or effectively integrated into, and built upon in molecular (and other) biology curricula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' presenting them explicitly in introductory biology courses seems necessary and appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Summary: There are a number of ways that do not involve complex mathematical (or chemical energy and entropy) calculations through which introductory level biology students can be introduced into the stochastic features of biological systems, including the complex relationship between genotype and phenotype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' These approaches can help students appreciate (and be 3 immunized against) the increasingly popular (apparently) illusion of genetic determinism, as illustrated by Harden’s (Harden, 2021) Genetic Lottery, reviewed by Feldman & Riskin (2022) and Coop & Przeworski (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' In the context of genetics and evolutionary mechanisms, the process of genetic drift seems an appropriate 3 context within which to introduce stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Students consider behavior of systems as a function of population size (see https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='be/B5M_C8gBvYo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Recently we have discovered a new genetic drift applet here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Making sense of noise Wednesday, January 11, 2023 of 9 13 "ln a now classic result (Gartner, 1990), 30 years of inbreeding experiments on laboratory mice and rats in shared environments eliminated only 20-30%ofobservedvarianceinanumberof phenotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content="Theremaining70-80%was referred to as the ‘intangible variance'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='" Honegger & de Bivort, 20181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='0 ↑cell1 totalpopulation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='6 Vilar et al measurement Lacz cell3→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='4 maximal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='2 cell2-→ % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content='0 0 50 100 150 200 time (minutes) additionof inducer(lactose)It may also help make sense of discussions of whether humans (and other organisms) have “free will”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Clearly the situation is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' From a scientific perspective we are analyzing systems without recourse to non-natural processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' At the same time, “Humans typically experience freely selecting between alternative courses of action” (Maoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2019) (Maoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' see also Maoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' It seems possible that recognizing the intrinsically unpredictable nature of many biological processes (including those of the central nervous system) may lead us to conclude that whether or not free will exists is in fact a non- scientific, unanswerable (and perhaps largely meaningless) question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Acknowledgment: Thanks to Melanie Cooper and Nick Galati for taking a look and Chhavinder Singh for getting it started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' A earlier version of this essay appear on the bioliteracy blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Making sense of noise Wednesday, January 11, 2023 of 10 13 Literature cited: Akieda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', Ogamino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', Furuie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', Ishitani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', Akiyoshi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=', Nogami, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} +page_content=' Making sense of noise Wednesday, January 11, 2023 of 13 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQf0gvl/content/2301.04739v1.pdf'} diff --git a/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf b/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4fe37f1f8d6ce89daf177923cd77f0bf725dc48c --- /dev/null +++ b/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:71befc537219ac39688bbf78ef5b79cfb65746e101c0424698d1d07b3633b6be +size 2556551 diff --git a/VdAyT4oBgHgl3EQfhfjt/vector_store/index.faiss b/VdAyT4oBgHgl3EQfhfjt/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4b0c3edc0d84f53bd9ff4ff0ce976a85a898eb46 --- /dev/null +++ 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XX, NO. XX, 2023 +1 +A Study on the Generality of Neural Network +Structures for Monocular Depth Estimation +Jinwoo Bae, Kyumin Hwang and Sunghoon Im +Abstract—Monocular depth estimation has been widely studied, and significant improvements in performance have been recently +reported. However, most previous works are evaluated on a few benchmark datasets, such as KITTI datasets, and none of the works +provide an in-depth analysis of the generalization performance of monocular depth estimation. In this paper, we deeply investigate the +various backbone networks (e.g.CNN and Transformer models) toward the generalization of monocular depth estimation. First, we +evaluate state-of-the-art models on both in-distribution and out-of-distribution datasets, which have never been seen during network +training. Then, we investigate the internal properties of the representations from the intermediate layers of CNN-/Transformer-based +models using synthetic texture-shifted datasets. Through extensive experiments, we observe that the Transformers exhibit a strong +shape-bias rather than CNNs, which have a strong texture-bias. We also discover that texture-biased models exhibit worse +generalization performance for monocular depth estimation than shape-biased models. We demonstrate that similar aspects are +observed in real-world driving datasets captured under diverse environments. Lastly, we conduct a dense ablation study with various +backbone networks which are utilized in modern strategies. The experiments demonstrate that the intrinsic locality of the CNNs and the +self-attention of the Transformers induce texture-bias and shape-bias, respectively. +Index Terms—Monocular depth estimation, Out-of-Distribution, Generalization, Transformer +! +1 +INTRODUCTION +Monocular depth estimation (MDE) is widely utilized for +spatial recognition technologies such as autonomous driv- +ing [1], [2], [3] or AR/VR [4], [5] because of its porta- +bility and cost-effectiveness. Various MDE processes have +achieved remarkable progress over the past decade. Most +previous works [6], [7], [8], [9], [10], [11], [12], [13] con- +centrate on boosting performance using limited benchmark +datasets, especially the KITTI dataset [14]. However, these +works have not provided a deep investigation into what +MDE networks have learned. This means that they cannot +guarantee the model’s behavior is correct. To examine the +interpretability of MDE networks, one previous work [15] +employs a target network for MDE model analysis. Another +approach [16] uses a synthetic dataset, which contains the +changes in image contents (e.g., camera pose). However, +universality questions about other networks still remain +because of the fixed specific network [6] in certain experi- +mental conditions. +Recently, several works [17], [18], [19], [20] aim to ana- +lyze interpretability, taking inspiration from convolutional +neural networks (CNNs) whose designs are based on hu- +man visual processes [21], [22]. Then, how can humans +efficiently extract and recognize important information from +complex scenes? Compared to other cues like texture or +color, the biological vision system considers the object’s +shape to be the single most important visual cue [23]. This +allows humans, including little kids, to easily distinguish +an object from a line drawing or a silhouette image. Many +• +J. Bae, K. Hwang and S. Im are with the Department of Electrical +Engineering and Computer Science, DGIST, Daegu, 42988, Republic of +Korea. {sjg02122, kyumin, sunghoonim}@dgist.ac.kr +Monodepth2 +PackNet-SfM +R-MSFM6 +MF-SLaK +BTS +AdaBins +MF-RegionViT +MF-Twins +MF-ConvNeXt +TransDepth +DepthFormer +MF-ViT +MF-Ours +GLPDepth +0.085 +0.135 +0.185 +0.235 +0.285 +0 +0.2 +0.4 +0.6 +0.8 +1 +Shape-biased +Texture-biased +Error (Abs Rel.) +CKA Similarity (Mean) +CNN-based +Transformer-based +Self-supervised: +Supervised: +Transformer-based +CNN-based +Fig. 1: Analysis on the generality of state-of-the-art models +and modernized network structures. The x-axis is the CKA +similarity indicating whether the network is shape biased +or texture biased. The y-axis shows Absolute Relative error +where a lower number is better performance. We use the +synthetic texture-shifted datasets described in Sec. 4.5. +researchers believe that CNN would also behave similarly +[24], [25], [26]. However, in contrast to belief, recent studies +[18], [20], [27], [28] have discovered that CNNs are heavily +predisposed to recognize textures rather than shapes. CNN- +based models accurately assign labels to images even when +the shapes of the structures are disturbed [29], [30]. On the +other hand, CNN models are unable to accurately predict +labels in a texture-removed image with a well-preserved +shape [31]. Transformers [32] achieve outstanding perfor- +mance on various computer vision tasks [33], [34], [35] and +have attracted much attention. Moreover, many works [27], +[36], [37], [38] reveal that Transformers have a strong shape +arXiv:2301.03169v1 [cs.CV] 9 Jan 2023 + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +2 +bias notwithstanding the lack of a spatial locality, compared +to CNNs. Due to its strong shape bias, Transformer is +considered more robust than a CNN and more similar to +human cognitive processes [39]. +Then, how does this observation affect the MDE? We +hypothesize two things. First, the network’s generality will +differ depending on the texture-/shape-bias. To verify the +generality, we evaluate state-of-the-art MDE models trained +on KITTI [14] using five public depth datasets (SUN3D [40], +RGBD [41], MVS [42], Scenes11 [42], and ETH3D [43]). We +also conduct experiments on six different texture-shifted +datasets, including three synthetic texture-shifted datasets +(Watercolor, Pencil-sketch, Style-transfer) and three real- +world texture-shifted datasets (Oxford Robotcar [44], Rainy +Cityscapes [45], Foggy Cityscapes [46]). Through these ex- +periments, we confirm that texture-bias is vulnerable to +generalization while shape-bias shows robust generalization +performance. Second, the texture-/shape-bias are related +to the intrinsic properties of CNN and Transformer struc- +tures. The modernized Transformer-based model [47], [48] +imitates the intrinsic locality inductive bias of the CNNs. +The modernized CNN-based model [49], [50] is designed +to mimic the self-attention of the Transformer. We finally +conduct ablation studies on the generalization performance, +and the texture-/shape-bias of various modernized back- +bone structures that originate from a specific design for each +CNN and Transformer (e.g., locality and self-attention) such +as RegionViT [47], Twins [48], ConvNeXt [49] and SLaK [50]. +This paper extends our previous work in [51] which +proposes a new self-supervised monocular depth estima- +tion network adopting Transformers [52]. The Transformer- +based network shows the generalization performance on +various environments on depth estimation. While the short +version [51] only addresses self-supervised MDE models, +the extended version deals with full MDE models, includ- +ing both self-supervised and supervised methods. More- +over, we deeply analyze the reason why the proposed net- +work achieves better-generalized performance rather than +CNN-based models by comparing the performance, and +intermediate-layer feature similarity [53], [54] on various +texture-shifted datasets. As shown in Fig. 1, we observe +that the Transformer structure has more shape-biased prop- +erties than the CNN structure, which has texture-biased +properties. It enables the Transformer-based models to +achieve better generalization performance than the CNN- +based models. We also experiment extensively on modern +backbone architectures (e.g., ConvNeXt [49], RegionViT [47]) +to investigate the origin of the texture-/shape-biased prop- +erties. Through these extensive experiments, we observe +that the intrinsic locality of CNNs induces texture-biased +characteristics, while the self-attention mechanism, the base +layer of Transformers, induces shape-biased properties. +2 +RELATED WORK +2.1 +Self-supervised monocular depth estimation +Self-supervised depth estimation methods [8], [9], [12], [55], +[56], [57] simultaneously train depth and motion network by +imposing photometric consistency loss between the target +and source images warped by the predicted depth and +motion. SfMLearner [55] first proposes a depth and ego- +motion estimation pipeline without the ground truth depth +and motion. Monodepth2 [8] presents a minimum repro- +jection loss to handle occlusions, a full-resolution multi- +scale sampling method to reduce visual artifacts, and an +auto-masking loss to ignore outlier pixels. PackNet-SfM [9] +introduces packing and unpacking blocks that leveraged 3D +convolutions to learn the dense appearance and geometric +information in real-time. HR-Depth [12] analyzes the reason +for the inaccurate depth prediction in large gradient regions +and designed a skip connection to extract representative +features in high resolution. +2.2 +Supervised monocular depth estimation +Supervised methods [10], [13], [58], [59], [60] use a ground +truth depth acquired from RGB-D cameras or LiDAR sen- +sors for supervision in training. They also estimate depth +maps given a single image as input. BTS [58] adopts a +local planar guidance layer to densely encoded features to +preserve local detail and create depth map sharpness at +multi-stages in the decoder. AdaBins [10] estimate the depth +by linear combinations of bin centers that are adaptively +decided per image. The bin building block divides the +depth range of the image into bins. LapDepth [59] employs +a Laplacian pyramid at the multi-level upscaling encoder +to preserve local detail, such as a boundary. It also trains +stably by utilizing weight standardization. GLPDepth [60] +proposes a hierarchical transformer encoder to capture the +global context of images and a selective feature fusion mod- +ule to connect multi-scale local features and global context +information at the decoder. The feature fusion module helps +the decoder become more powerful, even if the decoder +is lightweight. DepthFormer [13] proposes leveraging the +transformer’s effective attention mechanism and the spatial +inductive bias of the CNN to capture long-range correlation. +It also uses a hierarchical aggregation and heterogeneous +interaction module to enhance the affinity of the network. +2.3 +Vision Transformers +Recently, Transformers [52] has shown promise for solving +computer vision tasks such as image classification [32], [61], +object detection [33], and dense prediction. [11], [62], [63], +[64]. ViT [32] employs a Transformers architecture on fixed- +size image patches for image classification for the first time. +DeiT [61] utilizes Knowledge distinction on ViT architecture, +showing good performance only with the ImageNet dataset. +DETR [33] proposes the direct set prediction approach, +which simplifies the object detection pipeline, based on a +CNN-Transformer network and bipartite matching. Some +works [11], [63] have employed Transformers for monoc- +ular depth estimation in a supervised manner. DPT [63] +introduces a dense prediction using a Transformer as the +basic computational building block of the encoder. These +works show generalized performance, but they require +a large number of training datasets captured in diverse +environments with ground truth depth maps. TransDepth +[11] utilizes multi-scale information to capture local level +details. Previous works lack studies on whether models +behave as intended on another domain dataset. The work +[65] aggregates the attention map between a single frame + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +3 +L2-Norm +Channel Norm +tanh +× +× ++ ++ +× +𝜶 +𝜸 +𝜷 +E(𝜙) +Transformer +Feature Fusion Decoder +Norm +Multi-Head +Attention +Norm +MLP ++ ++ +Linear +Linear +Linear +Scaled Dot-Product +Attention +𝑸 +Concatenate +Linear +𝑽 +𝑲 +Transformer +Transformer +Transformer +Residual Block Unit +Transformer Encoder +ACM +ACM +ACM +ACM +Feature Fusion Decoder +Feature Fusion Decoder +Feature Fusion Decoder +Feature Fusion Decoder +Attention Connection Module +𝑍! +𝑍" +𝑍# +𝑍$ +𝑋! +𝑋" +𝑋# +𝑋$ +𝑋% +Image 𝐼 +Depth 𝐷 +Linear Projection +𝐹 +𝜙 +Embedding function +Element-wise multiplication +Element-wise addtion +Encoder +Feature +Fused +Feature +Attention +Map +Conv +Position +Attention +Channel +Attention +𝐴& +' +𝐴& +( +𝒘𝒑𝑨𝒑 +𝒘𝒄𝑨𝒄 +𝑍 +𝑋 +Conv ++ ++ +× +× +Fig. 2: Overall Architecture of MonoFormer (MF-ours). +and cross frames to refine the attraction map to improve +performance. The works [11], [65] only focus on improving +performance on benchmark datasets. +2.4 +Modernized Architecture +Despite the tremendous success of Transformers for vision +tasks, a Transformer requires a large model and data size to +achieve state-of-the-art performance. Recently, many works +[47], [48], [49], [50], [66] utilize local attention on the Trans- +former to alleviate problems. The Swin Transformer [66] de- +signs a pyramid structure network differently from a vision +transformer [32], which is an isotropic structure. It achieves +state-of-the-art performance in classification tasks with local +attention using the sliding window strategy. Twins [48] +employs global attention for the non-overlapping region +and local attention to perform better under limited condi- +tions. RegionViT [47] shows state-of-the-art performance on +several visual tasks using regional-to-local attention, which +alleviates the weakness of the standard attention mecha- +nism. ConvNeXt [49] modernizes convolution neural layers +such as depth-wise convolution and utilizes techniques such +as Patchify stem and achieves competitive performance with +the Transformer. SLaK [50] attempts to design the network +with an extremely large kernel size (e.g., 51×51) to leverage +the sparsity that is observed from the human visual system. +SLaK [50] repeats the Prune-and-Grow step in training to +optimize the sparse kernels. +3 +METHOD +This section describes MonoFormer, which is an encoder- +decoder structure with a multi-level feature fusion module +for self-supervised monocular depth estimation described in +Sec. 3.1. An Attention Connection Module (ACM) learns the +channel and position attentions in Sec. 3.2. A Feature Fusion +Decoder (FFD) adaptively fuses the encoder features with +the attention maps in Fig. 2. +3.1 +Transformer-based Encoder +MonoFormer [51] is composed of a CNN and Transformer +for an image encoder. The encoder employs ResNet50 [67] +as the CNN backbone (E(θ) in Fig. 2), and L number of +Transformers. MonoFormer sets the L to 4. The encoder is +used to extract a feature map F ∈ RC×H×W from an input +image I, and the map is divided into N (= H +16 × W +16 ) number +of patches pn ∈ RC×16×16, which is utilized as the input of +the first Transformer layer. Following the work [63], Mono- +Former additionally use a special token ts. MonoFormer +input the patch tokens pn, n ∈ {1, ..., N} and the special +token ts with a learnable linear projection layer E as follows: +Z0 = [ts; p1E; p2E; ... ; pNE], +(1) +where Z0 is the latent embedding vector. The CNN- +Transformer encoder comprises a Multi-head Self-Attention +(MSA) layer, a Multi-Layer Perceptron (MLP) layer, and +Layer Norm (LN) layers. The MLP is built with GELU non- +linearity [68]. The LN is used before each block, and residual +connections are used after every block. Self-Attention (SA) +at each layer l ∈ {1, ..., L} is processed with the learnable +parameters W m +Q , W m +K , W m +V +∈ RC×d of {query, key, value} +weight matrices, given the embedding vector Zl ∈ RN×C as +follows: +SAm +l−1 = softmax +�Qm +l−1(Km +l−1)T +√ +d +� +V m +l−1, m ∈ {1, ..., M}, +Qm +l−1 = Zl−1W m +Q , Km +l−1 = Zl−1W m +K , V m +l−1 = Zl−1W m +V , +(2) +where M and d are the number of SA blocks and the di- +mension of the self-attention block, which is the same as the +dimension of the weight matrices, respectively. The Multi- +head Self-Attention (MSA) consists of the M number of +SA blocks with the learnable parameters of weight matrices +W ∈ RMd×C as follows: +MSAl−1 = Zl−1 + concat(SA1 +l−1; SA2 +l−1; . . . ; SAM +l−1)W, +Zl = MLP(LN(MSAl−1)) + MSAl−1. +(3) + +tsIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +4 +16×16 +16×16 +16×16 +32×32 +16×16 +8×8 +Local +Patch +Region +Patch +16×16 +32×32 +64×64 +Other arrows are +omitted for simplicity. +4×4 +Local +Attention +Global +Attention +Other arrows are +omitted for simplicity. +𝐶 +4𝐶 +2𝐶 +𝑊 ∈ ℝ!"×$" +”patchify” Stem +Depth wise conv +Depth Conv +Inverted residual block +narrow +wide +narrow +Input +51×5 +5×5 +5×51 ++ +ViT [32] +RegionViT [47] +Twins [48] +ConvNeXt [49] +SLaK [50] +Fig. 3: Illustrations of various modernized architectures. +This Transformer layer is repeated L times with unique +learnable parameters. The outputs of the Transformers +{Z1, ..., ZL} are utilized as the input of the following layers +ACM and FFD. The hybrid encoder can be replaced with +the other backbones. We evaluate all the performance of +our depth estimation networks with various backbones by +changing the encoder to the modernized backbone struc- +tures, such as ViT [32], RegionViT [47], Twins [48], Con- +vNeXt [49] and SLaK [50] as shown in Fig. 3. +3.2 +Attention Connection Module (ACM) +The skip connection module of MonoFormer, ACM, that +extracts global context attention and a semantic presenta- +tion from the given features Zl, l ∈ {1, ..., L}. The skip +connection, widely utilized for dense prediction tasks [69], +helps keep the fine detail by directly transferring the spatial +information from the encoder to the decoder. However, +because of its simplicity, it is challenging for the naive +skip connection method to preserve local detail like object +boundaries [70]. To address the issue, The ACM is pro- +posed that extracts attention weight from the spatial and +channel domains inspired by [71] It consists of position +attention, channel attention modules, and a fusion block +that gathers important information from two attentions. +The position attention module produces a position attention +map Ap +l ∈ RC×N as follows: +Ap +l = softmax(Qp +l (Kp +l )T)V p +l , +(4) +where Qp +l , Kp +l and V p +l are the query, key, and value matrices +computed by passing Zl through a single convolutional +layer. The channel attention module directly calculate the +channel attention map Ac +l ∈ RC×N by computing the gram +matrix of Zl as follows: +Ac +l = softmax(ZlZT +l ). +(5) +The position attention map Ap +l and channel attention map +Ac +l enhance the feature representation by capturing long- +range context and exploiting the inter-dependencies be- +tween each channel map, respectively. These two attention +maps are utilized in the following section, which highlights +the importance of the features. +3.3 +Feature Fusion Decoder (FFD) +The FFD gets the encoder features Zl, the attention maps +Ap +l , Ac +l , and the output feature XL of the last Transformer +layer passed through a Residual convolutional layer. The +features XL−l+1, l ∈ {1, ..., L} are fused through the de- +coder with a single Convolutional layer (Conv) and Channel +Normalization (CN) with learnable parameters α, β and γ as +follows: +XL−l = ˆ +XL−l[1 + tanh(γ(CN(α|| ˆ +XL−l||2 + β)], +ˆ +XL−l = Conv(wpAp +l Zl + wcAc +l Zl + Zl) + XL−l+1, +(6) +where wp and wc are the learnable parameters that deter- +mine the importance of the position and channel attentions +[72]. The parameter α works so that each channel can +learn about each other individually, and γ and β control +the activation channel-wisely following the work in [73]. +Through this process, the FFD can assemble the local de- +tailed semantic representation and the global context from +the fused features to produce the fine depth map. +3.4 +Training loss and implementation detail +We train both depth and motion networks using pho- +tometric consistency (L2 loss and SSIM loss) and edge- +aware smoothness losses following the best practices of self- +supervised monocular depth estimation [8], [9], [55]. We set +the weight for SSIM, L2 photometric, and smoothness losses +as 0.85, 0.15 and 0.001, respectively. We use 7 convolution +layers for 6-DoF camera pose estimation following the work +in [55]. We implement our framework on PyTorch and train +it on 4 Titan RTX GPUs. We use the Adam optimizer [74] +with β1 = 0.9 and β2 = 0.999. Our model is trained for 50 +epochs with a batch size of 8. The learning rates for depth +and pose network are 2 × 10−5 and 5 × 10−4, respectively. +We will release the source code, the trained weights and the +datasets once the paper is accepted. +4 +EXPERIMENTAL RESULTS +We conduct extensive experiments to investigate the gen- +eralization performance of various network structures and +the effect of texture-/shape-bias for monocular depth es- +timation. First, we evaluate state-of-the-art KITTI-trained +models, including various modernized backbone architec- +tures on the KITTI dataset and various depth benchmark +datasets. Evaluations are conducted on the KITTI, an in- +distribution dataset in Sec. 4.2, and the other depth datasets, +out-of-distribution datasets in Sec. 4.4. We also conduct an +ablation study on the MonoFormer in Sec. 4.3. Then, we in- +vestigate the texture-bias and shape-bias of the MDE models + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +5 +Input images +Monodepth2 +PackNet-SfM +R-MSFM6 +MF-ConvNeXt +MF-SLaK +MF-ViT +MF-RegionViT +MF-Twins +MF-Ours +BTS +AdaBins +TransDepth +DepthFormer +GLPDepth +Input images +Monodepth2 +PackNet-SfM +R-MSFM6 +MF-ConvNeXt +MF-SLaK +MF-ViT +MF-RegionViT +MF-Twins +MF-Ours +BTS +AdaBins +TransDepth +DepthFormer +GLPDepth +Fig. 4: Depth map results on the in-distribution dataset. MonoFormer (MF-Ours) is from our previous work [51]. +Models +Supervision +Methods +CNN-based +Self-supervised +Monodepth2 [8], PackNet-SfM [9] +R-MSFM6 [75], MF-(ConvNeXt [49], SLaK [50]) +Supervised +BTS [58], AdaBins [10] +Transformer-based +Self-supervised +MF-(ViT [32], RegionViT [47], Twins [48], Ours [51]) +Supervised +TransDepth [11], DepthFormer [13], GLPDepth [60] +TABLE 1: Taxonomy of MDE methods w.r.t backbones. +using both the self-supervised and supervised methods with +texture-shifted datasets in Sec. 4.5. We analyze the depth es- +timation performance and the feature representation of each +model on the texture-shifted datasets, which are syntheti- +cally generated. Lastly, we demonstrate the intrinsic proper- +ties of the CNN-based and Transformer-based networks are +observed in the real-world texture-shifted datasets captured +from different driving environments, such as the weather +changes and illumination changes in Sec. 4.6. +4.1 +Competitive methods and evaluation setups +We conduct extensive experiments to compare the per- +formances of the monocular depth estimation methods. +For self-supervised setting, we compare our work, called +MF-Ours [51], with state-of-the-art methods, Monodepth2 +[8], PackNet-SfM [9], SGDepth [56], R-MSFM [75]. We +also evaluate state-of-the-art supervised methods, BTS [58], +AdaBins [10], TransDepth [11], DepthFormer [13], and +GLPDepth [60]. We note that some Transformer-based mod- +els, such as MF-Ours, TransDepth, and DepthFormer, use +both CNNs and Transformers for their encoders. We also +analyze the performance of various modernized architec- +tures such as RegionViT [47], Twins [48], ConvNeXt [49] and +SLaK [50]. We replace the encoder of MF-Ours with these +modernized backbones, whose names are defined as MF- +(backbone). Its taxonomy is described in Tab. 1. We illustrate +these basic network structures of the modernized CNN +and Transformer architectures in Fig. 3. eTransformer-based +models [47], [48] add the locality by using local attention to +compensate for the shortcomings of the ViT (e.g., necessity +Method +Train +Model +Lower is better ↓ +Higher is better ↑ +Abs Rel +Sq Rel +RMSE +RMSElog +δ < 1.25 +δ < 1.252 +δ < 1.253 +Monodepth2 +M +CNN +0.115 +0.903 +4.863 +0.193 +0.877 +0.959 +0.981 +PackNet-SfM +M +CNN +0.111 +0.785 +4.601 +0.189 +0.878 +0.960 +0.982 +R-MSFM6 +M +CNN +0.112 +0.806 +4.704 +0.191 +0.878 +0.960 +0.981 +MF-ConvNeXt +M +CNN +0.111 +0.760 +4.533 +0.184 +0.878 +0.961 +0.982 +MF-SLaK +M +CNN +0.117 +0.866 +4.811 +0.199 +0.866 +0.947 +0.976 +MF-ViT +M +Trans +0.118 +0.942 +4.840 +0.193 +0.873 +0.956 +0.981 +MF-RegionViT +M +Trans +0.113 +0.893 +4.756 +0.193 +0.875 +0.954 +0.979 +MF-Twins +M +Trans +0.125 +1.309 +4.973 +0.197 +0.866 +0.948 +0.974 +MF-Hybird +M +Trans +0.104 +0.846 +4.580 +0.183 +0.891 +0.962 +0.982 +BTS +D +CNN +0.061 +0.250 +2.803 +0.098 +0.954 +0.992 +0.998 +AdaBins +D +CNN +0.058 +0.190 +2.360 +0.088 +0.965 +0.995 +0.999 +TransDepth +D +Trans +0.064 +0.252 +2.755 +0.098 +0.956 +0.994 +0.999 +DepthFormer +D +Trans +0.052 +0.156 +2.133 +0.079 +0.974 +0.997 +0.999 +GLPDepth +D +Trans +0.059 +0.187 +2.133 +0.079 +0.974 +0.997 +0.999 +TABLE 2: Quantitative results on the in-distribution +dataset. M is Monocular images, and D is GT depth. Trans +means Transformer. Bold is the best performance. +of large dataset). CNN-based architectures [49], [50] aim +to extract global information while utilizing the intrinsic +locality inductive bias. They also imitate the self-attention +of the Transformer using improved strategies such as large +kernel size, patchify stem [49], and Layer Norm [76]. +We train all networks using the KITTI Eigen split [14], +[77] consisting of 39,810 training and 4,424 validation data +whose size is 640 × 192. All experiments in this paper are +conducted with this KITTI-trained model whose results are +reported in Sec. 4.2–4.6. Following a previous work [9], +we remove around 5% of the total data for training to +address the infinite-depth problems that commonly occur in +dynamic scenes. We use typical error and accuracy metrics +for depth, absolute relative (Abs Rel), square relative (Sq +Rel), root-mean-square-error (RMSE), its log (RMSElog), and +the ratio of inliers following the work [9]. +4.2 +Evaluation on the in-distribution dataset +We conduct the evaluations with an in-distribution dataset, +where the KITTI-trained models are evaluated on 697 KITTI +test data. The qualitative results in Fig. 4 show that our self- +supervised method precisely preserves object boundaries. +This indicates that the encoder captures global context and +informative local features and transfers them to the decoder + +ELBAIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +6 +Abs Rel ↓ +Sq Rel ↓ +RMSE↓ +RMSElog ↓ +δ < 1.25 ↑ +baseline +0.113 +0.899 +4.783 +0.189 +0.882 ++ACM +0.113 +0.879 +4.820 +0.189 +0.879 ++FFD +0.112 +0.860 +4.803 +0.186 +0.879 ++ACM&FFD +0.104 +0.846 +4.580 +0.183 +0.891 +TABLE 3: Ablation study on ACM and FFD. +(a) Input image +(b) Results with ACM +(c) Results with FFD +(d) Results with ACM and FFD +Fig. 5: Results of MF-Ours with/without ACM and FFD. +for pixel-wise prediction. The quantitative results in Tab. 2 +show that MF-Ours outperforms all competitive methods. +We also evaluate the performance of models with the other +backbones, MF-(ConvNeXt/SLaK/RegionViT/Twins/ViT). +We employ only the encoder part of our method as a +backbone without changing other parts in this experi- +ment. Supervised methods outperform all self-supervised +methods regardless of backbones. MF-Ours and Depth- +Former achieve the best performance among the self- +supervised methods and supervised methods on the in- +distribution datasets, respectively. Overall, the performance +gap among all the competitive methods is marginal for the +in-distribution dataset, but the Transformer-based models +generally outperform the CNN-based models. +4.3 +Ablation study +In this section, we conduct the ablation study on MF-Ours +proposed in our previous work [51]. We use the same +models and datasets for the experiments used in Sec. 4.2. +4.3.1 +Effectiveness of the proposed modules. +We conduct an ablation study to demonstrate the effective- +ness of the proposed modules, ACM and FFD in Tab. 3. The +baseline is DPT [63]. The models with only the ACM module +or FFD module marginally improve the depth estimation +performance due to the absence of proper attention map +fusions. On the other hand, our MonoFormer with both +ACM and FFD significantly improves the performance. The +results show the proposed model achieves the best perfor- +mance in all measurements. The qualitative comparison in +Fig. 5 shows that the model with both ACM and FFD keeps +clearer object boundaries, even a small car in far depth. +4.3.2 +Visualization of attention maps. +We visualize the attention maps from the lower to higher +layers of Transformers. As shown in Fig. 6, the encoder in +the shallow layer extracts local region features. The deeper +the layer, the more global shape contexts are extracted. +Another observation is that ACM captures more detailed +attention at different depths of the encoder features. FFD +enhances the encoder features by fusing them with the at- +tention map from ACM. The fused feature captures features +from coarse to fine details. These experiments show that +our model is capable of accurate pixel-wise prediction as it +secures adequate local details. +# of layers +Abs Rel ↓ +RMSE ↓ +δ < 1.25 ↑ +δ < 1.252 ↑ +L = 2 +0.148 +5.327 +0.810 +0.939 +L = 3 +0.112 +4.745 +0.881 +0.962 +L = 4 +0.104 +4.580 +0.873 +0.962 +L = 5 +0.111 +4.692 +0.884 +0.962 +TABLE 4: Ablation study on the number of layers. +Input image +GT depth +(a) Attention map of CNN-Transformer encoder +(b) Attention map of ACM +(c) Feature map of FFD +Fig. 6: Visualization of attention and feature maps. The +left column from the second row is the maps from shallow +layers, whereas the right is the maps from deep layers. +4.3.3 +The number of encoder and decoder layers. +We compare the performance of MF-Ours according to the +number of encoder and decoder layers in Tab. 4. Each layer +has ACM and FFD modules. We find out that the model +with four transformer layers achieves the best performance. +The results are slightly degraded with the MF-Ours with 3 +or 5 layers. Therefore, we set L as four for MF-Ours in all +experiments in this paper. +4.4 +Evaluation on out-of-distribution datasets +We compare the generalization performance of all the com- +petitive models using public depth datasets captured at +common indoor environments, including office workspaces, +meeting rooms, and kitchen areas (RGBD [78], SUN3D +[40]), man-made indoor and outdoor environments (MVS +[42], ETH3D [43]), and synthetic scenes from graphics tools +(Scenes11 [42]). Interestingly, both qualitative and quantita- +tive results in Fig. 7 and Tab. 5–7 show that Transformer- +based models better generalization performance even in +the out-of-distribution datasets that have never been seen +during network training. Meanwhile, CNN-based models +predict unreliable depth maps that have lost the detail of +object boundaries and produce significant errors in texture- +less regions, such as the wall of a building. The experiments +demonstrate that Transformers are more robust than CNNs +for environmental changes. Another interesting observation +is that MF-ConvNeXt generally outperforms all the other +CNN-based models and produces depth results comparable +to other Transformer-based models. On the other hand, MF- +RegionViT fails to estimate depth accurately, even though +Transformer-based. A detailed analysis of why Transformers +and MF-ConvNeXt show better generalization performance +is provided in the following section. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +7 +RGBD +SUN3D +MVS +ETH3D +Scenes11 +(a) Input images & GT Depth maps +(b) Self-supervised CNN-based methods: Monodepth2, PackNet-SfM, R-MSFM6, MF-ConvNeXt, MF-SLaK (Top-to-Bottom) +(c) Self-supervised Transformer-based methods: MF-ViT, MF-RegionViT, MF-Twins, MF-Ours (Top-to-Bottom) +(d) Supervised CNN-based methods: BTS, AdaBins (Top-to-Bottom) +(e) Supervised Transformer-based methods: TransDepth, DepthFormer, GLPDepth (Top-to-Bottom) +Fig. 7: Depth map results on out-of-distribution (RGBD, SUN3D, MVS, ETH3D, and Scenes11) datasets. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +8 +Datasets +Train +Backbone +RGBD +SUN3D +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Monodepth2 +M +CNN +0.610 +0.508 +0.488 +0.292 +0.520 +0.681 +0.554 +0.535 +0.576 +0.324 +0.556 +0.718 +PackNet-SfM +M +CNN +0.593 +0.416 +0.460 +0.318 +0.562 +0.731 +0.466 +0.336 +0.471 +0.350 +0.612 +0.792 +R-MSFM6 +M +CNN +0.695 +0.553 +0.490 +0.261 +0.471 +0.627 +0.523 +0.406 +0.506 +0.310 +0.544 +0.721 +MF-ConvNeXt +M +CNN +0.346 +0.094 +0.208 +0.438 +0.706 +0.860 +0.254 +0.061 +0.182 +0.538 +0.833 +0.942 +MF-SLaK +M +CNN +0.481 +0.198 +0.272 +0.349 +0.607 +0.770 +0.379 +0.134 +0.258 +0.396 +0.673 +0.838 +MF-ViT +M +Trans +0.337 +0.098 +0.207 +0.475 +0.747 +0.874 +0.244 +0.059 +0.1743 +0.570 +0.844 +0.946 +MF-RegionViT +M +Trans +0.484 +0.225 +0.276 +0.369 +0.626 +0.791 +0.358 +0.121 +0.247 +0.413 +0.696 +0.855 +MF-Twins +M +Trans +0.323 +0.085 +0.194 +0.522 +0.749 +0.876 +0.256 +0.058 +0.178 +0.537 +0.832 +0.941 +MF-Ours +M +Trans +0.302 +0.085 +0.193 +0.532 +0.766 +0.888 +0.232 +0.051 +0.167 +0.586 +0.860 +0.953 +BTS +D +CNN +0.551 +0.197 +0.293 +0.278 +0.504 +0.695 +0.403 +0.145 +0.268 +0.380 +0.664 +0.842 +AdaBins +D +CNN +0.448 +0.126 +0.261 +0.269 +0.539 +0.760 +0.353 +0.112 +0.240 +0.400 +0.683 +0.854 +TransDepth +D +Trans +0.416 +0.138 +0.237 +0.409 +0.663 +0.826 +0.297 +0.087 +0.218 +0.511 +0.793 +0.908 +DepthFormer +D +Trans +0.465 +0.164 +0.260 +0.358 +0.614 +0.789 +0.277 +0.083 +0.204 +0.563 +0.836 +0.935 +GLPDepth +D +Trans +0.381 +0.111 +0.225 +0.442 +0.680 +0.837 +0.273 +0.074 +0.207 +0.533 +0.819 +0.928 +TABLE 5: Quantitative results on the out-of-distribution (Common indoor environments - RGBD, SUN3D) datasets. +Datasets +Train +Backbone +MVS +ETH3D +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Monodepth2 +M +CNN +0.471 +0.407 +0.503 +0.408 +0.661 +0.806 +0.449 +0.219 +0.267 +0.397 +0.629 +0.753 +PackNet-SfM +M +CNN +0.449 +0.295 +0.429 +0.397 +0.670 +0.837 +0.359 +0.119 +0.218 +0.436 +0.700 +0.839 +R-MSFM6 +M +CNN +0.550 +0.603 +0.583 +0.352 +0.591 +0.756 +0.441 +0.168 +0.249 +0.379 +0.624 +0.757 +MF-ConvNeXt +M +CNN +0.259 +0.060 +0.171 +0.582 +0.863 +0.949 +0.271 +0.063 +0.178 +0.527 +0.802 +0.920 +MF-SLaK +M +CNN +0.424 +0.149 +0.264 +0.375 +0.651 +0.820 +0.386 +0.115 +0.226 +0.375 +0.652 +0.821 +MF-ViT +M +Trans +0.254 +0.061 +0.179 +0.589 +0.865 +0.952 +0.249 +0.056 +0.167 +0.559 +0.826 +0.933 +MF-RegionViT +M +Trans +0.376 +0.124 +0.238 +0.436 +0.714 +0.863 +0.364 +0.098 +0.136 +0.410 +0.695 +0.853 +MF-Twins +M +Trans +0.247 +0.056 +0.171 +0.590 +0.861 +0.952 +0.270 +0.062 +0.182 +0.531 +0.801 +0.921 +MF-Ours +M +Trans +0.231 +0.052 +0.167 +0.617 +0.883 +0.960 +0.232 +0.050 +0.160 +0.587 +0.851 +0.946 +BTS +D +CNN +0.530 +0.280 +0.333 +0.343 +0.635 +0.825 +0.508 +0.199 +0.287 +0.324 +0.582 +0.758 +AdaBins +D +CNN +0.409 +0.157 +0.246 +0.392 +0.689 +0.870 +0.478 +0.152 +0.256 +0.300 +0.571 +0.758 +TransDepth +D +Trans +0.416 +0.191 +0.280 +0.440 +0.714 +0.860 +0.370 +0.108 +0.220 +0.427 +0.696 +0.846 +DepthFormer +D +Trans +0.369 +0.158 +0.259 +0.516 +0.793 +0.906 +0.285 +0.079 +0.182 +0.573 +0.816 +0.921 +GLPDepth +D +Trans +0.279 +0.075 +0.195 +0.551 +0.839 +0.948 +0.290 +0.071 +0.191 +0.519 +0.786 +0.905 +TABLE 6: Quantitative results on the out-of-distribution (Man-made in/outdoor environments - MVS, ETH3D) datasets. +Datasets +Train +Backbone +Scenes11 +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Monodepth2 +M +CNN +1.647 +0.763 +0.356 +0.312 +0.529 +0.671 +PackNet-SfM +M +CNN +2.065 +0.837 +0.330 +0.310 +0.530 +0.674 +R-MSFM6 +M +CNN +1.727 +0.726 +0.361 +0.280 +0.494 +0.636 +MF-ConvNeXt +M +CNN +1.827 +0.620 +0.251 +0.350 +0.576 +0.718 +MF-SLaK +M +CNN +1.913 +0.533 +0.263 +0.295 +0.525 +0.680 +MF-ViT +M +Trans +1.476 +0.314 +0.219 +0.419 +0.657 +0.776 +MF-RegionViT +M +Trans +1.866 +0.486 +0.257 +0.316 +0.546 +0.695 +MF-Twins +M +Trans +1.503 +0.316 +0.233 +0.387 +0.615 +0.749 +MF-Ours +M +Trans +1.220 +0.213 +0.214 +0.433 +0.667 +0.780 +BTS +D +CNN +2.446 +0.844 +0.269 +0.269 +0.496 +0.663 +AdaBins +D +CNN +2.583 +0.992 +0.265 +0.277 +0.512 +0.675 +TransDepth +D +Trans +1.830 +0.464 +0.233 +0.313 +0.565 +0.723 +DepthFormer +D +Trans +2.470 +0.930 +0.251 +0.324 +0.560 +0.708 +GLPDepth +D +Trans +1.811 +0.439 +0.226 +0.391 +0.625 +0.757 +TABLE 7: Quantitative results on the out-of-distribution +(Synthetic from graphics tools - Scenes11) datasets. +4.5 +Analysis of texture-/shape-bias on state-of-the-art +methods and various backbone networks +In this section, we verify the intrinsic properties of CNNs +and Transformers that lead to the robustness of depth +estimation to environmental changes. We hypothesize that +CNNs and transformers identify texture and shape infor- +mation as key visual cues for depth estimation, respectively, +inspired by the work [20], [28]. Thus, we synthetically +generate the texture-shifted datasets in Sec. 4.5.1. Then, we +validate the texture-/shape-bias of the model by compar- +ing the performance of the competitive methods on the +generated datasets in Sec. 4.5.2. Finally, we analyze the +internal representation of each neural network structure by +measuring centered kernel alignment (CKA) in Sec. 4.5.3. +4.5.1 +Texture-shifted datasets generation +In general, the texture is defined as an image’s spatial color +or pixel intensity pattern [79], [80], [81]. Inspired by [18], +we use three different texture shift strategies to investigate +the impact of textures on the inference process in depth: +texture smoothing (Watercolor), texture removal (Pencil- +sketch), and texture transfer (Style-transfer). The generated +images and the corresponding results of each model are +shown in Fig. 8. The first two images are watercolored +images, the middle two images and the last two images are +pencil-sketch and style-transferred images, respectively. The +following is a summary of the image generation: +Watercolor +We smooth the texture details from original +images while preserving the color cues using +cv2.stylization. The image looks like a +watercolor pictures. +Pencil-sketch +We remove both textures and color from original +images using cv2.pencilSketch. The image +seems like a sketch drawn with pencils. +Style-transfer +We apply a new texture to the original image +(context) by utilizing other images (style) using a +style transfer [82]. The textures of the original +images are changed. +4.5.2 +Evaluation on synthetic texture-shifted datasets +We compare the performance of all the competitive methods +and the modernized backbones on the synthetic texture- +shifted datasets, the same as the experiments in Sec. 4.4. The +qualitative and quantitative results are shown in Fig. 8 and +Tab. 8, respectively. As seen in Sec. 4.4, both the Transformer- +based models produce better depth maps than pure CNN- +based methods regardless of supervised or self-supervised +methods. In particular, the depth results from CNN-based +models are unrecognizable with the strong texture-shifted +datasets, especially the style-transferred data. We also ob- +serve that MF-ConvNeXt shows a tolerable depth map on +texture shift datasets and has lower errors than other CNN- +based models, although it is purely CNN-based. These +experiments support our two findings observed in Sec. 4.4. +One is that networks whose encoder consists of Transform- +ers are generally robust to texture changes. However, MF- +ConvNeXt and MF-RegionViT show different aspects from +the CNN-based and Transformer-based models, which are +the respective backbone models. In the following section, +we deeply analyze the intermediate feature representations +of all backbones to verify the reason for these observations. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +9 +Watercolor +Pencil-sketch +Style-transfer +(a) Input images +(b) Self-supervised CNN-based methods: Monodepth2, PackNet-SfM, R-MSFM6, MF-ConvNeXt, MF-SLaK (Top-to-Bottom) +(c) Self-supervised Transformer-based methods: MF-ViT, MF-RegionViT, MF-Twins, MF-Ours (Top-to-Bottom) +(d) Supervised CNN-based methods: BTS, AdaBins (Top-to-Bottom) +(e) Supervised Transformer-based methods: TransDepth, DepthFormer, GLPDepth (Top-to-Bottom) +Fig. 8: Depth map results on synthetic texture-shifted (Watercolor, Pencil-sketch, Style-transfer) datasets. +Datasets +Watercolor +Pencil-sketch +Style-transfer +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Monodepth2 +0.170 +1.345 +6.175 +0.750 +0.909 +0.960 +0.196 +1.522 +6.232 +0.691 +0.898 +0.962 +0.247 +2.440 +7.525 +0.617 +0.828 +0.920 +PackNet-SfM +0.174 +1.364 +6.334 +0.742 +0.906 +0.961 +0.204 +1.569 +6.568 +0.670 +0.888 +0.957 +0.205 +1.606 +6.778 +0.672 +0.876 +0.948 +R-MSFM6 +0.194 +1.613 +7.173 +0.696 +0.876 +0.943 +0.217 +1.698 +6.719 +0.647 +0.872 +0.951 +0.294 +2.788 +8.579 +0.521 +0.770 +0.894 +MF-ConvNeXt +0.159 +1.175 +5.971 +0.774 +0.921 +0.969 +0.176 +1.280 +6.582 +0.728 +0.907 +0.965 +0.219 +1.722 +7.214 +0.634 +0.861 +0.942 +MF-SLaK +0.170 +1.272 +6.316 +0.755 +0.907 +0.961 +0.268 +2.268 +8.042 +0.553 +0.811 +0.915 +0.224 +1.859 +7.423 +0.648 +0.855 +0.934 +MF-ViT +0.152 +1.196 +5.668 +0.799 +0.932 +0.973 +0.174 +1.311 +5.770 +0.756 +0.920 +0.967 +0.186 +1.379 +6.652 +0.705 +0.898 +0.959 +MF-RegionViT +0.179 +1.341 +6.309 +0.732 +0.902 +0.958 +0.260 +2.012 +7.491 +0.545 +0.832 +0.939 +0.242 +2.030 +7.944 +0.587 +0.835 +0.927 +MF-Twins +0.166 +1.295 +6.558 +0.751 +0.911 +0.963 +0.205 +1.630 +7.512 +0.655 +0.871 +0.949 +0.232 +1.974 +8.045 +0.599 +0.839 +0.931 +MF-Ours +0.140 +1.053 +5.665 +0.815 +0.936 +0.975 +0.151 +1.084 +5.615 +0.786 +0.934 +0.976 +0.175 +1.307 +6.435 +0.728 +0.906 +0.962 +BTS +0.225 +2.469 +8.503 +0.599 +0.753 +0.844 +0.247 +2.180 +7.956 +0.509 +0.745 +0.871 +0.284 +3.233 +9.300 +0.512 +0.674 +0.770 +AdaBins +0.301 +2.938 +8.761 +0.415 +0.642 +0.782 +0.165 +1.133 +5.812 +0.721 +0.889 +0.957 +0.317 +3.500 +9.594 +0.414 +0.626 +0.751 +TransDepth +0.111 +0.728 +4.886 +0.855 +0.955 +0.985 +0.136 +0.987 +5.774 +0.775 +0.926 +0.975 +0.146 +1.127 +5.910 +0.761 +0.911 +0.964 +DepthFormer +0.135 +1.102 +5.983 +0.783 +0.905 +0.955 +0.162 +1.290 +6.498 +0.713 +0.885 +0.953 +0.274 +2.766 +8.366 +0.501 +0.691 +0.801 +GLPDepth +0.099 +0.573 +4.121 +0.872 +0.960 +0.986 +0.092 +0.513 +3.950 +0.874 +0.970 +0.992 +0.099 +0.513 +3.950 +0.874 +0.970 +0.992 +TABLE 8: Quantitative results on synthetic texture-shifted (Watercolor, Pencil-sketch, Style-transfer) datasets. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +10 +Monodepth2 +PackNet-SfM +R-MSFM6 +MF-ConvNeXt +MF-SLaK +BTS +AdaBins +MF-ViT +MF-RegionViT +MF-Twins +MF-Ours +TransDepth +DepthFormer +GLPDepth +Watercolor +1 +0.8 +0.6 +0.4 +0.2 +0 +Pencil-sketch +1 +0.8 +0.6 +0.4 +0.2 +0 +Style-transfer +1 +0.8 +0.6 +0.4 +0.2 +0 +CKA +Methods +CKA +Methods +CKA +Methods +Fig. 9: Box-and-Whisker plot of CKA similarity of all competitive methods on three different synthetic texture-shifted +datasets. The blue and red tones indicate CNN-based and Transformer-based models, respectively. The circle and triangle +indicate self-supervised and supervised methods, respectively. +Original Image (I𝑎) +Texture-shifted Image (I𝑏) +Encdoer (𝐸) +𝐸 I𝑎 = 𝑧𝑎 +CKA(𝑧𝑎, 𝑧𝑏) +𝐸 I𝑏 = 𝑧𝑏 +Fig. 10: Illustration of CKA [53] similarity computation. +4.5.3 +Analysis on feature representation from backbones +To analyze the internal properties of CNNs and Trans- +formers, we employ the centered kernel alignment (CKA), +which is the similarity measure of internal representations of +neural networks by following previous works [53], [54], [83]. +The CKA is widely used to analyze feature representations +of neural networks [53], [84], because of its invariant prop- +erties to the orthogonal transformation of representations +and isotropic scaling [54]. We freeze the encoder E of the +depth networks trained on the original KITTI dataset. We +extract features za, zb from the last layer of the encoder by +passing the original image Ia and texture-shifted image Ib, +respectively. Then, the CKA is computed given K = zazT +a +and L = zbzT +b with m number of samples as follows: +CKA(K, L) = +HSIC(K, L) +� +HSIC(K, K)HSIC(L, L), +(7) +HSIC(K, L) = (KHLH)/(m − 1)2, +(8) +where H is the centering matrix Hn = In − 1 +n11T . The +overall process is illustrated in Fig. 10. We use 697 image +pairs from the KITTI Eigen dataset and their correspond- +ing texture-shifted datasets. We compute the CKA with +three types of texture-shifted datasets, including watercolor, +pencil-sketch, and style-transfer, as shown in Fig. 9. +The results of feature similarity show a similar aspect of +the quantitative results on synthetic texture-shifted datasets +in Tab. 8. MF-Ours and GLPDepth extract the most similar +features from the original and texture-shifted datasets in +the self-supervised and supervised methods, respectively. +Generally, Transformer-based models (MF-ViT, MF-Twins, +GLPDepth, MF-Ours, TransDepth, DepthFormer) extract +features of texture-shifted images similar to the original +images. On the other hand, most CNN-based models (Mon- +odepth2, PackNet-SfM, R-MSFM6, BTS, AdaBins) extract +inconsistent features from original and texture-shifted im- +ages. The results are consistently observed regardless of the +types of texture changes. It demonstrates that the robustness +to environmental changes comes from consistent feature +extraction from encoders. These results also support our +observations in Sec. 4.5.2 that the Transformers have a +strong shape-bias whereas CNNs have a strong texture bias. +Moreover, models with shape-biased representation show +better generalization performance than models with texture +bias for monocular depth estimation. We also observe that +the pure Transformer-based methods (MF-ViT, MF-Twins, +GLPDepth) and CNN-Transformer hybrid methods (MF- +Ours, TransDepth, DepthFormer) show similar properties. +The experiments provide us with some interesting observa- +tions on modernized network structures. +First, it is noteworthy that MF-ConvNeXt, one of the +CNN-based models, shows a relatively higher CKA similar- +ity than the other CNN-based models. Even the similarity +of the models is on par with Transformer-based models. +ConvNeXt consists of a patchify stem, and a depth-wise +convolution [85] with a large receptive field that mimics +the self-attention mechanism using Transformer’s macro +design [49]. The design intuition of ConvNeXt makes that +ConvNeXt can focus on global information, similar to self- +attention. Because of the modernized macro design, MF- +ConvNeXt consisting of only convolutional layers is robust +to texture-shifted datasets and has a strong shape-bias. +Second, unlike other Transformer-based models, MF- +RegionViT is sensitive to texture changes, similar to CNN- +based models. It is highly biased toward texture informa- +tion rather than shape information, even though it is a +transformer-based model. RegionViT [47] employs region- +to-local attention, which is getting interaction between local +regions. It improves the locality of the networks through lo- +cal attention by capturing fine-grained spatial information. +We believe that the strong locality makes MF-RegionViT rely +heavily on spatial information. +Based on these experiments, we find that a self-attention +module capturing global information, strengthen the shape- +bias of the network. We also observe that the locality from +local attention or the intrinsic property of CNNs improves +the texture-bias of the network. In terms of network design, + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +11 +RobotCar +Foggy CityScapes +Rainy CityScapes +(a) Input images +(b) Self-supervised CNN-based methods: Monodepth2, PackNet-SfM, R-MSFM6, MF-ConvNeXt, MF-SLaK (Top-to-Bottom) +(c) Self-supervised Transformer-based methods: MF-ViT, MF-RegionViT, MF-Twins, MF-Ours (Top-to-Bottom) +(d) Supervised CNN-based methods: BTS, AdaBins (Top-to-Bottom) +(e) Supervised Transformer-based methods: TransDepth, DepthFormer, GLPDepth (Top-to-Bottom) +Fig. 11: Depth map results on real-world texture-shifted (RobotCar, Foggy and Rainy CityScapes) datasets. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +12 +Datasets +Oxford RobotCar +Foggy CityScapes +Rainy CityScapes +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Abs Rel +Sq Rel +RMSE +δ < 1.25 +δ < 1.252 +δ < 1.253 +Monodepth2 +1.039 +0.044 +0.159 +0.204 +0.406 +0.564 +0.438 +0.060 +0.109 +0.514 +0.757 +0.853 +1.215 +2.194 +1.289 +0.183 +0.348 +0.495 +PackNet-SfM +1.080 +0.045 +0.159 +0.193 +0.378 +0.536 +0.457 +0.055 +0.102 +0.545 +0.782 +0.873 +1.515 +1.771 +1.169 +0.234 +0.444 +0.609 +R-MSFM6 +0.951 +0.043 +0.160 +0.235 +0.362 +0.469 +0.435 +0.062 +0.124 +0.442 +0.697 +0.809 +1.310 +3.028 +1.509 +0.171 +0.315 +0.441 +MF-ConvNeXt +0.714 +0.037 +0.159 +0.310 +0.546 +0.705 +0.353 +0.031 +0.090 +0.638 +0.864 +0.922 +0.658 +0.332 +0.396 +0.475 +0.739 +0.900 +MF-SLaK +0.916 +0.041 +0.160 +0.239 +0.456 +0.623 +0.839 +0.151 +0.205 +0.209 +0.440 +0.652 +0.856 +0.510 +0.460 +0.356 +0.631 +0.822 +MF-ViT +0.935 +0.042 +0.159 +0.235 +0.444 +0.602 +0.368 +0.033 +0.097 +0.591 +0.845 +0.917 +0.803 +0.891 +0.868 +0.298 +0.545 +0.709 +MF-RegionViT +1.067 +0.041 +0.160 +0.239 +0.456 +0.623 +0.643 +0.088 +0.167 +0.337 +0.605 +0.782 +0.798 +0.773 +0.805 +0.251 +0.493 +0.676 +MF-Twins +0.831 +0.037 +0.159 +0.300 +0.516 +0.686 +0.350 +0.029 +0.087 +0.648 +0.862 +0.920 +0.795 +0.825 +0.829 +0.285 +0.526 +0.688 +MF-Ours +0.869 +0.040 +0.159 +0.292 +0.555 +0.691 +0.316 +0.025 +0.080 +0.674 +0.879 +0.929 +0.788 +0.707 +0.752 +0.323 +0.574 +0.738 +BTS +0.707 +0.039 +0.160 +0.340 +0.604 +0.723 +1.092 +0.294 +0.117 +0.572 +0.716 +0.794 +1.366 +1.632 +0.710 +0.379 +0.623 +0.756 +AdaBins +0.637 +0.038 +0.160 +0.371 +0.640 +0.771 +1.441 +0.524 +0.185 +0.516 +0.667 +0.744 +1.299 +1.452 +0.668 +0.330 +0.585 +0.771 +TransDepth +0.531 +0.037 +0.159 +0.430 +0.645 +0.768 +0.631 +0.075 +0.069 +0.631 +0.797 +0.860 +1.034 +1.003 +0.793 +0.485 +0.698 +0.771 +DepthFormer +0.550 +0.037 +0.159 +0.420 +0.629 +0.756 +0.669 +0.083 +0.070 +0.636 +0.794 +0.857 +1.039 +1.016 +0.816 +0.473 +0.690 +0.764 +GLPDepth +0.538 +0.037 +0.159 +0.422 +0.673 +0.780 +0.594 +0.070 +0.103 +0.561 +0.799 +0.872 +1.030 +0.928 +0.560 +0.513 +0.742 +0.822 +TABLE 9: Quantitative results on real-world texture-shifted (RobotCar, Foggy and Rainy CityScapes) datasets. +a self-attention mechanism is ultimately key to generaliza- +tion. Additionally, the locality of the network weakens gen- +erality, even if it helps to optimize in the training process. +4.6 +Evaluation on real-world texture-shifted datasets +In addition, we conduct experiments on practical texture +changes that can occur in real driving scenarios to demon- +strate their applicability to the real-world. We evaluate all +models using practical texture-shifted datasets consisting of +the night (Oxford RobotCar [44]), foggy (Foggy CityScapes +[46]), and rainy driving scenes (Rainy CityScapes [45]). +Overall, the experiment demonstrates that the properties of +CNNs and Transformers identified in Sec. 4.5 are similarly +observed in real-world texture-shifted scenarios. Quantita- +tive and qualitative results and detailed analyzes for each +network are described in the following paragraphs. +The depth map results of each model are shown in +Fig. 11. Similar to the experiments in Sec. 4.5.2, Transformer- +based models show plausible depth maps, except MF- +RegionViT. On the other hand, CNN-based models fail to es- +timate depth even for objects such as a car in scenes similar +to the training dataset, with the exception of MF-ConvNeXt. +The quantitative results in Tab. 9 also show that CNN- +based models have higher errors than Transformer-based +models. MF-SLaK is no different from other CNN models, +but MF-ConvNeXt shows low errors and feasible depth +maps by mimicking Transformers, as shown in Fig. 11. +MF-Twins show similar results to other Transformer-based +models because only a weak locality is added compared to +MF-RegionViT. However, MF-RegionViT indicates texture- +biased results like CNN-based models due to its strong +locality. +The results on rainy and foggy datasets (Rainy and +Foggy CityScapes) also indicate similar aspects to the night +scenes. Transformer-based models estimate the plausible +depth and distinguish even tiny objects such as bollards. +On the contrary, CNN-based models predict incorrect depth +maps. In particular, CNN-based models exhibit significant +errors in areas such as roads despite similar texture to +training datasets. MF-ConvNeXt and MF-Twins also infer +plausible depth maps regardless of changes in the weather +environment, but MF-RegionViT and MF-SLaK estimate +bizarre depth maps. In these experiments, we observe that +textural shifts like weather and illumination changes con- +fuse CNN-based models. We also find that CNN-based +models are affected by changes in texture as well as loss +of texture information, which can be a fatal problem in real- +world driving applications. +5 +CONCLUSION +In this paper, we have proposed a self-supervised monoc- +ular depth estimation method called MonoFormer (MF- +Ours). More importantly, we have presented an in-depth +analysis of the generalization performance of various mod- +ernized backbone structures as well as state-of-the-art +self-supervised and supervised methods for monocular +depth estimation using various datasets. The in-distribution +datasets are used to compare common performance on the +KITTI benchmark, while the out-of-distribution and texture- +shifted datasets are used to compare the generalization +performance. We deeply analyze the properties of the fea- +tures extracted from each model using the synthetic texture- +shifted datasets. Finally, we demonstrate the applicability of +the model in real-world scenarios of changing environments +using day-night, foggy and rainy datasets. +Through extensive experiments, we observe that MF- +Ours and GLPDepth have the best generalization perfor- +mance among all self-supervised and supervised methods, +respectively. The supervised method, GLPDepth, achieves +the best generalization performance among all monocular +depth estimation methods. More interestingly, we provide +three important observations about the generality of monoc- +ular depth estimation. First, CNN-based models heavily +rely on texture information to recognize scenes and objects, +while Transformer-based models use shape information to +a greater degree for the monocular depth estimation task. +Second, texture-biased representations result in poor gen- +eralization performance for environmental changes such as +differences in illumination and weather. In contrast, shape- +biased representations are more robust to such texture- +shifted environments. Lastly, ConvNeXt and RegionViT are +CNN-based and Transformer-based, respectively, but have +properties different from those of the backbone structure. It +shows that the texture-bias and shape-bias are not the intrin- +sic properties of CNNs and Transformers, respectively. In- +stead, the intrinsic locality of CNNs induces texture-biased +characteristics, while the self-attention mechanism, the base +layer of Transformers, induces shape-biased properties. +We believe our observations provide valuable insights +into best practices in network design not only for monocular +depth estimation but also for a variety of dense prediction +tasks such as semantic segmentation, depth completion, +normal estimation, etc. Based on these observations, the best +way to design generalized networks for dense prediction +is to utilize multi-self-attention (MSA) from Transformers +and supplement the local information loss of global self- +attention by using weak local attention as an auxiliary. + +IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. XX, NO. XX, 2023 +13 +REFERENCES +[1] +F. Xue, G. Zhuo, Z. Huang, W. Fu, Z. Wu, and M. H. 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Rosing, “Depthwise convolution is +all you need for learning multiple visual domains,” in AAAI, 2019, +pp. 8368–8375. +Jinwoo Bae received the B.S. degree in the +Department of Electrical and Information Engi- +neering from the Seoul National University of +Science and Technology in 2020. He is currently +pursuing the M.S. degree in the Department of +Electrical Engineering and Computer Science at +DGIST. His research interests include 3D vision +such as depth estimation, robot vision, and multi- +camera framework. He was a research intern at +KIST (Seoul, Korea) in 2020 and ETRI (Pangyo, +Korea) in 2019. +Kyumin Hwang received the B.S. degree in the +Department of Computer Science and Engineer- +ing from the Kyungpook National University in +2020, and the M.S. degree in the Department +of Information and Communication Engineering +from DGIST in 2022. He is currently pursuing +the Ph.D. degree in the Department of Electrical +Engineering and Computer Science at DGIST. +His research interests include 3D computer vi- +sion, including multi-view stereo for 3D recon- +struction, and deep learning with geometry for +autonomous driving. He was a research intern at ETRI (Daegu, Korea) +in 2019. +Sunghoon Im received the B.S. degree in the +Department of Electronic Engineering from So- +gang University in 2014, and the M.S. and Ph.D. +degree in the School of Electrical Engineering +from KAIST in 2016 and 2019. He joined the +Department of Electrical Engineering and Com- +puter Science at DGIST, Daegu, Korea, in 2019, +where he is currently working as an assistant +professor. He was a recipient of Microsoft Re- +search Asia fellowship and Global Ph.D. fellow- +ship from NRF of Korea. His research interests +include computer vision, robot vision, and machine learning. + diff --git a/VdE1T4oBgHgl3EQfbATf/content/tmp_files/load_file.txt b/VdE1T4oBgHgl3EQfbATf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2969850e99476751e636b20104c6506a77a6745d --- /dev/null +++ b/VdE1T4oBgHgl3EQfbATf/content/tmp_files/load_file.txt @@ -0,0 +1,2252 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf,len=2251 +page_content='IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 1 A Study on the Generality of Neural Network Structures for Monocular Depth Estimation Jinwoo Bae, Kyumin Hwang and Sunghoon Im Abstract—Monocular depth estimation has been widely studied, and significant improvements in performance have been recently reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' However, most previous works are evaluated on a few benchmark datasets, such as KITTI datasets, and none of the works provide an in-depth analysis of the generalization performance of monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' In this paper, we deeply investigate the various backbone networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='CNN and Transformer models) toward the generalization of monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' First, we evaluate state-of-the-art models on both in-distribution and out-of-distribution datasets, which have never been seen during network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Then, we investigate the internal properties of the representations from the intermediate layers of CNN-/Transformer-based models using synthetic texture-shifted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Through extensive experiments, we observe that the Transformers exhibit a strong shape-bias rather than CNNs, which have a strong texture-bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also discover that texture-biased models exhibit worse generalization performance for monocular depth estimation than shape-biased models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We demonstrate that similar aspects are observed in real-world driving datasets captured under diverse environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Lastly, we conduct a dense ablation study with various backbone networks which are utilized in modern strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The experiments demonstrate that the intrinsic locality of the CNNs and the self-attention of the Transformers induce texture-bias and shape-bias, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Index Terms—Monocular depth estimation, Out-of-Distribution, Generalization, Transformer !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 1 INTRODUCTION Monocular depth estimation (MDE) is widely utilized for spatial recognition technologies such as autonomous driv- ing [1], [2], [3] or AR/VR [4], [5] because of its porta- bility and cost-effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Various MDE processes have achieved remarkable progress over the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Most previous works [6], [7], [8], [9], [10], [11], [12], [13] con- centrate on boosting performance using limited benchmark datasets, especially the KITTI dataset [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' However, these works have not provided a deep investigation into what MDE networks have learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' This means that they cannot guarantee the model’s behavior is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' To examine the interpretability of MDE networks, one previous work [15] employs a target network for MDE model analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Another approach [16] uses a synthetic dataset, which contains the changes in image contents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', camera pose).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' However, universality questions about other networks still remain because of the fixed specific network [6] in certain experi- mental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Recently, several works [17], [18], [19], [20] aim to ana- lyze interpretability, taking inspiration from convolutional neural networks (CNNs) whose designs are based on hu- man visual processes [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Then, how can humans efficiently extract and recognize important information from complex scenes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Compared to other cues like texture or color, the biological vision system considers the object’s shape to be the single most important visual cue [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' This allows humans, including little kids, to easily distinguish an object from a line drawing or a silhouette image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Many J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Bae, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Hwang and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Im are with the Department of Electrical Engineering and Computer Science, DGIST, Daegu, 42988, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' {sjg02122, kyumin, sunghoonim}@dgist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='kr Monodepth2 PackNet-SfM R-MSFM6 MF-SLaK BTS AdaBins MF-RegionViT MF-Twins MF-ConvNeXt TransDepth DepthFormer MF-ViT MF-Ours GLPDepth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='285 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='8 1 Shape-biased Texture-biased Error (Abs Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=') CKA Similarity (Mean) CNN-based Transformer-based Self-supervised: Supervised: Transformer-based CNN-based Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 1: Analysis on the generality of state-of-the-art models and modernized network structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The x-axis is the CKA similarity indicating whether the network is shape biased or texture biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The y-axis shows Absolute Relative error where a lower number is better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We use the synthetic texture-shifted datasets described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' researchers believe that CNN would also behave similarly [24], [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' However, in contrast to belief, recent studies [18], [20], [27], [28] have discovered that CNNs are heavily predisposed to recognize textures rather than shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' CNN- based models accurately assign labels to images even when the shapes of the structures are disturbed [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' On the other hand, CNN models are unable to accurately predict labels in a texture-removed image with a well-preserved shape [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Transformers [32] achieve outstanding perfor- mance on various computer vision tasks [33], [34], [35] and have attracted much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Moreover, many works [27], [36], [37], [38] reveal that Transformers have a strong shape arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='03169v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='CV] 9 Jan 2023 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 2 bias notwithstanding the lack of a spatial locality, compared to CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Due to its strong shape bias, Transformer is considered more robust than a CNN and more similar to human cognitive processes [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Then, how does this observation affect the MDE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We hypothesize two things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' First, the network’s generality will differ depending on the texture-/shape-bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' To verify the generality, we evaluate state-of-the-art MDE models trained on KITTI [14] using five public depth datasets (SUN3D [40], RGBD [41], MVS [42], Scenes11 [42], and ETH3D [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also conduct experiments on six different texture-shifted datasets, including three synthetic texture-shifted datasets (Watercolor, Pencil-sketch, Style-transfer) and three real- world texture-shifted datasets (Oxford Robotcar [44], Rainy Cityscapes [45], Foggy Cityscapes [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Through these ex- periments, we confirm that texture-bias is vulnerable to generalization while shape-bias shows robust generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Second, the texture-/shape-bias are related to the intrinsic properties of CNN and Transformer struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The modernized Transformer-based model [47], [48] imitates the intrinsic locality inductive bias of the CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The modernized CNN-based model [49], [50] is designed to mimic the self-attention of the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We finally conduct ablation studies on the generalization performance, and the texture-/shape-bias of various modernized back- bone structures that originate from a specific design for each CNN and Transformer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', locality and self-attention) such as RegionViT [47], Twins [48], ConvNeXt [49] and SLaK [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' This paper extends our previous work in [51] which proposes a new self-supervised monocular depth estima- tion network adopting Transformers [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The Transformer- based network shows the generalization performance on various environments on depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' While the short version [51] only addresses self-supervised MDE models, the extended version deals with full MDE models, includ- ing both self-supervised and supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' More- over, we deeply analyze the reason why the proposed net- work achieves better-generalized performance rather than CNN-based models by comparing the performance, and intermediate-layer feature similarity [53], [54] on various texture-shifted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 1, we observe that the Transformer structure has more shape-biased prop- erties than the CNN structure, which has texture-biased properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' It enables the Transformer-based models to achieve better generalization performance than the CNN- based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also experiment extensively on modern backbone architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', ConvNeXt [49], RegionViT [47]) to investigate the origin of the texture-/shape-biased prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Through these extensive experiments, we observe that the intrinsic locality of CNNs induces texture-biased characteristics, while the self-attention mechanism, the base layer of Transformers, induces shape-biased properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='1 Self-supervised monocular depth estimation Self-supervised depth estimation methods [8], [9], [12], [55], [56], [57] simultaneously train depth and motion network by imposing photometric consistency loss between the target and source images warped by the predicted depth and motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' SfMLearner [55] first proposes a depth and ego- motion estimation pipeline without the ground truth depth and motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Monodepth2 [8] presents a minimum repro- jection loss to handle occlusions, a full-resolution multi- scale sampling method to reduce visual artifacts, and an auto-masking loss to ignore outlier pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' PackNet-SfM [9] introduces packing and unpacking blocks that leveraged 3D convolutions to learn the dense appearance and geometric information in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' HR-Depth [12] analyzes the reason for the inaccurate depth prediction in large gradient regions and designed a skip connection to extract representative features in high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 Supervised monocular depth estimation Supervised methods [10], [13], [58], [59], [60] use a ground truth depth acquired from RGB-D cameras or LiDAR sen- sors for supervision in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' They also estimate depth maps given a single image as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' BTS [58] adopts a local planar guidance layer to densely encoded features to preserve local detail and create depth map sharpness at multi-stages in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' AdaBins [10] estimate the depth by linear combinations of bin centers that are adaptively decided per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The bin building block divides the depth range of the image into bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' LapDepth [59] employs a Laplacian pyramid at the multi-level upscaling encoder to preserve local detail, such as a boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' It also trains stably by utilizing weight standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' GLPDepth [60] proposes a hierarchical transformer encoder to capture the global context of images and a selective feature fusion mod- ule to connect multi-scale local features and global context information at the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The feature fusion module helps the decoder become more powerful, even if the decoder is lightweight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' DepthFormer [13] proposes leveraging the transformer’s effective attention mechanism and the spatial inductive bias of the CNN to capture long-range correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' It also uses a hierarchical aggregation and heterogeneous interaction module to enhance the affinity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3 Vision Transformers Recently, Transformers [52] has shown promise for solving computer vision tasks such as image classification [32], [61], object detection [33], and dense prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' [11], [62], [63], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' ViT [32] employs a Transformers architecture on fixed- size image patches for image classification for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' DeiT [61] utilizes Knowledge distinction on ViT architecture, showing good performance only with the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' DETR [33] proposes the direct set prediction approach, which simplifies the object detection pipeline, based on a CNN-Transformer network and bipartite matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Some works [11], [63] have employed Transformers for monoc- ular depth estimation in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' DPT [63] introduces a dense prediction using a Transformer as the basic computational building block of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' These works show generalized performance, but they require a large number of training datasets captured in diverse environments with ground truth depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' TransDepth [11] utilizes multi-scale information to capture local level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Previous works lack studies on whether models behave as intended on another domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The work [65] aggregates the attention map between a single frame IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 3 L2-Norm Channel Norm tanh × × + + × 𝜶 𝜸 𝜷 E(𝜙) Transformer Feature Fusion Decoder Norm Multi-Head Attention Norm MLP + + Linear Linear Linear Scaled Dot-Product Attention 𝑸 Concatenate Linear 𝑽 𝑲 Transformer Transformer Transformer Residual Block Unit Transformer Encoder ACM ACM ACM ACM Feature Fusion Decoder Feature Fusion Decoder Feature Fusion Decoder Feature Fusion Decoder Attention Connection Module 𝑍!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 𝑍" 𝑍# 𝑍$ 𝑋!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 𝑋" 𝑋# 𝑋$ 𝑋% Image 𝐼 Depth 𝐷 Linear Projection 𝐹 𝜙 Embedding function Element-wise multiplication Element-wise addtion Encoder Feature Fused Feature Attention Map Conv Position Attention Channel Attention 𝐴& \' 𝐴& ( 𝒘𝒑𝑨𝒑 𝒘𝒄𝑨𝒄 𝑍 𝑋 Conv + + × × Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 2: Overall Architecture of MonoFormer (MF-ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' and cross frames to refine the attraction map to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The works [11], [65] only focus on improving performance on benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4 Modernized Architecture Despite the tremendous success of Transformers for vision tasks, a Transformer requires a large model and data size to achieve state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Recently, many works [47], [48], [49], [50], [66] utilize local attention on the Trans- former to alleviate problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The Swin Transformer [66] de- signs a pyramid structure network differently from a vision transformer [32], which is an isotropic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' It achieves state-of-the-art performance in classification tasks with local attention using the sliding window strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Twins [48] employs global attention for the non-overlapping region and local attention to perform better under limited condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' RegionViT [47] shows state-of-the-art performance on several visual tasks using regional-to-local attention, which alleviates the weakness of the standard attention mecha- nism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' ConvNeXt [49] modernizes convolution neural layers such as depth-wise convolution and utilizes techniques such as Patchify stem and achieves competitive performance with the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' SLaK [50] attempts to design the network with an extremely large kernel size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', 51×51) to leverage the sparsity that is observed from the human visual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' SLaK [50] repeats the Prune-and-Grow step in training to optimize the sparse kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3 METHOD This section describes MonoFormer, which is an encoder- decoder structure with a multi-level feature fusion module for self-supervised monocular depth estimation described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' An Attention Connection Module (ACM) learns the channel and position attentions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' A Feature Fusion Decoder (FFD) adaptively fuses the encoder features with the attention maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='1 Transformer-based Encoder MonoFormer [51] is composed of a CNN and Transformer for an image encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The encoder employs ResNet50 [67] as the CNN backbone (E(θ) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 2), and L number of Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' MonoFormer sets the L to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The encoder is used to extract a feature map F ∈ RC×H×W from an input image I, and the map is divided into N (= H 16 × W 16 ) number of patches pn ∈ RC×16×16, which is utilized as the input of the first Transformer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Following the work [63], Mono- Former additionally use a special token ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' MonoFormer input the patch tokens pn, n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', N} and the special token ts with a learnable linear projection layer E as follows: Z0 = [ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' p1E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' p2E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' pNE], (1) where Z0 is the latent embedding vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The CNN- Transformer encoder comprises a Multi-head Self-Attention (MSA) layer, a Multi-Layer Perceptron (MLP) layer, and Layer Norm (LN) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The MLP is built with GELU non- linearity [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The LN is used before each block, and residual connections are used after every block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Self-Attention (SA) at each layer l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', L} is processed with the learnable parameters W m Q , W m K , W m V ∈ RC×d of {query, key, value} weight matrices, given the embedding vector Zl ∈ RN×C as follows: SAm l−1 = softmax �Qm l−1(Km l−1)T √ d � V m l−1, m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', M}, Qm l−1 = Zl−1W m Q , Km l−1 = Zl−1W m K , V m l−1 = Zl−1W m V , (2) where M and d are the number of SA blocks and the di- mension of the self-attention block, which is the same as the dimension of the weight matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The Multi- head Self-Attention (MSA) consists of the M number of SA blocks with the learnable parameters of weight matrices W ∈ RMd×C as follows: MSAl−1 = Zl−1 + concat(SA1 l−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' SA2 l−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' SAM l−1)W, Zl = MLP(LN(MSAl−1)) + MSAl−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' (3) tsIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 4 16×16 16×16 16×16 32×32 16×16 8×8 Local Patch Region Patch 16×16 32×32 64×64 Other arrows are omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4×4 Local Attention Global Attention Other arrows are omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 𝐶 4𝐶 2𝐶 𝑊 ∈ ℝ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' "×$" ”patchify” Stem Depth wise conv Depth Conv Inverted residual block narrow wide narrow Input 51×5 5×5 5×51 + ViT [32] RegionViT [47] Twins [48] ConvNeXt [49] SLaK [50] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3: Illustrations of various modernized architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' This Transformer layer is repeated L times with unique learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The outputs of the Transformers {Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', ZL} are utilized as the input of the following layers ACM and FFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The hybrid encoder can be replaced with the other backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We evaluate all the performance of our depth estimation networks with various backbones by changing the encoder to the modernized backbone struc- tures, such as ViT [32], RegionViT [47], Twins [48], Con- vNeXt [49] and SLaK [50] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 Attention Connection Module (ACM) The skip connection module of MonoFormer, ACM, that extracts global context attention and a semantic presenta- tion from the given features Zl, l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The skip connection, widely utilized for dense prediction tasks [69], helps keep the fine detail by directly transferring the spatial information from the encoder to the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' However, because of its simplicity, it is challenging for the naive skip connection method to preserve local detail like object boundaries [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' To address the issue, The ACM is pro- posed that extracts attention weight from the spatial and channel domains inspired by [71] It consists of position attention, channel attention modules, and a fusion block that gathers important information from two attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The position attention module produces a position attention map Ap l ∈ RC×N as follows: Ap l = softmax(Qp l (Kp l )T)V p l , (4) where Qp l , Kp l and V p l are the query, key, and value matrices computed by passing Zl through a single convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The channel attention module directly calculate the channel attention map Ac l ∈ RC×N by computing the gram matrix of Zl as follows: Ac l = softmax(ZlZT l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' (5) The position attention map Ap l and channel attention map Ac l enhance the feature representation by capturing long- range context and exploiting the inter-dependencies be- tween each channel map, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' These two attention maps are utilized in the following section, which highlights the importance of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3 Feature Fusion Decoder (FFD) The FFD gets the encoder features Zl, the attention maps Ap l , Ac l , and the output feature XL of the last Transformer layer passed through a Residual convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The features XL−l+1, l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', L} are fused through the de- coder with a single Convolutional layer (Conv) and Channel Normalization (CN) with learnable parameters α, β and γ as follows: XL−l = ˆ XL−l[1 + tanh(γ(CN(α|| ˆ XL−l||2 + β)], ˆ XL−l = Conv(wpAp l Zl + wcAc l Zl + Zl) + XL−l+1, (6) where wp and wc are the learnable parameters that deter- mine the importance of the position and channel attentions [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The parameter α works so that each channel can learn about each other individually, and γ and β control the activation channel-wisely following the work in [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Through this process, the FFD can assemble the local de- tailed semantic representation and the global context from the fused features to produce the fine depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4 Training loss and implementation detail We train both depth and motion networks using pho- tometric consistency (L2 loss and SSIM loss) and edge- aware smoothness losses following the best practices of self- supervised monocular depth estimation [8], [9], [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We set the weight for SSIM, L2 photometric, and smoothness losses as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='001, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We use 7 convolution layers for 6-DoF camera pose estimation following the work in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We implement our framework on PyTorch and train it on 4 Titan RTX GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We use the Adam optimizer [74] with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='9 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Our model is trained for 50 epochs with a batch size of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The learning rates for depth and pose network are 2 × 10−5 and 5 × 10−4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We will release the source code, the trained weights and the datasets once the paper is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4 EXPERIMENTAL RESULTS We conduct extensive experiments to investigate the gen- eralization performance of various network structures and the effect of texture-/shape-bias for monocular depth es- timation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' First, we evaluate state-of-the-art KITTI-trained models, including various modernized backbone architec- tures on the KITTI dataset and various depth benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Evaluations are conducted on the KITTI, an in- distribution dataset in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2, and the other depth datasets, out-of-distribution datasets in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also conduct an ablation study on the MonoFormer in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Then, we in- vestigate the texture-bias and shape-bias of the MDE models IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 5 Input images Monodepth2 PackNet-SfM R-MSFM6 MF-ConvNeXt MF-SLaK MF-ViT MF-RegionViT MF-Twins MF-Ours BTS AdaBins TransDepth DepthFormer GLPDepth Input images Monodepth2 PackNet-SfM R-MSFM6 MF-ConvNeXt MF-SLaK MF-ViT MF-RegionViT MF-Twins MF-Ours BTS AdaBins TransDepth DepthFormer GLPDepth Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4: Depth map results on the in-distribution dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' MonoFormer (MF-Ours) is from our previous work [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Models Supervision Methods CNN-based Self-supervised Monodepth2 [8], PackNet-SfM [9] R-MSFM6 [75], MF-(ConvNeXt [49], SLaK [50]) Supervised BTS [58], AdaBins [10] Transformer-based Self-supervised MF-(ViT [32], RegionViT [47], Twins [48], Ours [51]) Supervised TransDepth [11], DepthFormer [13], GLPDepth [60] TABLE 1: Taxonomy of MDE methods w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='t backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' using both the self-supervised and supervised methods with texture-shifted datasets in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We analyze the depth es- timation performance and the feature representation of each model on the texture-shifted datasets, which are syntheti- cally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Lastly, we demonstrate the intrinsic proper- ties of the CNN-based and Transformer-based networks are observed in the real-world texture-shifted datasets captured from different driving environments, such as the weather changes and illumination changes in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='1 Competitive methods and evaluation setups We conduct extensive experiments to compare the per- formances of the monocular depth estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' For self-supervised setting, we compare our work, called MF-Ours [51], with state-of-the-art methods, Monodepth2 [8], PackNet-SfM [9], SGDepth [56], R-MSFM [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also evaluate state-of-the-art supervised methods, BTS [58], AdaBins [10], TransDepth [11], DepthFormer [13], and GLPDepth [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We note that some Transformer-based mod- els, such as MF-Ours, TransDepth, and DepthFormer, use both CNNs and Transformers for their encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also analyze the performance of various modernized architec- tures such as RegionViT [47], Twins [48], ConvNeXt [49] and SLaK [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We replace the encoder of MF-Ours with these modernized backbones, whose names are defined as MF- (backbone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Its taxonomy is described in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We illustrate these basic network structures of the modernized CNN and Transformer architectures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' eTransformer-based models [47], [48] add the locality by using local attention to compensate for the shortcomings of the ViT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=', necessity Method Train Model Lower is better ↓ Higher is better ↑ Abs Rel Sq Rel RMSE RMSElog δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='25 δ < 1.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='999 GLPDepth D Trans 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='187 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='999 TABLE 2: Quantitative results on the in-distribution dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' M is Monocular images, and D is GT depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Trans means Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Bold is the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' of large dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' CNN-based architectures [49], [50] aim to extract global information while utilizing the intrinsic locality inductive bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' They also imitate the self-attention of the Transformer using improved strategies such as large kernel size, patchify stem [49], and Layer Norm [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We train all networks using the KITTI Eigen split [14], [77] consisting of 39,810 training and 4,424 validation data whose size is 640 × 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' All experiments in this paper are conducted with this KITTI-trained model whose results are reported in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Following a previous work [9], we remove around 5% of the total data for training to address the infinite-depth problems that commonly occur in dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We use typical error and accuracy metrics for depth, absolute relative (Abs Rel), square relative (Sq Rel), root-mean-square-error (RMSE), its log (RMSElog), and the ratio of inliers following the work [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 Evaluation on the in-distribution dataset We conduct the evaluations with an in-distribution dataset, where the KITTI-trained models are evaluated on 697 KITTI test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The qualitative results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4 show that our self- supervised method precisely preserves object boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' This indicates that the encoder captures global context and informative local features and transfers them to the decoder ELBAIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 6 Abs Rel ↓ Sq Rel ↓ RMSE↓ RMSElog ↓ δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='25 ↑ baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='899 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='882 +ACM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='879 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='879 +FFD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='860 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='879 +ACM&FFD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='846 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='580 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='891 TABLE 3: Ablation study on ACM and FFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' (a) Input image (b) Results with ACM (c) Results with FFD (d) Results with ACM and FFD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 5: Results of MF-Ours with/without ACM and FFD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' for pixel-wise prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The quantitative results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 2 show that MF-Ours outperforms all competitive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also evaluate the performance of models with the other backbones, MF-(ConvNeXt/SLaK/RegionViT/Twins/ViT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We employ only the encoder part of our method as a backbone without changing other parts in this experi- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Supervised methods outperform all self-supervised methods regardless of backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' MF-Ours and Depth- Former achieve the best performance among the self- supervised methods and supervised methods on the in- distribution datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Overall, the performance gap among all the competitive methods is marginal for the in-distribution dataset, but the Transformer-based models generally outperform the CNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3 Ablation study In this section, we conduct the ablation study on MF-Ours proposed in our previous work [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We use the same models and datasets for the experiments used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='1 Effectiveness of the proposed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We conduct an ablation study to demonstrate the effective- ness of the proposed modules, ACM and FFD in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The baseline is DPT [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The models with only the ACM module or FFD module marginally improve the depth estimation performance due to the absence of proper attention map fusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' On the other hand, our MonoFormer with both ACM and FFD significantly improves the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The results show the proposed model achieves the best perfor- mance in all measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The qualitative comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 5 shows that the model with both ACM and FFD keeps clearer object boundaries, even a small car in far depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 Visualization of attention maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We visualize the attention maps from the lower to higher layers of Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 6, the encoder in the shallow layer extracts local region features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The deeper the layer, the more global shape contexts are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Another observation is that ACM captures more detailed attention at different depths of the encoder features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' FFD enhances the encoder features by fusing them with the at- tention map from ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The fused feature captures features from coarse to fine details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' These experiments show that our model is capable of accurate pixel-wise prediction as it secures adequate local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' # of layers Abs Rel ↓ RMSE ↓ δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='25 ↑ δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='252 ↑ L = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='148 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='327 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='939 L = 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='112 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='962 L = 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='580 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='962 L = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='111 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='692 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='962 TABLE 4: Ablation study on the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Input image GT depth (a) Attention map of CNN-Transformer encoder (b) Attention map of ACM (c) Feature map of FFD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 6: Visualization of attention and feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The left column from the second row is the maps from shallow layers, whereas the right is the maps from deep layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3 The number of encoder and decoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We compare the performance of MF-Ours according to the number of encoder and decoder layers in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Each layer has ACM and FFD modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We find out that the model with four transformer layers achieves the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The results are slightly degraded with the MF-Ours with 3 or 5 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Therefore, we set L as four for MF-Ours in all experiments in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4 Evaluation on out-of-distribution datasets We compare the generalization performance of all the com- petitive models using public depth datasets captured at common indoor environments, including office workspaces, meeting rooms, and kitchen areas (RGBD [78], SUN3D [40]), man-made indoor and outdoor environments (MVS [42], ETH3D [43]), and synthetic scenes from graphics tools (Scenes11 [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Interestingly, both qualitative and quantita- tive results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 7 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 5–7 show that Transformer- based models better generalization performance even in the out-of-distribution datasets that have never been seen during network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Meanwhile, CNN-based models predict unreliable depth maps that have lost the detail of object boundaries and produce significant errors in texture- less regions, such as the wall of a building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The experiments demonstrate that Transformers are more robust than CNNs for environmental changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Another interesting observation is that MF-ConvNeXt generally outperforms all the other CNN-based models and produces depth results comparable to other Transformer-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' On the other hand, MF- RegionViT fails to estimate depth accurately, even though Transformer-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' A detailed analysis of why Transformers and MF-ConvNeXt show better generalization performance is provided in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 7 RGBD SUN3D MVS ETH3D Scenes11 (a) Input images & GT Depth maps (b) Self-supervised CNN-based methods: Monodepth2, PackNet-SfM, R-MSFM6, MF-ConvNeXt, MF-SLaK (Top-to-Bottom) (c) Self-supervised Transformer-based methods: MF-ViT, MF-RegionViT, MF-Twins, MF-Ours (Top-to-Bottom) (d) Supervised CNN-based methods: BTS, AdaBins (Top-to-Bottom) (e) Supervised Transformer-based methods: TransDepth, DepthFormer, GLPDepth (Top-to-Bottom) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 7: Depth map results on out-of-distribution (RGBD, SUN3D, MVS, ETH3D, and Scenes11) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 8 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results on the out-of-distribution (Synthetic from graphics tools - Scenes11) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5 Analysis of texture-/shape-bias on state-of-the-art methods and various backbone networks In this section, we verify the intrinsic properties of CNNs and Transformers that lead to the robustness of depth estimation to environmental changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We hypothesize that CNNs and transformers identify texture and shape infor- mation as key visual cues for depth estimation, respectively, inspired by the work [20], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Thus, we synthetically generate the texture-shifted datasets in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Then, we validate the texture-/shape-bias of the model by compar- ing the performance of the competitive methods on the generated datasets in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Finally, we analyze the internal representation of each neural network structure by measuring centered kernel alignment (CKA) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='1 Texture-shifted datasets generation In general, the texture is defined as an image’s spatial color or pixel intensity pattern [79], [80], [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Inspired by [18], we use three different texture shift strategies to investigate the impact of textures on the inference process in depth: texture smoothing (Watercolor), texture removal (Pencil- sketch), and texture transfer (Style-transfer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The generated images and the corresponding results of each model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The first two images are watercolored images, the middle two images and the last two images are pencil-sketch and style-transferred images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The following is a summary of the image generation: Watercolor We smooth the texture details from original images while preserving the color cues using cv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='stylization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The image looks like a watercolor pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Pencil-sketch We remove both textures and color from original images using cv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='pencilSketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The image seems like a sketch drawn with pencils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Style-transfer We apply a new texture to the original image (context) by utilizing other images (style) using a style transfer [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The textures of the original images are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 Evaluation on synthetic texture-shifted datasets We compare the performance of all the competitive methods and the modernized backbones on the synthetic texture- shifted datasets, the same as the experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The qualitative and quantitative results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 8 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' As seen in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4, both the Transformer- based models produce better depth maps than pure CNN- based methods regardless of supervised or self-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' In particular, the depth results from CNN-based models are unrecognizable with the strong texture-shifted datasets, especially the style-transferred data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also ob- serve that MF-ConvNeXt shows a tolerable depth map on texture shift datasets and has lower errors than other CNN- based models, although it is purely CNN-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' These experiments support our two findings observed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' One is that networks whose encoder consists of Transform- ers are generally robust to texture changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' However, MF- ConvNeXt and MF-RegionViT show different aspects from the CNN-based and Transformer-based models, which are the respective backbone models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' In the following section, we deeply analyze the intermediate feature representations of all backbones to verify the reason for these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 9 Watercolor Pencil-sketch Style-transfer (a) Input images (b) Self-supervised CNN-based methods: Monodepth2, PackNet-SfM, R-MSFM6, MF-ConvNeXt, MF-SLaK (Top-to-Bottom) (c) Self-supervised Transformer-based methods: MF-ViT, MF-RegionViT, MF-Twins, MF-Ours (Top-to-Bottom) (d) Supervised CNN-based methods: BTS, AdaBins (Top-to-Bottom) (e) Supervised Transformer-based methods: TransDepth, DepthFormer, GLPDepth (Top-to-Bottom) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 8: Depth map results on synthetic texture-shifted (Watercolor, Pencil-sketch, Style-transfer) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Datasets Watercolor Pencil-sketch Style-transfer Abs Rel Sq Rel RMSE δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='25 δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='252 δ < 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='801 GLPDepth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='573 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='872 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='513 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='992 TABLE 8: Quantitative results on synthetic texture-shifted (Watercolor, Pencil-sketch, Style-transfer) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 10 Monodepth2 PackNet-SfM R-MSFM6 MF-ConvNeXt MF-SLaK BTS AdaBins MF-ViT MF-RegionViT MF-Twins MF-Ours TransDepth DepthFormer GLPDepth Watercolor 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 0 Pencil-sketch 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 0 Style-transfer 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 0 CKA Methods CKA Methods CKA Methods Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 9: Box-and-Whisker plot of CKA similarity of all competitive methods on three different synthetic texture-shifted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The blue and red tones indicate CNN-based and Transformer-based models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The circle and triangle indicate self-supervised and supervised methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Original Image (I𝑎) Texture-shifted Image (I𝑏) Encdoer (𝐸) 𝐸 I𝑎 = 𝑧𝑎 CKA(𝑧𝑎, 𝑧𝑏) 𝐸 I𝑏 = 𝑧𝑏 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 10: Illustration of CKA [53] similarity computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='3 Analysis on feature representation from backbones To analyze the internal properties of CNNs and Trans- formers, we employ the centered kernel alignment (CKA), which is the similarity measure of internal representations of neural networks by following previous works [53], [54], [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The CKA is widely used to analyze feature representations of neural networks [53], [84], because of its invariant prop- erties to the orthogonal transformation of representations and isotropic scaling [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We freeze the encoder E of the depth networks trained on the original KITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We extract features za, zb from the last layer of the encoder by passing the original image Ia and texture-shifted image Ib, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Then, the CKA is computed given K = zazT a and L = zbzT b with m number of samples as follows: CKA(K, L) = HSIC(K, L) � HSIC(K, K)HSIC(L, L), (7) HSIC(K, L) = (KHLH)/(m − 1)2, (8) where H is the centering matrix Hn = In − 1 n11T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The overall process is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We use 697 image pairs from the KITTI Eigen dataset and their correspond- ing texture-shifted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We compute the CKA with three types of texture-shifted datasets, including watercolor, pencil-sketch, and style-transfer, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The results of feature similarity show a similar aspect of the quantitative results on synthetic texture-shifted datasets in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' MF-Ours and GLPDepth extract the most similar features from the original and texture-shifted datasets in the self-supervised and supervised methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Generally, Transformer-based models (MF-ViT, MF-Twins, GLPDepth, MF-Ours, TransDepth, DepthFormer) extract features of texture-shifted images similar to the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' On the other hand, most CNN-based models (Mon- odepth2, PackNet-SfM, R-MSFM6, BTS, AdaBins) extract inconsistent features from original and texture-shifted im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The results are consistently observed regardless of the types of texture changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' It demonstrates that the robustness to environmental changes comes from consistent feature extraction from encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' These results also support our observations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2 that the Transformers have a strong shape-bias whereas CNNs have a strong texture bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Moreover, models with shape-biased representation show better generalization performance than models with texture bias for monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also observe that the pure Transformer-based methods (MF-ViT, MF-Twins, GLPDepth) and CNN-Transformer hybrid methods (MF- Ours, TransDepth, DepthFormer) show similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The experiments provide us with some interesting observa- tions on modernized network structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' First, it is noteworthy that MF-ConvNeXt, one of the CNN-based models, shows a relatively higher CKA similar- ity than the other CNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Even the similarity of the models is on par with Transformer-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' ConvNeXt consists of a patchify stem, and a depth-wise convolution [85] with a large receptive field that mimics the self-attention mechanism using Transformer’s macro design [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The design intuition of ConvNeXt makes that ConvNeXt can focus on global information, similar to self- attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Because of the modernized macro design, MF- ConvNeXt consisting of only convolutional layers is robust to texture-shifted datasets and has a strong shape-bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Second, unlike other Transformer-based models, MF- RegionViT is sensitive to texture changes, similar to CNN- based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' It is highly biased toward texture informa- tion rather than shape information, even though it is a transformer-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' RegionViT [47] employs region- to-local attention, which is getting interaction between local regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' It improves the locality of the networks through lo- cal attention by capturing fine-grained spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We believe that the strong locality makes MF-RegionViT rely heavily on spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Based on these experiments, we find that a self-attention module capturing global information, strengthen the shape- bias of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also observe that the locality from local attention or the intrinsic property of CNNs improves the texture-bias of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' In terms of network design, IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 11 RobotCar Foggy CityScapes Rainy CityScapes (a) Input images (b) Self-supervised CNN-based methods: Monodepth2, PackNet-SfM, R-MSFM6, MF-ConvNeXt, MF-SLaK (Top-to-Bottom) (c) Self-supervised Transformer-based methods: MF-ViT, MF-RegionViT, MF-Twins, MF-Ours (Top-to-Bottom) (d) Supervised CNN-based methods: BTS, AdaBins (Top-to-Bottom) (e) Supervised Transformer-based methods: TransDepth, DepthFormer, GLPDepth (Top-to-Bottom) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 11: Depth map results on real-world texture-shifted (RobotCar, Foggy and Rainy CityScapes) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 12 Datasets Oxford RobotCar Foggy CityScapes Rainy CityScapes Abs Rel Sq Rel RMSE δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='25 δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='252 δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='253 Abs Rel Sq Rel RMSE δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='25 δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='252 δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='253 Abs Rel Sq Rel RMSE δ < 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='822 TABLE 9: Quantitative results on real-world texture-shifted (RobotCar, Foggy and Rainy CityScapes) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' a self-attention mechanism is ultimately key to generaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Additionally, the locality of the network weakens gen- erality, even if it helps to optimize in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='6 Evaluation on real-world texture-shifted datasets In addition, we conduct experiments on practical texture changes that can occur in real driving scenarios to demon- strate their applicability to the real-world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We evaluate all models using practical texture-shifted datasets consisting of the night (Oxford RobotCar [44]), foggy (Foggy CityScapes [46]), and rainy driving scenes (Rainy CityScapes [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Overall, the experiment demonstrates that the properties of CNNs and Transformers identified in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5 are similarly observed in real-world texture-shifted scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Quantita- tive and qualitative results and detailed analyzes for each network are described in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The depth map results of each model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Similar to the experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='2, Transformer- based models show plausible depth maps, except MF- RegionViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' On the other hand, CNN-based models fail to es- timate depth even for objects such as a car in scenes similar to the training dataset, with the exception of MF-ConvNeXt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The quantitative results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 9 also show that CNN- based models have higher errors than Transformer-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' MF-SLaK is no different from other CNN models, but MF-ConvNeXt shows low errors and feasible depth maps by mimicking Transformers, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' MF-Twins show similar results to other Transformer-based models because only a weak locality is added compared to MF-RegionViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' However, MF-RegionViT indicates texture- biased results like CNN-based models due to its strong locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The results on rainy and foggy datasets (Rainy and Foggy CityScapes) also indicate similar aspects to the night scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Transformer-based models estimate the plausible depth and distinguish even tiny objects such as bollards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' On the contrary, CNN-based models predict incorrect depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' In particular, CNN-based models exhibit significant errors in areas such as roads despite similar texture to training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' MF-ConvNeXt and MF-Twins also infer plausible depth maps regardless of changes in the weather environment, but MF-RegionViT and MF-SLaK estimate bizarre depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' In these experiments, we observe that textural shifts like weather and illumination changes con- fuse CNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We also find that CNN-based models are affected by changes in texture as well as loss of texture information, which can be a fatal problem in real- world driving applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 5 CONCLUSION In this paper, we have proposed a self-supervised monoc- ular depth estimation method called MonoFormer (MF- Ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' More importantly, we have presented an in-depth analysis of the generalization performance of various mod- ernized backbone structures as well as state-of-the-art self-supervised and supervised methods for monocular depth estimation using various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The in-distribution datasets are used to compare common performance on the KITTI benchmark, while the out-of-distribution and texture- shifted datasets are used to compare the generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We deeply analyze the properties of the fea- tures extracted from each model using the synthetic texture- shifted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Finally, we demonstrate the applicability of the model in real-world scenarios of changing environments using day-night, foggy and rainy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Through extensive experiments, we observe that MF- Ours and GLPDepth have the best generalization perfor- mance among all self-supervised and supervised methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' The supervised method, GLPDepth, achieves the best generalization performance among all monocular depth estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' More interestingly, we provide three important observations about the generality of monoc- ular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' First, CNN-based models heavily rely on texture information to recognize scenes and objects, while Transformer-based models use shape information to a greater degree for the monocular depth estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Second, texture-biased representations result in poor gen- eralization performance for environmental changes such as differences in illumination and weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' In contrast, shape- biased representations are more robust to such texture- shifted environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Lastly, ConvNeXt and RegionViT are CNN-based and Transformer-based, respectively, but have properties different from those of the backbone structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' It shows that the texture-bias and shape-bias are not the intrin- sic properties of CNNs and Transformers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' In- stead, the intrinsic locality of CNNs induces texture-biased characteristics, while the self-attention mechanism, the base layer of Transformers, induces shape-biased properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' We believe our observations provide valuable insights into best practices in network design not only for monocular depth estimation but also for a variety of dense prediction tasks such as semantic segmentation, depth completion, normal estimation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Based on these observations, the best way to design generalized networks for dense prediction is to utilize multi-self-attention (MSA) from Transformers and supplement the local information loss of global self- attention by using weak local attention as an auxiliary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' XX, 2023 13 REFERENCES [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Xue, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Zhuo, Z.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Wang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Rosing, “Depthwise convolution is all you need for learning multiple visual domains,” in AAAI, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' 8368–8375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Jinwoo Bae received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' degree in the Department of Electrical and Information Engi- neering from the Seoul National University of Science and Technology in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' He is currently pursuing the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' degree in the Department of Electrical Engineering and Computer Science at DGIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' His research interests include 3D vision such as depth estimation, robot vision, and multi- camera framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' He was a research intern at KIST (Seoul, Korea) in 2020 and ETRI (Pangyo, Korea) in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Kyumin Hwang received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' degree in the Department of Computer Science and Engineer- ing from the Kyungpook National University in 2020, and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' degree in the Department of Information and Communication Engineering from DGIST in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' He is currently pursuing the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' degree in the Department of Electrical Engineering and Computer Science at DGIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' His research interests include 3D computer vi- sion, including multi-view stereo for 3D recon- struction, and deep learning with geometry for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' He was a research intern at ETRI (Daegu, Korea) in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' Sunghoon Im received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' degree in the Department of Electronic Engineering from So- gang University in 2014, and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' degree in the School of Electrical Engineering from KAIST in 2016 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' He joined the Department of Electrical Engineering and Com- puter Science at DGIST, Daegu, Korea, in 2019, where he is currently working as an assistant professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' He was a recipient of Microsoft Re- search Asia fellowship and Global Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' fellow- ship from NRF of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} +page_content=' His research interests include computer vision, robot vision, and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE1T4oBgHgl3EQfbATf/content/2301.03169v1.pdf'} diff --git a/VdE5T4oBgHgl3EQfbw_i/content/tmp_files/2301.05599v1.pdf.txt b/VdE5T4oBgHgl3EQfbw_i/content/tmp_files/2301.05599v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a89458c4f3f6f89cdaf9f23e51466335343772f1 --- /dev/null +++ b/VdE5T4oBgHgl3EQfbw_i/content/tmp_files/2301.05599v1.pdf.txt @@ -0,0 +1,1282 @@ +arXiv:2301.05599v1 [q-bio.NC] 13 Jan 2023 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +1 +Short-time SSVEP data extension by a novel generative +adversarial networks based framework +Yudong Pan ID,Student Member, IEEE, Ning Li ID, Yangsong Zhang ID +Abstract— Objective: +Steady-state +visual +evoked +potentials +(SSVEPs) based brain-computer interface (BCI) has received con- +siderable attention due to its high transfer rate and available quan- +tity of targets. However, the performance of frequency identifica- +tion methods heavily hinges on the amount of user calibration data +and signal data length, which hinders the deployment in real-world +applications. Recently, generative adversarial networks (GANs)- +based data generation methods have been widely adopted to cre- +ate supplementary synthetic electroencephalography (EEG) data, +holds promise to address these issues. Methods: In this paper, we +proposed a GAN-based end-to-end signal transformation network +for data length window extension, termed as TEGAN. TEGAN trans- +forms short-time SSVEP signals into long-time artificial SSVEP +signals. By incorporating a novel U-Net generator architecture +and auxiliary classifier into the network design, the TEGAN could +produce conditioned features in the synthetic data. Additionally, +to regularize the training process of GAN, we introduced a two- +stage training strategy and the LeCam-divergence regularization +term during the network implementation. Results: The proposed +framework was evaluated on two public SSVEP datasets. With the +assistance of TEGAN, the performance of traditional frequency +recognition methods and deep learning-based methods have been +significantly improved under limited calibration data. Conclusion: +This study substantiates the feasibility of the proposed method to +extend the data length for short-time SSVEP signals to develop a +high-performance BCI system. Significance: The proposed GAN- +based methods have the great potential of shortening the calibra- +tion time for various real-world BCI-based applications, while the +novelty of our augmentation strategies shed some value light on +understanding the subject-invariant properties of SSVEPs. +Index Terms— brain-computer interface (BCI), steady- +state visual evoked potential (SSVEP), electroencephalog- +raphy (EEG), generative adversarial network (GAN) +I. INTRODUCTION +B +RAIN-COMPUTER interface (BCI) has shown to become a +promising technology that can provide its users with communi- +cation channels that do not depend on conventional output channels of +peripheral nerves and muscles by decoding their neural activities into +specific control commands [1]. Among various neuroimaging modal- +ities to implement a BCI system, electroencephalography (EEG) has +been the most prominent signal accounting for such the advantages +as non-invasiveness, high temporal resolution, affordability, ease of +implementation, portability, and convenience of use [2], [3]. +Several most popular paradigms can be employed to build EEG- +based BCI systems, such as motor imaginary (MI) [4], P300 event +This work was supported in part by the National Natural Science +Foundation of China under Grant No.62076209. +Y. Pan is with the School of Computer Science and Technology, +Southwest University of Science and Technology, Mianyang 621010, +China (e-mail: panydacademy@163.com). +N. Li is with the School of Computer Science and Technology, South- +west University of Science and Technology, Mianyang 621010, China +(e-mail: liningacademy@163.com). +Y. Zhang is with the the School of Computer Science and Technology, +Southwest University of Science and Technology, Mianyang 621010, +China, and also with MOE Key Laboratory for Neuroinformation, Clin- +ical Hospital of Chengdu Brain Science Institute, University of Elec- +tronic Science and Technology of China, Chengdu 610054, China (e- +mail:zhangysacademy@gmail.com). +related potentials (P300) [5], auditory steady-state response (ASSR) +[6], and steady-state visual evoked potential (SSVEP) [7]. Among +them, SSVEP-based BCI systems have received considerable atten- +tion due to its high transfer rate (ITR) and available quantity of +targets. SSVEPs refer to periodic evoked potentials over occipital +scalp areas, in response to rapidly repetitive visual stimulation flicking +or reversing at a specific frequency [8]. The SSVEP signal consists of +a number of discrete frequency components, normally including the +fundamental frequency of the visual stimulus and its harmonics. On +the strengths and characteristics of SSVEP, numerous SSVEP-based +BCI applications have been developed, such as bionic mechanical +leg [9], unmanned aerial vehicle [10], dial interface [11], high-speed +mental speller [12], smart homes [13], and games [14]. To design +a high-performance BCI system based on SSVEP signal, the most +crucial aspect is to develop a fast and accurate frequency recognition +method that can distinguish the stimulus frequency of the target gazed +by the users through analyzing the EEG signal in the shortest possible +time. Thus, various cutting-edge algorithms have been proposed based +on different perspectives. +Generally, the frequency identification algorithms can be divided +into three categories: training-free methods, user-dependent (UD) +training methods and user-independent (UI) training methods [15]. +Training-free identification methods do not require any training data +from the user of the BCI, from which the user can interact directly +with the system. The representative training-free methods are canon- +ical correlation analysis (CCA) [7] and multivariate synchronization +index (MSI) [16]. CCA seeks to capture the underlying correlation +between EEG data and a series of sinusoidal reference templates +corresponding to the stimuli frequencies, while MSI adopts the S- +estimator to estimate the synchronization index. CCA and MSI are +able to achieve comparable performance when only a few stimulus +targets are available and the EEG signals are sufficiently long, but +their performance drops dramatically when encountering a large +number of targets and short-time signals [17]. By incorporating the +user-specific training data into the algorithm design, the UD methods +could derive better performance than training-free methods under +these harsh conditions. The UD approaches take the discrimination +of recorded signals from different subjects into account, namely +subject-variant features, such as magnitude, phase, and visual la- +tency. Draw support from individual training data to learn these +properties, the performance of recognition algorithm would be greatly +improved. The typical UD algorithms mainly comprise individual +template CCA (ITCCA) [18], multiway CCA (MCCA) [19], task- +related component analysis (TRCA) [20], and correlated component +analysis (CORCA) [21], etc. In contrast to the UD methods, UI +methods are envisaged to build a generalized model for detection +of unseen subjects via studying the subject-invariant features that +learning from the existing subject datasets or human prior knowledge. +For UI algorithms, subject-specific training data is not required. +However, their performance usually is worse than UD algorithms. +Theoretically, training-free algorithms are all belong to UI methods +for their user-independent properties. The prevalent UI approaches +include filter bank canonical correlation analysis (FBCCA) [22], +transfer template-based canonical correlation analysis (ttCCA) [23] + +LOGO2 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +and adaptive combined CCA (A3C) [24], etc. +Although UD approaches commonly exhibit better performance +than UI algorithms, they are heavily dependent on the amount +of collected user-sepcific calibration data. Actually, collecting user +calibration data is a time-consuming and laborious process, and +prolonged experiments would cause user’s fatigue, leading to the +decreased quality of induced SSVEP signals. Hence, how to de- +velop a high-performance frequency recognition algorithm that needs +only a little calibration data or calibration-free has become a hot +research topic in recent years [23], [25]–[28]. Benefit from the +rapid development of deep learning (DL) in the past decade, there +has been an increased interest in applying DL algorithms to detect +SSVEPs for BCI researchers. In view of powerful feature represen- +tation capacity and flexibility, the DL-based methods hold promise +for reducing the gap between UD approaches and UI approaches. +For instance, Waytowich et al. employed a compact convolutional +neural network (Compact-CNN) entitled EEGNet to conduct inter- +subject classification, which yielded about 80% accuracy in a 12- +class SSVEP dataset without any user calibration data [29]. Ravi +et al. utilized fast Fourier transform (FFT) to design a complex +spectrum CNN (C-CNN) which could be trained on both UD and UI +schemes, achieving about 92.5% (UD), 81.6% (UI) and 92.3% (UD), +81.6% (UI) accuracy on a 12-class and a 7-class SSVEP dataset, +respectively [30]. Guney et al. proposed a two-stage training strategy +and a deep neural network (DNN) to improve performance of intra- +subject classification [31]. The proposed framework was evaluated +on Benchmark SSVEP dataset that contains 40 stimulus targets, +and has obtained approximately 95% recognition accuracy. Pan et +al. developed an efficient CNN-LSTM (Long short-term memory) +network with spectral normalization and label smoothing technolo- +gies, termed as SSVEPNet, for SSVEP classification under the small +sample size and short-time window scenarios [32]. SSVEPNet was +verified on a 4-class and a 12-class SSVEP dataset, yielding about +88.5% identification accuracy when a few trials of each stimulus are +available. Chen et al. introduced a Transformer-based deep neural +network model named SSVEPformer for enhancing the performance +of zero-calibration SSVEP-BCI [33]. The experimental results have +shown that SSVEPformer could achieve high accuracy of about 84% +and 80% on a 12-class and Benchmark SSVEP dataset. +On the other hand, in addition to explicitly modelling an efficient +frequency recognition method that requires less calibration data, +recent studies have substantiated the potential of using the generative +models to address the issues of data shortage in the SSVEP classi- +fication tasks [34]. Generative models aim to learn the distribution +of real data by constructing a probabilistic statistical model given +real data and using it to generate synthetic data that approximate the +distribution of real data [35]. Autoregressive models [36], variational +auto-encoder (VAE) [37], generative adversarial network (GAN) [38], +and denoising diffusion probabilistic model (DDPM) [39] are the +most commonly generative models. Among them, GAN has been the +most widely applied technique to synthesize fake data and overcome +the problem of limited data. Since EEG-GAN [40], the first GAN- +based generative model of EEG signal, was proposed in 2018, BCI +researchers have successively developed several GAN-based SSVEP +generation models. In 2019, Aznan et al. firstly exploited the GAN- +based generative model in circumventing the limited calibration data +via generating supplementary synthetic data to enlarge the size of +training data [41]. Only one year later, they also proposed a subject- +invariant SSVEP GAN (SIS-GAN) to generate artificial EEG data +that learns the subject-invariant features from the multiple SSVEP +categories [42]. After two years, inspired by StarGAN v2, which has +been used to solve multidomain image-to-image conversion, Kwon +et al. proposed a novel multidomain signal-to-signal transformation +method which is capable of generating artificial SSVEP signals from +resting EEG [43]. +Although these GANs based on SSVEP signals have made notice- +able progress, these studies only focus on generating simulated data +to enlarge the amount of the training dataset, without considering the +extension of signal length. However, longer SSVEP signal length +would often achieve more accurate recognition results under the +same conditions [30]–[33]. Hence, in this study, we proposed a +GAN-based end-to-end signal transformation network for data length +window extension, termed as TEGAN. TEGAN transforms short- +time SSVEP signals into long-time artificial SSVEP signals. By +incorporating a novel U-Net generator architecture and auxiliary +classifier into the network design, the TEGAN could produce con- +ditioned features in the synthetic data. Additionally, to regularize +the training process of GAN, we introduced a two-stage training +strategy and the LeCam-divergence regularization term during the +network implementation. The proposed menthods were evaluated +on two public SSVEP datasets. With the assistance of TEGAN, +the performance of traditional frequency recognition methods and +DL-based methods have been significantly improved under limited +calibration data. The extensive experimental analysis demonstrates +the effectinveness of the proposed methods, while the novelty of our +augmentation strategies shed some value light on understanding the +subject-invariant properties of SSVEPs. +II. METHODOLOGY +A. Dataset +In this study, two public SSVEP datasets were employed to +evaluate the proposed augmentation methods. According to the de- +sign purpose of these two datasets, we hereinafter termed them as +Direction SSVEP dataset and Dial SSVEP dataset, respectively. The +specific details of each dataset are described as follows: +1) Direction SSVEP dataset: This dataset was published by Lee +et al. in 2019 [44]. Fifty-four healthy subjects (25 females, aged 24-35 +years) participated in the experiment. The experiment collected EEG +data of subjects in two different periods (Session1 and Session2), and +the data in each period was divided into offline analysis stage and +online testing stage. For the sake of simplicity, the offline data from +Session1 was chosen for experimental evaluation in this study. +In the process of data acquisition, the four target stimuli were +coded by 5.45 Hz, 6.67 Hz, 8.57 Hz and 12 Hz , and played in the +lower, right, left and upper directions of the personal computer (PC) +display, respectively. Participants were asked to focus on the center +of the black screen, and then on the direction of the target stimulus +highlighted in different colors. Each SSVEP stimulus was presented +for 4 s, and the interval between two stimuli was 6 s. Each target +frequency was presented 25 times, leading to a total of 100 trials (4 +classes × 25 trial). EEG data of 62 Ag/AgCl electrodes collected +at a sampling rate of 1000 Hz were recorded in the experiment. Ten +electrodes (P7, P3, Pz, P4, P8, PO9, PO10, O1, Oz and O2) covering +the occipital lobe area were selected for our research. All data was +down sampled to 100 Hz and band-pass filtered between 4 and 40 +Hz through a fourth-order Butterworth band-pass filter. +2) Dial SSVEP dataset: This open access dataset was provided +by Nakanishi et al. in 2015 [11]. In this dataset, ten healthy subjects +(1 female, mean age:28 years) participated in the experiment. The +subjects were instructed to sit in a comfortable chair 60 cm in front of +a liquid crystal display (LCD) monitor in a dim room. The stimuli was +arranged in a 4×3 grid space as simulation a dial interface. Twelve +flickering stimuli (f0 = 9.25Hz, ∆f = 0.5Hz) were presented on the +monitor. The EEG data of eight Ag/AgCl electrodes (PO7, PO3, POZ, +PO4, PO8, O1, Oz and O2) covering the occipital were acquired using + +PAN et al.:SHORT-TIME SSVEP DATA EXTENSION BY A NOVEL GENERATIVE ADVERSARIAL NETWORKS BASED FRAMEWORK +3 +the BioSemi ActiveTwo EEG system with a sampling rate of 2048 Hz. +For each subject, there were 15-run experiments. In each run, 12 trials +corresponding to all 12 stimuli were generated in a random order. +Thus, a total of 180 trials were collected in the experiment. Each +trial was composed of 1 s cuing period and 4 s targeted flickering +period. +All data was down sampled to 256 Hz and band-pass filtered +between 6 and 80 Hz through a fourth-order Butterworth band-pass +filter. To suppress the adverse effect of visual latency, the data was +extracted from the 135 ms after the stimulus onset. +B. Frequency recognition methods +In this subsection, we briefly introduce two traditional frequency +recognition methods and two DL-based methods. All of these state- +of-the-art methods were adopted as baselines to verify the effective- +ness of the proposed methods. +1) Traditional Methods: +• IT-CCA:CCA is a multivariate statistical technique to search the +underlying correlation between two multidimensional variables, +by calculating a pair of weight vectors to maximize the Pear- +son’s correlation coefficients between their linear combinations. +For CCA-based SSVEP recognition methods, the coefficient +is calculated between the multi-channel EEG data and the +artificial reference signal [7]. However, the artificial reference +signals lack the specific characteristics of subjects. Then, IT- +CCA was proposed to substitute the artificial reference signal +for individual template reference signal obtained by averaging +multiple training trials of specific subject [18]. This strategy +has been proved to be able to suppress spontaneous EEG +interference and achieve better classification performance than +CCA algorithm. +• TRCA:TRCA is a method which learns the spatial filters to +extract task related components by maximizing the reproducibil- +ity of neuroimaging data during the task period [45]. For +SSVEP data, TRCA seeks to find a linear weight vector to +maximize the inter-trial correlation of its linear combinations, +referring as task-related components. In 2018, Nakanish et al. +first applied TRCA algorithm to build a high-speed SSVEP- +based BCI system [20]. In the literature, TRCA is extended +with the filter bank technique, and the selection of filter banks +follows the principle proposed by Chen et al. [22]. For better +comparison with other methods, the original TRCA algorithm +without filter bank technology was adpoted in the current study. +2) DL-based methods: +• EEGNet: EEGNet is a robust DL model which could yield +comparable performance across multiple EEG tasks and datasets +[46]. It is mainly comprised of a temporal filtering layer, a +depthwise convolution layer, a separable convolution layer, and +a fully connected layer. Among these network components, +depthwise and separable convolution make the model become +compact and efficient. Due to its effectiveness, Waytowich et +al. used EEGNet to decode SSVEP signals for inter-subject +classification on Dial SSVEP dataset, and has achieved about +80% accuracy [29]. +• C-CNN: The frequency domain of SSVEP data contains abun- +dant frequency and phase information relevant to the recognition +task. If this information could be adequately exploited, the +classification performance could be further improved. Therefore, +Ravi et al. utilized FFT to transform SSVEP signals from the +time domain to the frequency domain and designed a shallow +CNN consisting of a spatial filter layer, a convolutional layer, +and a fully connected layer to handle complex spectral data [30]. +C-CNN was evaluated on a 7-class and Dial SSVEP dataset, +yielding satisfactory results for both intra- and inter-subject +classification. +C. The proposed augmentation methods +Fig. 1: The procedure flowchart of the proposed augmentation +method. The whole process is divided into two steps. In the first +step, real short EEG and real long EEG of training dataset are used +to train the GAN model. In the second step, the pretrained generator is +employed to transform all input short EEG into synthetic long EEG. +Then the synthetic long EEG are used to train the EEG classifier and +conduct classification. +1) Overall framework: The procedure flowchart of the proposed +augmentation method is illustrated in Fig. 1. The whole process +is divided into two steps. In the first step, we train the TEGAN, +which could transform short time-window natural EEG into long +time-window artificial EEG. Concretely, followed by the auxiliary +classifier GAN (ACGAN) paradigm [47], TEGAN mainly includes +two components, i.e., a generator and a discriminator, competing in a +zero-sum game. The generator receives the real short EEG as network +input, then output the extended EEG data, namely long artifical EEG +data. The discriminator is responsible for distinguishing between long +real EEG data and long fake EEG data and simultaneously identifying +their respective classes. Let VG and VD denote the training objectives +of the generator G and discriminator D, respectively. Then the +training of the proposed GAN frameworks can be generally expressed +as: +min +G VG = +E +x∼pxs +[−D(G(x))] − +E +x∼pxs +[log(D(G(x) ∈ C))] +(1) +max +D VD = +E +x∼pxl +[1 − D(x)] + +E +x∼pxs +[1 + D(G(x))] +(2) ++ +E +x∼pxl +[log(D(x) ∈ C))] + +E +x∼pxs +[log(D(G(x) ∈ C))] +where pxs and pxl is the data distribution of short EEG data and +long EEG data from the training dataset, respectively. The notation +D(x ∈ C) represents the probability of the class label being correctly +identified. +To optimize these two objective functions, we use gradient descent +algorithm to train discriminator and generator alternately, and obtain +the optimal parameters of generator θ∗ +G. Hence, a short-to-long signal +converter ζ(·) could be constructed using θ∗ +G as follows: +ζ(xs) = G(xs|θ = θ∗ +G) = xl +(3) + +Procedure +InputEEG +FakeExtensionEEG +Flowchart +G +F(0) +Adv +RealExtensionEEG +R(1) +C=2 +AUX +C=3 +Pretrain +C=K +Train EEG +TrainExtensionEEG +EEG +Classifer +TEGAN +TestEEG +TestExtensionEEG +Validation4 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +In the second step, the converter ζ(·) is employed to transform all +short EEG into synthetic long EEG. Finally, the synthetic long EEG +are used to train the EEG classifier and conduct classification. It is +worth noting that since the converter is trained using only the training +dataset and does not require label information as input, there is no +data leakage problem during the entire operation. The implementation +code of TEGAN would be available in Github platform. +2) The architecture of generator: The detailed architecture of +generator is presented in Fig. 2. The whole architecture of the +generator follows the U-Net architecture proposed by Ronneberger +et al. [51], which is divided into down sampling stages and up +sampling stages. Moreover, considering the dependency between +spatial-temporal features of EEG data, we utilize a bidirectional +long short-term memory (Bi-LSTM) network to encode the features +derived by the last down-sampling module, thus connecting the down +sampling stages and up sampling stages. Furthermore, we added a +conditional batch normalization (cBN) layer in each up-sampling +module to mitigate the adverse effects of homogenization between +different synthetic samples [50]. However, due to the inaccessibility +of label information in the input end of generator, we adopted a fully +connected layer to create high-confidence pseudo label, which can be +improved via minimizing the following objective function LG: +min +G LG = VG − +K +� +c=1 +yc log pc +(4) +where yc is realistic label with one-hot encoding, pc is the +predicted probability of class c, and K is number of classes. +The network parameters of generator are exhibited in Table I. +Among the parameters, the kernel size DK for down-sampling +module could be determined through the following formula: +DKi = ⌈Di ∗ ws⌉, i = 1, 2, 3 +(5) +where D = {20, 12, 8}, and ws represents the window size of +input short EEG signal. Then the kernel size UK for up-sampling +module could be calculated as: +UKi+1 = DT2−i − (DT3−i) ∗ Si, i = 0, 1, 2 +(6) +Here, S = {1, 2, 2}, and DT represents the number of time steps +for down sampling module, which is presented in Table I. +3) The architecture of discriminator: The minute architecture of +discriminator is shown in Fig. 3. Overall, the discriminator follows +the design of our previously proposed SSVEPNet model [32]. It is +mainly comprised of three modules, namely spatial filtering module, +temporal filtering module, and decision module. In the spatial filtering +module, a one-dimensional convolution (1D Conv) layer is used to +fuse different channel information of EEG. In the temporal filtering +layer, 1D Conv layer is employed to extract temporal features of +EEG. In the decision module, a Bi-LSTM layer is leveraged to +learn the dependence between spatial-temporal features, while fully +connected layers are employed for classification and authentication. +The distinction between SSVEPNet and proposed discriminator only +lies in two points. One is that a max pooling layer with a kernel size +of 2 has been added to the temporal filtering module. Another is that +the fully connection layer in the decision module has changed from +three layers to two layers, and the neurons in the first layer is equal to +twentieth of encoded spatial-temporal features outputted by BiLSTM +layer. The remaining network parameters of the discriminator are +identical with SSVEPNet. +4) Regularizing TEGAN on limited data: GAN is notoriously +difficult to train, it is highly susceptible to encounter with mode col- +lapse phenomenon - particularly from datasets with high variability. +Furthermore, this problem may deteriorate with a limited training +data. Generators are inclined to merely memorize limited training +samples, rather than learning the sophisticated data distribution of +training dataset [47]. To moderate this issue, we introduce a two- +stage training strategy and LeCam divergence regularization term to +regularize the training process of TEGAN. +Firstly, inspired by the previous study [31], we utilized transfer +learning technique to design a two-stage training strategy for the +proposed GAN framework. In the first stage, we train two global +models Ds and Gs using the cross-subject data. In the second stage, +the network parameters of Ds and Gs are directly duplicated to two +target models, Dt and Gt. Then we freeze the parameters in Dt +in addition to the fully connection layers, and fine-tune Dt and Gt +using a limited amount of training data of target subjects. +Secondly, we supplemented a LeCam regularization term in the +training objective of discriminator, which has been substantiated that +could offer meaningful constraints under limited training data [52]: +min +D LD = VD + λRLC(D) +(7) +where λ represents regularization term, and LeCam regularization +term RLC(D) is expressed as: +RLC(D) = +E +x∼pxl +[∥D(x) − αF ∥2] − +E +x∼pxs +[∥D(G(x)) − αR∥2] +(8) +where αF and αR are anchors obtained by the exponential moving +average variables, which is aiming at tracking the discriminator +predictions. Assuming that there are T E training epochs in total, +then αF and αR could be calculated using the following equations: +� +αF (i + 1) = γD(G(xs)) + (1 − γ)αF (i) +αR(i + 1) = γD(xl) + (1 − γ)αR(i) , SE ≤ i ≤ T E (9) +Here, γ is decay coefficient, and SE represents the starting epoch +to implement LeCam regularization term, which is conducive to avoid +the excessive regularization in the initial stage with under fitting. +D. Experimental evaluation +To validate the efficacy of the proposed augmentation method, +we conducted intra-subject classification experiments with limited +training data. Specifically, the original data from a target subject +is divided into training set Ttr(Xs) and testing set Ttt(Xs). We +exploit the two-stage training strategy to optimize the parameters of +the generator Gt, and then employ it to transform the Ttr(Xs) and +Ttt(Xs) into Ttr(Xl) and Ttt(Xl), respectively. Finally, the Ttr(Xl) +is used to train four SSVEP classifiers as mentioned in Section 2.2, +and undertake evaluation on Ttt(Xl). To eliminate the randomness of +data partitioning, the K-Fold evaluation strategy was adopted in the +experiments, under which the parameters of Gs remained unchanged +while Gt was updated K times in the whole process. +In this study, we implemented the proposed augmentation method +in PyTorch framework. The hyperparameters on two datasets are set +as follows. +• Direction: For the first training stage, mini-batch (B) = 64, +epochs (E) = 200, learning rate (lr) = 0.001, optimizer (Opt) = +Adam (beta1 = 0.9, beta2 = 0.999), weight decay (wd) = 0.0001. +For the second training stage, B = 20, E = 500, lr = 0.01, Opt += Adam (beta1 = 0.9, beta2 = 0.999) + Cosine Annealing, wd + +PAN et al.:SHORT-TIME SSVEP DATA EXTENSION BY A NOVEL GENERATIVE ADVERSARIAL NETWORKS BASED FRAMEWORK +5 +Fig. 2: The architecture of generator. The whole architecture of the generator follows the design of U-Net, which is divided into two stages: +down sampling and up sampling. Where Nc is the number of EEG channels, Nt denotes the number of sample points. K, S represents the +size of convolution/deconvolution kernel and stride, respectively. DKi, UKi (i=1, 2, 3) corresponding to kernel size for down sampling stage +and up sampling stage, respectively. +Fig. 3: The architecture of discriminator. The discriminator follows the design of SSVEPNet, which mainly comprised of a spatial filtering +module, a temporal filtering module, and a decision module. += 0.0003. For the regularization term, SE = 50, λ = 0.6, γ = +0.90. +• Dial:All hyperparameters are identical with Direction SSVEP +dataset, except for the min-batch in the second training stage, +which is set to 24 in this dataset. +E. Statistical analysis +In this study, classification accuracy was adopted as the evaluation +metric. The average classification results across all subjects for K- +times validation were presented in the form of mean ± standard +deviation. Two transformation scenarios, i.e. 0.5 s to 1 s and 1 +s to 2 s, were analyzed in the experiments. Paired t-tests were +implemented to investigate whether there were significant differences +in the classification accuracy between all pairs of methods at each +condition. +III. RESULTS +Firstly, we investigated the results of four baseline methods using +limited training data. In this setting, the TEGAN that used to extend +signal length was trained by 20% SSVEP data. Fig. 4 shows the +averaged classification results of original signals and augmented +signals with different signal lengths across all subjects. We could +observe that augmented signals yielded better classification perfor- +mance than original signals on both two transformation scenarios, and +the results of two SSVEP datasets manifest the consistent tendency. +Interestingly, we could find that the augmented signals at 1.0 s have +achieved better results than original signals at 1.0 s for three methods +(ITCCA, EEGNet, and C-CNN). +Moreover, we investigated the results how the performance of four +baseline methods varied with different scales of dataset. Specifically, +the SSVEP dataset from each subject was divided into the training + +Spatial Filtering Module +Temporal Filtering Module +Decision Module +InputEEG +FC3 ++Label +BatchNorm +1DConv +1D Conv +BatchNorm +PReLU +Dropout +PReLU +MaxPool +ISTM +FC2 +Real/FakeOutputEEG (Nc×2Nt) +Spatial 1DConvolution +Temporal1DConvolution +Temporal1DDeconvolution +Spatial1DDeconvolution +Up Stage5 +DataFlow +-LabelEmbedding--SkipConnection +K=2,S=2 +InputEEG- +Down Stage1 +Up Stage4 +(Nc × Nt) +K = Nc, S= Nc +K=Nc,S=Nc +Down Stage2 +Up Stage3 +K=DK1,S =2 +K=UK3,S=2 +Down Stage3 +Up Stage2 +K =DK2,S=2 +K=UK2,S=2 +Down Stage4 +Up Stage1 +K=DK3,S=1 +K=UK1,S=1 +Reshape +Reshape +BiLSTM Stage +PseudoLabel +FcStage6 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +TABLE I: The detailed network parameters of generator +Block +Module +Layer +Output +Description +Down Sampling +Input Module +Input +(b, 1, Nc, Nt) +(batch size, 1, num channels, num time steps) +DownStage Module(× 4) +Conv1D +(b, DCi, 1, DTi) +kernel = {Nc, DK1, DK2, DK3} +stride = {Nc, 2, 2, 1} +Ci = 2i+1 ∗ Nc, i = {0, 1, 2, 3} +DT0 = Nt, DTi+1 = (DTi − kernel[i]) // stride[i] + 1 +Norm +(b, DCi, 1, DTi) +max norm (= 1.0) + BN, i = 0 +SN [48] + BN, i = 1, 2, 3 +Activation +(b, DCi, 1, DTi) +PReLU [49] +Dropout +(b, DCi, 1, DTi) +p = 0.5 +Reshape Module +Reshape +(b, DT3, DC3) +BiLSTM Encoding +BiLSTM Module +LSTM +(b, 2 ∗ DT3, DC3) +input size, hidden size = DC3, num layers = 1 +batch first = True, bidirectional = True +Reshape +(b, 1, 2 ∗ DT3 ∗ DC3) +AvgPool +(b, 1, DT3 ∗ DC3) +kernel = 2, stride = 2 +Pesudo Label Generation +FC Module +Flatten +(b, DT3 ∗ DC3) +Linear +(b, K) +Norm +(b, K) +SN +Selection Module +Argmax +(b, 1) +select the highest probability as pseudo label +Up Sampling +Reshape Module +Reshape +(b, DC3, 1, DT3) +UpStage Module(× 5) +DeConv1D +(b, UCi, 1, UTi) +kernel = {UK1, UK2, UK3, Nc, 2} +stride = {1, 2, 2, Nc, 2} +UC = {8 ∗ Nc, 8 ∗ Nc, 4 ∗ Nc, 2 ∗ Nc, Nc} +UT = {DT2, DT1, DT0, Nt, 2 ∗ Nt} +Norm +(b, UCi, 1, UTi) +SN + cBN [50] +Activation +(b, UCi, 1, UTi) +PReLU +Conv1D +(b, UCi // Mi, 1, UTi) +kernel = 1, stride = 1, M = {0, 2, 2, 2, Nc} +only exist when i ≥ 1 +Norm +(b, UCi // Mi, 1, UTi) +SN + cBN, only exist when i ≥ 1 +Activation +(b, UCi // Mi, 1, UTi) +PReLU, only exist when i ≥ 1 +Output Module +Output +(b, 1, Nc, 2 ∗ Nt) +dataset and testing dataset in the portion of 2:8, 5:5 and 8:2 respec- +tively. In this study, we marked these three scales of datasets as small- +scale, middle-scale and large-scale dataset. Fig. 5 illustrates averaged +classification results across subjects of four baseline classification +methods using original signals and augmented signals at different +scales of datasets on two SSVEP datasets. With the continuous +expansion of training data, the average classification performance of +each baseline algorithm for the original signal and the augmented +signal has been gradually improved, and the trend is consistent on +both two SSVEP datasets. In addition, the augmented signal improves +the classification performance on a small-scale training dataset more +significantly. With the continuous expansion of training data, the im- +provement effect of classification performance of augmented signals +is gradually weakened. More interestingly, when original signals were +converted to augmented signal by TEGAN on any scale of dataset, +the classification performance gap of all algorithms is significantly +reduced. +Furthermore, we conducted ablation experiments on the Dial +SSVEP dataset to explore the contribution of pivotal component to +implement the augmentation framework. The augmented data at 2.0 +s was generated by original data at 1.0 s, and the signal converter +TEGAN was trained by 20% training data. Four modules, i.e. +auxiliary classifier in the discriminator, the pseudo-label generation +in the generator, the two-stage training strategy, and the LeCam +divergence regularization term were investigated in this study. As +shown in Table II, the averaged classification results indicate that +four important components of TEGAN are all helpful to improve +the classification performance of augmented data. It is worth noting +that after removing the two-stage training strategy from TEGAN, the +model would be trained using only 20% data of target subject. +IV. DISCUSSION +In recent years, enhancing the classification performance of +SSVEPs with limited calibration data is a hot research topic, which +empower the practicality of various BCI applications [32], [53]–[55]. +To this end, modelling an efficient classification method requiring +less calibration data or create supplemented synthetic data to enlarge +the size of the training dataset are most commonly strategies. Never- +theless, in this study, we proposed a novel pipeline that leverage +the subject-invariant properties of SSVEPs to address this issue. +Specifically, we developed a GAN-based model, i.e., TEGAN, which +could be used to extend the data length of SSVEP data. TEGAN seeks +to learn the mapping relationship between short-time and long-time +signals through an adversarial game training paradigm. Concretely, +the generator is responsible for extracting signal features from the +short-time SSVEP data and reconstructing the long-time SSVEP data. +The discriminator assists the generator to improve the quality of the +generated data by learning the discrepancy information between the +real and fake long-time signal. +After analyzing the result in Table II, we could conclude that the +two-stage training strategy plays an indispensable role in enhanc- +ing the performance of TEGAN. However, although the two-stage +training strategy can greatly improve BCI performance with only +a small amount of target subject data, it still suffers the laborious +calibration procedure. Implementing a high-performance BCI system +with zero calibration for new users has always been the ultimate +goal of the BCI community. In this study, we verify the feasibility +via improving the performance of training-free methods. Concretely, +we employ the cross-subject data to train TEGAN, and then use it to +extend the signal length of target data. Two representative methods as + +PAN et al.:SHORT-TIME SSVEP DATA EXTENSION BY A NOVEL GENERATIVE ADVERSARIAL NETWORKS BASED FRAMEWORK +7 +TABLE II: Albation study on Dial SSVEP Dataset. +Module Selction +Accuracy(%) +Case +Auxiliary Classifier +Pseudo Label +Two Stage +LeCam Divergence +ITCCA +TRCA +EEGNet +C-CNN +(a) +– +✓ +✓ +✓ +80.56±18.86** +72.26±15.23*** +79.70±17.90** +79.68±18.63** +(b) +✓ +– +✓ +✓ +84.24±18.40 +78.61±19.82 +71.57±23.59*** +83.93±18.53 +(c) +✓ +✓ +– +✓ +74.40±23.34** +76.92±22.24 +78.14±22.47* +78.24±22.47* +(d) +✓ +✓ +✓ +– +79.22±19.92** +74.43±19.44*** +78.72±20.02** +79.41±18.91** +(e) +✓ +✓ +✓ +✓ +84.61±16.54 +82.44±17.19 +83.54±17.40 +84.15±16.91 +‘–’ denotes which module is deleted from the proposed GAN model, and ‘✓’ denotes which module is remained. The asterisk in the table indicate +significant difference between each pair of the two methods by paired t-tests (*p < 0.05, **p < 0.01, ***p < 0.001) +Fig. 4: Averaged classification results of original signals and augmented signals with different signal lengths across all subjects. Four baseline +classification methods as ITCCA, TRCA, EEGNet and C-CNN were validated on (a) Direction SSVEP Dataset, (b) Dial SSVEP Dataset. +On each dataset, the original signal length was set to 0.5 s and 1.0 s, corresponding to their augmented signal length 1.0 s and 2.0 s. +Fig. 5: Averaged classification results across subjects of four baseline classification methods as ITCCA, TRCA, EEGNet and C-CNN +using original signals and augmented signals at different scales of datasets on (a) Direction SSVEP Dataset, (b) Dial SSVEP Dataset. On +each dataset, the original signal length was set to 1.0 s, corresponding to its augmented signal length 2.0 s. The colored asterisk in the +figure indicates significant difference between original signals and augmented signals at this scale of dataset by paired t-tests (*p < 0.05, +**p < 0.01, ***p < 0.001). + +*** +*** +*** +*** +*** +*** +*** +*** +*** +100 +100 +90 +90 +Accuracy(%) +Accuracy(%) +80 +80 +70 +70 +60 +60 +50 +50 +40 +40 +ITCCA +TRCA +EEGNet +C-CNN +ITCCA +TRCA +EEGNet +C-CNN +30 +30 +small_org +small_aug +midle_org +middle_aug +large_org +large_aug +small_org +small_aug +middle_org +middle_aug +large_org +large_aug +(a) +(b)0.5s (Original) +1.0s (Augmented) +1.0s (Original) +2.0s (Augmented) +0.5s (Original) +1.0s (Augmented) +1.0s (Original) +12.0s (Augmented) +ITCCA +ITCCA +100 +100 +20 +20 +10 +10 +TRCA +0 +C-CNN +TRCA +C-CNN +EEGNet +EEGNet +(a) +(b)8 +GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2022 +CCA and MSI were selected to conduct evaluation. Fig. 6 illustrates +the averaged classification result across all subjects of two methods +on two SSVEP datasets. The results demonstrate the TEGAN could +significantly improve the performance of CCA and MSI on both +SSVEP datasets. Especially in the transformation scenario of 0.5 s to +1.0 s on Direction SSVEP dataset, the classification performance of +CCA and MSI has nearly doubled. The impressive result substantiates +the great potential of using TEGAN to assist unsupervised algorithms +for implementing high-performance zero-calibration BCI systems. +From Fig. 4 and Fig. 5, we can find an interesting phenomenon that +reconstructed long-time SSVEP signals are capable of reducing the +performance gap among various classification methods. To explain +this phenomenon, we visualize extracted features of a representative +subject (subject 5 on Dial SSVEP Dataset) for four baseline clas- +sification methods. It can be observed from Fig. 7 that there is a +large gap in the features extracted from each category on the original +signals, while the gap is gradually narrowed on the augmented +signals. Specifically, for the classification algorithms that previously +had not obvious features of the target stimulus on the original signal, +the features were further amplified on the augmented signals. In +other words, the augmented signals possess more discriminative or +representative features of the target stimulus compared to the original +signal, which makes it easier for the classification algorithm to +distinguish between target and non-target stimuli and thus facilitates +the improvement of the classification performance. +Extensive studies have proved the significance of filter bank +technologies in enhancing the recognition performance of SSVEPs +[10], [22], [33], [56]–[58]. The filter bank technology is based on +the premise that the brain-generated SSVEP signal has a distinctive +peak at the harmonic or subharmonic frequency of the flash stimulus. +Generally, the amplitude of the fundamental frequency peak in the +SSVEP signal is higher than that of the harmonic and subharmonic +peaks [59]. For the information fusion process of different frequency +bands of filter bank technology, the frequency band where the funda- +mental wave is located has a higher weight, while the frequency band +where the harmonic wave is located has a lower weight. However, +based on the particularity that the harmonic component or subhar- +monic component of SSVEP signal is not obvious, we speculate +that this feature would bring great trouble to GAN in the process +of learning to generate SSVEP data. Deep neural networks may +tend to preferentially capture the fundamental frequency information +with dominated characteristics, while ignoring the harmonic and +subharmonic information with less obvious characteristics. To verify +this hypothesis, we made a comparison of time and frequency domain +representation between real SSVEP data and generated SSVEP data. +The TEGAN was trained by 20% SSVEP data, and two frequencies +(6.67 Hz and 12.0 Hz) were chosen for evaluation. As shown +in Fig. 8, we could observe that the amplitude and trend of the +original SSVEP signal and the generated SSVEP signal at the two +frequencies differ sufficiently in the time domain. In addition, both +the original signal and the generated signal have significant peaks on +the fundamental frequency in the frequency domain characterization, +while the identifiable harmonic components contained in the original +signal are hard to reproduce in the generated signal. +According to the limitations and challenges of current study, at +least the following aspects could be further improved in the future. +Firstly, to the best of our knowledge, this is the first research using +GAN technology to generate more than 10 categories of SSVEP data +(Dial SSVEP dataset). However, with the increasing number of cate- +gories, it would become extremely difficult for GAN to simulate the +real data distribution. Therefore, we should strive to generate SSVEP +data with more categories, such as Benchmark [60] and BETA dataset +[61]. Secondly, in this paper, our GAN model could only expand the +length of the original signal twice. In the subsequent research, we +could further improve the network architecture of TEGAN, enabling +it to expand more multiples of the signal length. Moreover, filter bank +technology should be incorporated in the framework design as well, +which could help the GAN model excavate the harmonic information +that has an important contribution to the identification process, thus +improving the quality of generated data. Last but not least, the most +advanced transfer learning technique, such as improved two-stage +training strategy [62], could be utilized to build a high-performance +BCI system with shorter calibration time. +Fig. 6: Averaged classification results across subjects of two training- +free classification methods as CCA, MSI using original signals and +augmented signals on Direction and Dial SSVEP dataset. The red +asterisk in the figure indicates significant difference between original +signals and augmented signals at this transformation scenario by +paired t-tests (**p < 0.01, ***p < 0.001) +V. CONCLUSION +Building a high-performance SSVEP-BCI system with limited +calibration data is an urgent demand for the BCI community. In +this study, we proposed a novel GAN-based augmentation strategy to +enhance the performance of SSVEP-based BCI. Rather than simply +enlarging the training dataset, the proposed augmentation method +served as a signal converter that could be employed to extend +data length. Experimental results based on a 4-class and a 12-class +SSVEP datasets demonstrate that the proposed augmentation method +could significantly improve the recognition accuracy of traditional +methods and deep learning methods with limited subject-specific +data. Furthermore, it holds transformative promise to build a high- +performance SSVEP-BCI system with zero-calibration procedure +for new users. Overall, the proposed augmentation method could +significantly shorten the calibration time and uncover the underlying +subject-invariant properties of SSVEP data, which facilitates the +implementation of various SSVEP-based BCI applications. +ACKNOWLEDGEMENT +This work was supported in part by the National Natural Science +Foundation of China under Grant No.62076209. +REFERENCES +[1] J. R. Wolpaw et al., “Brain-computer interface technology: a review +of the first international meeting,” IEEE transactions on rehabilitation +engineering, vol. 8, no. 2, pp. 164–173, 2000. +[2] H.-J. Hwang, S. Kim, S. Choi, and C.-H. Im, “Eeg-based brain- +computer interfaces: a thorough literature survey,” International Journal +of Human-Computer Interaction, vol. 29, no. 12, pp. 814–826, 2013. +[3] R. Abiri, S. Borhani, E. W. Sellers, Y. Jiang, and X. 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Guney et al., “Transfer learning of an ensemble of dnns for +ssvep bci spellers without user-specific training,” Journal of Neural +Engineering, 2022. + diff --git a/VdE5T4oBgHgl3EQfbw_i/content/tmp_files/load_file.txt b/VdE5T4oBgHgl3EQfbw_i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb345b54ad07ed57eb9e4554d58041362113a05f --- /dev/null +++ b/VdE5T4oBgHgl3EQfbw_i/content/tmp_files/load_file.txt @@ -0,0 +1,910 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf,len=909 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='05599v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='NC] 13 Jan 2023 GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, XXXX 2022 1 Short-time SSVEP data extension by a novel generative adversarial networks based framework Yudong Pan ID,Student Member, IEEE, Ning Li ID, Yangsong Zhang ID Abstract— Objective: Steady-state visual evoked potentials (SSVEPs) based brain-computer interface (BCI) has received con- siderable attention due to its high transfer rate and available quan- tity of targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' However, the performance of frequency identifica- tion methods heavily hinges on the amount of user calibration data and signal data length, which hinders the deployment in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Recently, generative adversarial networks (GANs)- based data generation methods have been widely adopted to cre- ate supplementary synthetic electroencephalography (EEG) data, holds promise to address these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Methods: In this paper, we proposed a GAN-based end-to-end signal transformation network for data length window extension, termed as TEGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' TEGAN trans- forms short-time SSVEP signals into long-time artificial SSVEP signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' By incorporating a novel U-Net generator architecture and auxiliary classifier into the network design, the TEGAN could produce conditioned features in the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Additionally, to regularize the training process of GAN, we introduced a two- stage training strategy and the LeCam-divergence regularization term during the network implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Results: The proposed framework was evaluated on two public SSVEP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' With the assistance of TEGAN, the performance of traditional frequency recognition methods and deep learning-based methods have been significantly improved under limited calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Conclusion: This study substantiates the feasibility of the proposed method to extend the data length for short-time SSVEP signals to develop a high-performance BCI system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Significance: The proposed GAN- based methods have the great potential of shortening the calibra- tion time for various real-world BCI-based applications, while the novelty of our augmentation strategies shed some value light on understanding the subject-invariant properties of SSVEPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Index Terms— brain-computer interface (BCI), steady- state visual evoked potential (SSVEP), electroencephalog- raphy (EEG), generative adversarial network (GAN) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' INTRODUCTION B RAIN-COMPUTER interface (BCI) has shown to become a promising technology that can provide its users with communi- cation channels that do not depend on conventional output channels of peripheral nerves and muscles by decoding their neural activities into specific control commands [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Among various neuroimaging modal- ities to implement a BCI system, electroencephalography (EEG) has been the most prominent signal accounting for such the advantages as non-invasiveness, high temporal resolution, affordability, ease of implementation, portability, and convenience of use [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Several most popular paradigms can be employed to build EEG- based BCI systems, such as motor imaginary (MI) [4], P300 event This work was supported in part by the National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='62076209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Pan is with the School of Computer Science and Technology, Southwest University of Science and Technology, Mianyang 621010, China (e-mail: panydacademy@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Li is with the School of Computer Science and Technology, South- west University of Science and Technology, Mianyang 621010, China (e-mail: liningacademy@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Zhang is with the the School of Computer Science and Technology, Southwest University of Science and Technology, Mianyang 621010, China, and also with MOE Key Laboratory for Neuroinformation, Clin- ical Hospital of Chengdu Brain Science Institute, University of Elec- tronic Science and Technology of China, Chengdu 610054, China (e- mail:zhangysacademy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' related potentials (P300) [5], auditory steady-state response (ASSR) [6], and steady-state visual evoked potential (SSVEP) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Among them, SSVEP-based BCI systems have received considerable atten- tion due to its high transfer rate (ITR) and available quantity of targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' SSVEPs refer to periodic evoked potentials over occipital scalp areas, in response to rapidly repetitive visual stimulation flicking or reversing at a specific frequency [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The SSVEP signal consists of a number of discrete frequency components, normally including the fundamental frequency of the visual stimulus and its harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' On the strengths and characteristics of SSVEP, numerous SSVEP-based BCI applications have been developed, such as bionic mechanical leg [9], unmanned aerial vehicle [10], dial interface [11], high-speed mental speller [12], smart homes [13], and games [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' To design a high-performance BCI system based on SSVEP signal, the most crucial aspect is to develop a fast and accurate frequency recognition method that can distinguish the stimulus frequency of the target gazed by the users through analyzing the EEG signal in the shortest possible time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Thus, various cutting-edge algorithms have been proposed based on different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Generally, the frequency identification algorithms can be divided into three categories: training-free methods, user-dependent (UD) training methods and user-independent (UI) training methods [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Training-free identification methods do not require any training data from the user of the BCI, from which the user can interact directly with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The representative training-free methods are canon- ical correlation analysis (CCA) [7] and multivariate synchronization index (MSI) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' CCA seeks to capture the underlying correlation between EEG data and a series of sinusoidal reference templates corresponding to the stimuli frequencies, while MSI adopts the S- estimator to estimate the synchronization index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' CCA and MSI are able to achieve comparable performance when only a few stimulus targets are available and the EEG signals are sufficiently long, but their performance drops dramatically when encountering a large number of targets and short-time signals [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' By incorporating the user-specific training data into the algorithm design, the UD methods could derive better performance than training-free methods under these harsh conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The UD approaches take the discrimination of recorded signals from different subjects into account, namely subject-variant features, such as magnitude, phase, and visual la- tency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Draw support from individual training data to learn these properties, the performance of recognition algorithm would be greatly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The typical UD algorithms mainly comprise individual template CCA (ITCCA) [18], multiway CCA (MCCA) [19], task- related component analysis (TRCA) [20], and correlated component analysis (CORCA) [21], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In contrast to the UD methods, UI methods are envisaged to build a generalized model for detection of unseen subjects via studying the subject-invariant features that learning from the existing subject datasets or human prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For UI algorithms, subject-specific training data is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' However, their performance usually is worse than UD algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Theoretically, training-free algorithms are all belong to UI methods for their user-independent properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The prevalent UI approaches include filter bank canonical correlation analysis (FBCCA) [22], transfer template-based canonical correlation analysis (ttCCA) [23] LOGO2 GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, XXXX 2022 and adaptive combined CCA (A3C) [24], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Although UD approaches commonly exhibit better performance than UI algorithms, they are heavily dependent on the amount of collected user-sepcific calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Actually, collecting user calibration data is a time-consuming and laborious process, and prolonged experiments would cause user’s fatigue, leading to the decreased quality of induced SSVEP signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Hence, how to de- velop a high-performance frequency recognition algorithm that needs only a little calibration data or calibration-free has become a hot research topic in recent years [23], [25]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Benefit from the rapid development of deep learning (DL) in the past decade, there has been an increased interest in applying DL algorithms to detect SSVEPs for BCI researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In view of powerful feature represen- tation capacity and flexibility, the DL-based methods hold promise for reducing the gap between UD approaches and UI approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For instance, Waytowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' employed a compact convolutional neural network (Compact-CNN) entitled EEGNet to conduct inter- subject classification, which yielded about 80% accuracy in a 12- class SSVEP dataset without any user calibration data [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' utilized fast Fourier transform (FFT) to design a complex spectrum CNN (C-CNN) which could be trained on both UD and UI schemes, achieving about 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5% (UD), 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='6% (UI) and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='3% (UD), 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='6% (UI) accuracy on a 12-class and a 7-class SSVEP dataset, respectively [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Guney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' proposed a two-stage training strategy and a deep neural network (DNN) to improve performance of intra- subject classification [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The proposed framework was evaluated on Benchmark SSVEP dataset that contains 40 stimulus targets, and has obtained approximately 95% recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' developed an efficient CNN-LSTM (Long short-term memory) network with spectral normalization and label smoothing technolo- gies, termed as SSVEPNet, for SSVEP classification under the small sample size and short-time window scenarios [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' SSVEPNet was verified on a 4-class and a 12-class SSVEP dataset, yielding about 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5% identification accuracy when a few trials of each stimulus are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' introduced a Transformer-based deep neural network model named SSVEPformer for enhancing the performance of zero-calibration SSVEP-BCI [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The experimental results have shown that SSVEPformer could achieve high accuracy of about 84% and 80% on a 12-class and Benchmark SSVEP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' On the other hand, in addition to explicitly modelling an efficient frequency recognition method that requires less calibration data, recent studies have substantiated the potential of using the generative models to address the issues of data shortage in the SSVEP classi- fication tasks [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Generative models aim to learn the distribution of real data by constructing a probabilistic statistical model given real data and using it to generate synthetic data that approximate the distribution of real data [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Autoregressive models [36], variational auto-encoder (VAE) [37], generative adversarial network (GAN) [38], and denoising diffusion probabilistic model (DDPM) [39] are the most commonly generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Among them, GAN has been the most widely applied technique to synthesize fake data and overcome the problem of limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Since EEG-GAN [40], the first GAN- based generative model of EEG signal, was proposed in 2018, BCI researchers have successively developed several GAN-based SSVEP generation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In 2019, Aznan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' firstly exploited the GAN- based generative model in circumventing the limited calibration data via generating supplementary synthetic data to enlarge the size of training data [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Only one year later, they also proposed a subject- invariant SSVEP GAN (SIS-GAN) to generate artificial EEG data that learns the subject-invariant features from the multiple SSVEP categories [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' After two years, inspired by StarGAN v2, which has been used to solve multidomain image-to-image conversion, Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' proposed a novel multidomain signal-to-signal transformation method which is capable of generating artificial SSVEP signals from resting EEG [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Although these GANs based on SSVEP signals have made notice- able progress, these studies only focus on generating simulated data to enlarge the amount of the training dataset, without considering the extension of signal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' However, longer SSVEP signal length would often achieve more accurate recognition results under the same conditions [30]–[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Hence, in this study, we proposed a GAN-based end-to-end signal transformation network for data length window extension, termed as TEGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' TEGAN transforms short- time SSVEP signals into long-time artificial SSVEP signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' By incorporating a novel U-Net generator architecture and auxiliary classifier into the network design, the TEGAN could produce con- ditioned features in the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Additionally, to regularize the training process of GAN, we introduced a two-stage training strategy and the LeCam-divergence regularization term during the network implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The proposed menthods were evaluated on two public SSVEP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' With the assistance of TEGAN, the performance of traditional frequency recognition methods and DL-based methods have been significantly improved under limited calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The extensive experimental analysis demonstrates the effectinveness of the proposed methods, while the novelty of our augmentation strategies shed some value light on understanding the subject-invariant properties of SSVEPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Dataset In this study, two public SSVEP datasets were employed to evaluate the proposed augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' According to the de- sign purpose of these two datasets, we hereinafter termed them as Direction SSVEP dataset and Dial SSVEP dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The specific details of each dataset are described as follows: 1) Direction SSVEP dataset: This dataset was published by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' in 2019 [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Fifty-four healthy subjects (25 females, aged 24-35 years) participated in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The experiment collected EEG data of subjects in two different periods (Session1 and Session2), and the data in each period was divided into offline analysis stage and online testing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For the sake of simplicity, the offline data from Session1 was chosen for experimental evaluation in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the process of data acquisition, the four target stimuli were coded by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='45 Hz, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='67 Hz, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='57 Hz and 12 Hz , and played in the lower, right, left and upper directions of the personal computer (PC) display, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Participants were asked to focus on the center of the black screen, and then on the direction of the target stimulus highlighted in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Each SSVEP stimulus was presented for 4 s, and the interval between two stimuli was 6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Each target frequency was presented 25 times, leading to a total of 100 trials (4 classes × 25 trial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' EEG data of 62 Ag/AgCl electrodes collected at a sampling rate of 1000 Hz were recorded in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Ten electrodes (P7, P3, Pz, P4, P8, PO9, PO10, O1, Oz and O2) covering the occipital lobe area were selected for our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' All data was down sampled to 100 Hz and band-pass filtered between 4 and 40 Hz through a fourth-order Butterworth band-pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2) Dial SSVEP dataset: This open access dataset was provided by Nakanishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' in 2015 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In this dataset, ten healthy subjects (1 female, mean age:28 years) participated in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The subjects were instructed to sit in a comfortable chair 60 cm in front of a liquid crystal display (LCD) monitor in a dim room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The stimuli was arranged in a 4×3 grid space as simulation a dial interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Twelve flickering stimuli (f0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='25Hz, ∆f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5Hz) were presented on the monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The EEG data of eight Ag/AgCl electrodes (PO7, PO3, POZ, PO4, PO8, O1, Oz and O2) covering the occipital were acquired using PAN et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' :SHORT-TIME SSVEP DATA EXTENSION BY A NOVEL GENERATIVE ADVERSARIAL NETWORKS BASED FRAMEWORK 3 the BioSemi ActiveTwo EEG system with a sampling rate of 2048 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For each subject, there were 15-run experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In each run, 12 trials corresponding to all 12 stimuli were generated in a random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Thus, a total of 180 trials were collected in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Each trial was composed of 1 s cuing period and 4 s targeted flickering period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' All data was down sampled to 256 Hz and band-pass filtered between 6 and 80 Hz through a fourth-order Butterworth band-pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' To suppress the adverse effect of visual latency, the data was extracted from the 135 ms after the stimulus onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Frequency recognition methods In this subsection, we briefly introduce two traditional frequency recognition methods and two DL-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' All of these state- of-the-art methods were adopted as baselines to verify the effective- ness of the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1) Traditional Methods: IT-CCA:CCA is a multivariate statistical technique to search the underlying correlation between two multidimensional variables, by calculating a pair of weight vectors to maximize the Pear- son’s correlation coefficients between their linear combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For CCA-based SSVEP recognition methods, the coefficient is calculated between the multi-channel EEG data and the artificial reference signal [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' However, the artificial reference signals lack the specific characteristics of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Then, IT- CCA was proposed to substitute the artificial reference signal for individual template reference signal obtained by averaging multiple training trials of specific subject [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' This strategy has been proved to be able to suppress spontaneous EEG interference and achieve better classification performance than CCA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' TRCA:TRCA is a method which learns the spatial filters to extract task related components by maximizing the reproducibil- ity of neuroimaging data during the task period [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For SSVEP data, TRCA seeks to find a linear weight vector to maximize the inter-trial correlation of its linear combinations, referring as task-related components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In 2018, Nakanish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' first applied TRCA algorithm to build a high-speed SSVEP- based BCI system [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the literature, TRCA is extended with the filter bank technique, and the selection of filter banks follows the principle proposed by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For better comparison with other methods, the original TRCA algorithm without filter bank technology was adpoted in the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2) DL-based methods: EEGNet: EEGNet is a robust DL model which could yield comparable performance across multiple EEG tasks and datasets [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' It is mainly comprised of a temporal filtering layer, a depthwise convolution layer, a separable convolution layer, and a fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Among these network components, depthwise and separable convolution make the model become compact and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Due to its effectiveness, Waytowich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' used EEGNet to decode SSVEP signals for inter-subject classification on Dial SSVEP dataset, and has achieved about 80% accuracy [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' C-CNN: The frequency domain of SSVEP data contains abun- dant frequency and phase information relevant to the recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' If this information could be adequately exploited, the classification performance could be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Therefore, Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' utilized FFT to transform SSVEP signals from the time domain to the frequency domain and designed a shallow CNN consisting of a spatial filter layer, a convolutional layer, and a fully connected layer to handle complex spectral data [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' C-CNN was evaluated on a 7-class and Dial SSVEP dataset, yielding satisfactory results for both intra- and inter-subject classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The proposed augmentation methods Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1: The procedure flowchart of the proposed augmentation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The whole process is divided into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the first step, real short EEG and real long EEG of training dataset are used to train the GAN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the second step, the pretrained generator is employed to transform all input short EEG into synthetic long EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Then the synthetic long EEG are used to train the EEG classifier and conduct classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1) Overall framework: The procedure flowchart of the proposed augmentation method is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The whole process is divided into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the first step, we train the TEGAN, which could transform short time-window natural EEG into long time-window artificial EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Concretely, followed by the auxiliary classifier GAN (ACGAN) paradigm [47], TEGAN mainly includes two components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=', a generator and a discriminator, competing in a zero-sum game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The generator receives the real short EEG as network input, then output the extended EEG data, namely long artifical EEG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The discriminator is responsible for distinguishing between long real EEG data and long fake EEG data and simultaneously identifying their respective classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Let VG and VD denote the training objectives of the generator G and discriminator D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Then the training of the proposed GAN frameworks can be generally expressed as: min G VG = E x∼pxs [−D(G(x))] − E x∼pxs [log(D(G(x) ∈ C))] (1) max D VD = E x∼pxl [1 − D(x)] + E x∼pxs [1 + D(G(x))] (2) + E x∼pxl [log(D(x) ∈ C))] + E x∼pxs [log(D(G(x) ∈ C))] where pxs and pxl is the data distribution of short EEG data and long EEG data from the training dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The notation D(x ∈ C) represents the probability of the class label being correctly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' To optimize these two objective functions, we use gradient descent algorithm to train discriminator and generator alternately, and obtain the optimal parameters of generator θ∗ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Hence, a short-to-long signal converter ζ(·) could be constructed using θ∗ G as follows: ζ(xs) = G(xs|θ = θ∗ G) = xl (3) Procedure InputEEG FakeExtensionEEG Flowchart G F(0) Adv RealExtensionEEG R(1) C=2 AUX C=3 Pretrain C=K Train EEG TrainExtensionEEG EEG Classifer TEGAN TestEEG TestExtensionEEG Validation4 GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, XXXX 2022 In the second step, the converter ζ(·) is employed to transform all short EEG into synthetic long EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Finally, the synthetic long EEG are used to train the EEG classifier and conduct classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' It is worth noting that since the converter is trained using only the training dataset and does not require label information as input, there is no data leakage problem during the entire operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The implementation code of TEGAN would be available in Github platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2) The architecture of generator: The detailed architecture of generator is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The whole architecture of the generator follows the U-Net architecture proposed by Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' [51], which is divided into down sampling stages and up sampling stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Moreover, considering the dependency between spatial-temporal features of EEG data, we utilize a bidirectional long short-term memory (Bi-LSTM) network to encode the features derived by the last down-sampling module, thus connecting the down sampling stages and up sampling stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Furthermore, we added a conditional batch normalization (cBN) layer in each up-sampling module to mitigate the adverse effects of homogenization between different synthetic samples [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' However, due to the inaccessibility of label information in the input end of generator, we adopted a fully connected layer to create high-confidence pseudo label, which can be improved via minimizing the following objective function LG: min G LG = VG − K � c=1 yc log pc (4) where yc is realistic label with one-hot encoding, pc is the predicted probability of class c, and K is number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The network parameters of generator are exhibited in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Among the parameters, the kernel size DK for down-sampling module could be determined through the following formula: DKi = ⌈Di ∗ ws⌉, i = 1, 2, 3 (5) where D = {20, 12, 8}, and ws represents the window size of input short EEG signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Then the kernel size UK for up-sampling module could be calculated as: UKi+1 = DT2−i − (DT3−i) ∗ Si, i = 0, 1, 2 (6) Here, S = {1, 2, 2}, and DT represents the number of time steps for down sampling module, which is presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 3) The architecture of discriminator: The minute architecture of discriminator is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Overall, the discriminator follows the design of our previously proposed SSVEPNet model [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' It is mainly comprised of three modules, namely spatial filtering module, temporal filtering module, and decision module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the spatial filtering module, a one-dimensional convolution (1D Conv) layer is used to fuse different channel information of EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the temporal filtering layer, 1D Conv layer is employed to extract temporal features of EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the decision module, a Bi-LSTM layer is leveraged to learn the dependence between spatial-temporal features, while fully connected layers are employed for classification and authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The distinction between SSVEPNet and proposed discriminator only lies in two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' One is that a max pooling layer with a kernel size of 2 has been added to the temporal filtering module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Another is that the fully connection layer in the decision module has changed from three layers to two layers, and the neurons in the first layer is equal to twentieth of encoded spatial-temporal features outputted by BiLSTM layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The remaining network parameters of the discriminator are identical with SSVEPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 4) Regularizing TEGAN on limited data: GAN is notoriously difficult to train, it is highly susceptible to encounter with mode col- lapse phenomenon - particularly from datasets with high variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Furthermore, this problem may deteriorate with a limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Generators are inclined to merely memorize limited training samples, rather than learning the sophisticated data distribution of training dataset [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' To moderate this issue, we introduce a two- stage training strategy and LeCam divergence regularization term to regularize the training process of TEGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Firstly, inspired by the previous study [31], we utilized transfer learning technique to design a two-stage training strategy for the proposed GAN framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the first stage, we train two global models Ds and Gs using the cross-subject data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the second stage, the network parameters of Ds and Gs are directly duplicated to two target models, Dt and Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Then we freeze the parameters in Dt in addition to the fully connection layers, and fine-tune Dt and Gt using a limited amount of training data of target subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Secondly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' we supplemented a LeCam regularization term in the training objective of discriminator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' which has been substantiated that could offer meaningful constraints under limited training data [52]: min D LD = VD + λRLC(D) (7) where λ represents regularization term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' and LeCam regularization term RLC(D) is expressed as: RLC(D) = E x∼pxl [∥D(x) − αF ∥2] − E x∼pxs [∥D(G(x)) − αR∥2] (8) where αF and αR are anchors obtained by the exponential moving average variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' which is aiming at tracking the discriminator predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Assuming that there are T E training epochs in total, then αF and αR could be calculated using the following equations: � αF (i + 1) = γD(G(xs)) + (1 − γ)αF (i) αR(i + 1) = γD(xl) + (1 − γ)αR(i) , SE ≤ i ≤ T E (9) Here, γ is decay coefficient, and SE represents the starting epoch to implement LeCam regularization term, which is conducive to avoid the excessive regularization in the initial stage with under fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Experimental evaluation To validate the efficacy of the proposed augmentation method, we conducted intra-subject classification experiments with limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Specifically, the original data from a target subject is divided into training set Ttr(Xs) and testing set Ttt(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' We exploit the two-stage training strategy to optimize the parameters of the generator Gt, and then employ it to transform the Ttr(Xs) and Ttt(Xs) into Ttr(Xl) and Ttt(Xl), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Finally, the Ttr(Xl) is used to train four SSVEP classifiers as mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='2, and undertake evaluation on Ttt(Xl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' To eliminate the randomness of data partitioning, the K-Fold evaluation strategy was adopted in the experiments, under which the parameters of Gs remained unchanged while Gt was updated K times in the whole process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In this study, we implemented the proposed augmentation method in PyTorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The hyperparameters on two datasets are set as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Direction: For the first training stage, mini-batch (B) = 64, epochs (E) = 200, learning rate (lr) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='001, optimizer (Opt) = Adam (beta1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='9, beta2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='999), weight decay (wd) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For the second training stage, B = 20, E = 500, lr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='01, Opt = Adam (beta1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='9, beta2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='999) + Cosine Annealing, wd PAN et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' :SHORT-TIME SSVEP DATA EXTENSION BY A NOVEL GENERATIVE ADVERSARIAL NETWORKS BASED FRAMEWORK 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2: The architecture of generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The whole architecture of the generator follows the design of U-Net, which is divided into two stages: down sampling and up sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Where Nc is the number of EEG channels, Nt denotes the number of sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' K, S represents the size of convolution/deconvolution kernel and stride, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DKi, UKi (i=1, 2, 3) corresponding to kernel size for down sampling stage and up sampling stage, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 3: The architecture of discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The discriminator follows the design of SSVEPNet, which mainly comprised of a spatial filtering module, a temporal filtering module, and a decision module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For the regularization term, SE = 50, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='6, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Dial:All hyperparameters are identical with Direction SSVEP dataset, except for the min-batch in the second training stage, which is set to 24 in this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Statistical analysis In this study, classification accuracy was adopted as the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The average classification results across all subjects for K- times validation were presented in the form of mean ± standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Two transformation scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5 s to 1 s and 1 s to 2 s, were analyzed in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Paired t-tests were implemented to investigate whether there were significant differences in the classification accuracy between all pairs of methods at each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' RESULTS Firstly, we investigated the results of four baseline methods using limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In this setting, the TEGAN that used to extend signal length was trained by 20% SSVEP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 4 shows the averaged classification results of original signals and augmented signals with different signal lengths across all subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' We could observe that augmented signals yielded better classification perfor- mance than original signals on both two transformation scenarios, and the results of two SSVEP datasets manifest the consistent tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Interestingly, we could find that the augmented signals at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s have achieved better results than original signals at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s for three methods (ITCCA, EEGNet, and C-CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Moreover, we investigated the results how the performance of four baseline methods varied with different scales of dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' the SSVEP dataset from each subject was divided into the training Spatial Filtering Module Temporal Filtering Module Decision Module InputEEG FC3 +Label BatchNorm 1DConv 1D Conv BatchNorm PReLU Dropout PReLU MaxPool ISTM FC2 Real/FakeOutputEEG (Nc×2Nt) Spatial 1DConvolution Temporal1DConvolution Temporal1DDeconvolution Spatial1DDeconvolution Up Stage5 DataFlow LabelEmbedding--SkipConnection K=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='S=2 InputEEG- Down Stage1 Up Stage4 (Nc × Nt) K = Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' S= Nc K=Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='S=Nc Down Stage2 Up Stage3 K=DK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='S =2 K=UK3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='S=2 Down Stage3 Up Stage2 K =DK2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='S=2 K=UK2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='S=2 Down Stage4 Up Stage1 K=DK3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='S=1 K=UK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='S=1 Reshape Reshape BiLSTM Stage PseudoLabel FcStage6 GENERIC COLORIZED JOURNAL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, XXXX 2022 TABLE I: The detailed network parameters of generator Block Module Layer Output Description Down Sampling Input Module Input (b, 1, Nc, Nt) (batch size, 1, num channels, num time steps) DownStage Module(× 4) Conv1D (b, DCi, 1, DTi) kernel = {Nc, DK1, DK2, DK3} stride = {Nc, 2, 2, 1} Ci = 2i+1 ∗ Nc, i = {0, 1, 2, 3} DT0 = Nt, DTi+1 = (DTi − kernel[i]) // stride[i] + 1 Norm (b, DCi, 1, DTi) max norm (= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0) + BN, i = 0 SN [48] + BN, i = 1, 2, 3 Activation (b, DCi, 1, DTi) PReLU [49] Dropout (b, DCi, 1, DTi) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5 Reshape Module Reshape (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DT3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DC3) BiLSTM Encoding BiLSTM Module LSTM (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2 ∗ DT3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DC3) input size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' hidden size = DC3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' num layers = 1 batch first = True,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' bidirectional = True Reshape (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2 ∗ DT3 ∗ DC3) AvgPool (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DT3 ∗ DC3) kernel = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' stride = 2 Pesudo Label Generation FC Module Flatten (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DT3 ∗ DC3) Linear (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' K) Norm (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' K) SN Selection Module Argmax (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1) select the highest probability as pseudo label Up Sampling Reshape Module Reshape (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DC3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DT3) UpStage Module(× 5) DeConv1D (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UCi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UTi) kernel = {UK1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UK2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UK3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2} stride = {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2} UC = {8 ∗ Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 8 ∗ Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 4 ∗ Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2 ∗ Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Nc} UT = {DT2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DT1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DT0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Nt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2 ∗ Nt} Norm (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UCi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UTi) SN + cBN [50] Activation (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UCi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UTi) PReLU Conv1D (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UCi // Mi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UTi) kernel = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' stride = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' M = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Nc} only exist when i ≥ 1 Norm (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UCi // Mi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UTi) SN + cBN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' only exist when i ≥ 1 Activation (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UCi // Mi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' UTi) PReLU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' only exist when i ≥ 1 Output Module Output (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 2 ∗ Nt) dataset and testing dataset in the portion of 2:8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 5:5 and 8:2 respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In this study, we marked these three scales of datasets as small- scale, middle-scale and large-scale dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 5 illustrates averaged classification results across subjects of four baseline classification methods using original signals and augmented signals at different scales of datasets on two SSVEP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' With the continuous expansion of training data, the average classification performance of each baseline algorithm for the original signal and the augmented signal has been gradually improved, and the trend is consistent on both two SSVEP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In addition, the augmented signal improves the classification performance on a small-scale training dataset more significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' With the continuous expansion of training data, the im- provement effect of classification performance of augmented signals is gradually weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' More interestingly, when original signals were converted to augmented signal by TEGAN on any scale of dataset, the classification performance gap of all algorithms is significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Furthermore, we conducted ablation experiments on the Dial SSVEP dataset to explore the contribution of pivotal component to implement the augmentation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The augmented data at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s was generated by original data at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s, and the signal converter TEGAN was trained by 20% training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Four modules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' auxiliary classifier in the discriminator, the pseudo-label generation in the generator, the two-stage training strategy, and the LeCam divergence regularization term were investigated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' As shown in Table II, the averaged classification results indicate that four important components of TEGAN are all helpful to improve the classification performance of augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' It is worth noting that after removing the two-stage training strategy from TEGAN, the model would be trained using only 20% data of target subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' DISCUSSION In recent years, enhancing the classification performance of SSVEPs with limited calibration data is a hot research topic, which empower the practicality of various BCI applications [32], [53]–[55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' To this end, modelling an efficient classification method requiring less calibration data or create supplemented synthetic data to enlarge the size of the training dataset are most commonly strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Never- theless, in this study, we proposed a novel pipeline that leverage the subject-invariant properties of SSVEPs to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Specifically, we developed a GAN-based model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=', TEGAN, which could be used to extend the data length of SSVEP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' TEGAN seeks to learn the mapping relationship between short-time and long-time signals through an adversarial game training paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Concretely, the generator is responsible for extracting signal features from the short-time SSVEP data and reconstructing the long-time SSVEP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The discriminator assists the generator to improve the quality of the generated data by learning the discrepancy information between the real and fake long-time signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' After analyzing the result in Table II, we could conclude that the two-stage training strategy plays an indispensable role in enhanc- ing the performance of TEGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' However, although the two-stage training strategy can greatly improve BCI performance with only a small amount of target subject data, it still suffers the laborious calibration procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Implementing a high-performance BCI system with zero calibration for new users has always been the ultimate goal of the BCI community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In this study, we verify the feasibility via improving the performance of training-free methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Concretely, we employ the cross-subject data to train TEGAN, and then use it to extend the signal length of target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Two representative methods as PAN et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' :SHORT-TIME SSVEP DATA EXTENSION BY A NOVEL GENERATIVE ADVERSARIAL NETWORKS BASED FRAMEWORK 7 TABLE II: Albation study on Dial SSVEP Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Module Selction Accuracy(%) Case Auxiliary Classifier Pseudo Label Two Stage LeCam Divergence ITCCA TRCA EEGNet C-CNN (a) – ✓ ✓ ✓ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='56±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='86** 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='26±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='23*** 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='70±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='90** 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='68±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='63** (b) ✓ – ✓ ✓ 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='24±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='40 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='61±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='82 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='57±23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='59*** 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='93±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='53 (c) ✓ ✓ – ✓ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='40±23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='34** 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='92±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='24 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='14±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='47* 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='24±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='47* (d) ✓ ✓ ✓ – 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='22±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='92** 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='43±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='44*** 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='72±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='02** 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='41±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='91** (e) ✓ ✓ ✓ ✓ 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='61±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='54 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='44±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='19 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='54±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='40 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='15±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='91 ‘–’ denotes which module is deleted from the proposed GAN model, and ‘✓’ denotes which module is remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The asterisk in the table indicate significant difference between each pair of the two methods by paired t-tests (*p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='05, **p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='01, ***p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='001) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 4: Averaged classification results of original signals and augmented signals with different signal lengths across all subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Four baseline classification methods as ITCCA, TRCA, EEGNet and C-CNN were validated on (a) Direction SSVEP Dataset, (b) Dial SSVEP Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' On each dataset, the original signal length was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5 s and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s, corresponding to their augmented signal length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 5: Averaged classification results across subjects of four baseline classification methods as ITCCA, TRCA, EEGNet and C-CNN using original signals and augmented signals at different scales of datasets on (a) Direction SSVEP Dataset, (b) Dial SSVEP Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' On each dataset, the original signal length was set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s, corresponding to its augmented signal length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The colored asterisk in the figure indicates significant difference between original signals and augmented signals at this scale of dataset by paired t-tests (*p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='05, **p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='01, ***p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' *** *** *** *** *** *** *** *** *** 100 100 90 90 Accuracy(%) Accuracy(%) 80 80 70 70 60 60 50 50 40 40 ITCCA TRCA EEGNet C-CNN ITCCA TRCA EEGNet C-CNN 30 30 small_org small_aug midle_org middle_aug large_org large_aug small_org small_aug middle_org middle_aug large_org large_aug (a) (b)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5s (Original) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s (Augmented) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s (Original) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s (Augmented) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5s (Original) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s (Augmented) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s (Original) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s (Augmented) ITCCA ITCCA 100 100 20 20 10 10 TRCA 0 C-CNN TRCA C-CNN EEGNet EEGNet (a) (b)8 GENERIC COLORIZED JOURNAL, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' XX, XXXX 2022 CCA and MSI were selected to conduct evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 6 illustrates the averaged classification result across all subjects of two methods on two SSVEP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The results demonstrate the TEGAN could significantly improve the performance of CCA and MSI on both SSVEP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Especially in the transformation scenario of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5 s to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s on Direction SSVEP dataset, the classification performance of CCA and MSI has nearly doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The impressive result substantiates the great potential of using TEGAN to assist unsupervised algorithms for implementing high-performance zero-calibration BCI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 5, we can find an interesting phenomenon that reconstructed long-time SSVEP signals are capable of reducing the performance gap among various classification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' To explain this phenomenon, we visualize extracted features of a representative subject (subject 5 on Dial SSVEP Dataset) for four baseline clas- sification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' It can be observed from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 7 that there is a large gap in the features extracted from each category on the original signals, while the gap is gradually narrowed on the augmented signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Specifically, for the classification algorithms that previously had not obvious features of the target stimulus on the original signal, the features were further amplified on the augmented signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In other words, the augmented signals possess more discriminative or representative features of the target stimulus compared to the original signal, which makes it easier for the classification algorithm to distinguish between target and non-target stimuli and thus facilitates the improvement of the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Extensive studies have proved the significance of filter bank technologies in enhancing the recognition performance of SSVEPs [10], [22], [33], [56]–[58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The filter bank technology is based on the premise that the brain-generated SSVEP signal has a distinctive peak at the harmonic or subharmonic frequency of the flash stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Generally, the amplitude of the fundamental frequency peak in the SSVEP signal is higher than that of the harmonic and subharmonic peaks [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' For the information fusion process of different frequency bands of filter bank technology, the frequency band where the funda- mental wave is located has a higher weight, while the frequency band where the harmonic wave is located has a lower weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' However, based on the particularity that the harmonic component or subhar- monic component of SSVEP signal is not obvious, we speculate that this feature would bring great trouble to GAN in the process of learning to generate SSVEP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Deep neural networks may tend to preferentially capture the fundamental frequency information with dominated characteristics, while ignoring the harmonic and subharmonic information with less obvious characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' To verify this hypothesis, we made a comparison of time and frequency domain representation between real SSVEP data and generated SSVEP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The TEGAN was trained by 20% SSVEP data, and two frequencies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='67 Hz and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 Hz) were chosen for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 8, we could observe that the amplitude and trend of the original SSVEP signal and the generated SSVEP signal at the two frequencies differ sufficiently in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In addition, both the original signal and the generated signal have significant peaks on the fundamental frequency in the frequency domain characterization, while the identifiable harmonic components contained in the original signal are hard to reproduce in the generated signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' According to the limitations and challenges of current study, at least the following aspects could be further improved in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Firstly, to the best of our knowledge, this is the first research using GAN technology to generate more than 10 categories of SSVEP data (Dial SSVEP dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' However, with the increasing number of cate- gories, it would become extremely difficult for GAN to simulate the real data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Therefore, we should strive to generate SSVEP data with more categories, such as Benchmark [60] and BETA dataset [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Secondly, in this paper, our GAN model could only expand the length of the original signal twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the subsequent research, we could further improve the network architecture of TEGAN, enabling it to expand more multiples of the signal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Moreover, filter bank technology should be incorporated in the framework design as well, which could help the GAN model excavate the harmonic information that has an important contribution to the identification process, thus improving the quality of generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Last but not least, the most advanced transfer learning technique, such as improved two-stage training strategy [62], could be utilized to build a high-performance BCI system with shorter calibration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 6: Averaged classification results across subjects of two training- free classification methods as CCA, MSI using original signals and augmented signals on Direction and Dial SSVEP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The red asterisk in the figure indicates significant difference between original signals and augmented signals at this transformation scenario by paired t-tests (**p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='01, ***p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='001) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' CONCLUSION Building a high-performance SSVEP-BCI system with limited calibration data is an urgent demand for the BCI community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In this study, we proposed a novel GAN-based augmentation strategy to enhance the performance of SSVEP-based BCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Rather than simply enlarging the training dataset, the proposed augmentation method served as a signal converter that could be employed to extend data length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Experimental results based on a 4-class and a 12-class SSVEP datasets demonstrate that the proposed augmentation method could significantly improve the recognition accuracy of traditional methods and deep learning methods with limited subject-specific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Furthermore, it holds transformative promise to build a high- performance SSVEP-BCI system with zero-calibration procedure for new users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Overall, the proposed augmentation method could significantly shorten the calibration time and uncover the underlying subject-invariant properties of SSVEP data, which facilitates the implementation of various SSVEP-based BCI applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work was supported in 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s Dial-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='5s to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s Dial-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0s to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0sPAN et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' :SHORT-TIME SSVEP DATA EXTENSION BY A NOVEL GENERATIVE ADVERSARIAL NETWORKS BASED FRAMEWORK 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 7: The features of a representative subject (subject 5 on Dial SSVEP Dataset) for four baseline classification methods as ITCCA, TRCA, EEGNet and C-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' ITCCA, TRCA use the correlation coefficients as features, while the prediction probabilities are treated as features by EEGNet and C-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' (a) Original signals at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s, (b) Augmented signals at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Red dot marks the correlation coefficient or prediction probability at the frequency of gaze following target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' 8: Comparison of time and frequency domain representation at Oz channel of a representative subject (subject 30 on Direction SSVEP Dataset) between real EEG data and generated EEG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' (a) SSVEP signal at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='67 Hz, (b) SSVEP signal at 12 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' The time window for the real EEG data is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s, while the generated data is transformed from the real EEG data at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE5T4oBgHgl3EQfbw_i/content/2301.05599v1.pdf'} +page_content=' In the frequency domain representation, red circles mark the fundamental frequency or harmonics.' metadata={'source': 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ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS +1 +A New Noncoherent Gaussian Signaling Scheme +for Low Probability of Detection Communications +Yuma Katsuki, Graduate Student Member, IEEE, Giuseppe Thadeu Freitas de Abreu, Senior Member, IEEE, +Koji Ishibashi, Senior Member, IEEE, and Naoki Ishikawa, Senior Member, IEEE. +Abstract—We propose a novel, Gaussian signaling mechanism +for low probability of detection (LPD) communication systems +with either single or multiple antennas. The new scheme is de- +signed to allow the noncoherent detection of Gaussian-distributed +signals, enabling LPD communications using signals that follow +the complex Gaussian distribution in the time and frequency +domains. It is demonstrated via simulations that the proposed +scheme achieves better performance than a comparable conven- +tional scheme over the entire SNR region, with the advantage +becoming more significant in scenarios with lower overhead. +Index Terms—Multiple-input multiple-output (MIMO), low +probability of detection (LPD), Gaussian signaling. +I. INTRODUCTION +L +OW probability of detection (LPD) systems are a new +type of covert wireless communication technology [1] +that is expected to play a key role in applications such as +stealth IoT and military networks [2], which require high levels +of security. In the LPD communication systems, a legitimate +transmitter attempts to communicate with a legitimate receiver +without being detected by an illegitimate adversary. +Low-power zero-mean complex-valued Gaussian signals are +ideal for LPD, as it has been recently shown [3] that such +signals minimize the probability of detection by illegitimate +adversaries. This is unlike most current wireless systems, many +of which rely on orthogonal frequency division multiplexing +(OFDM) waveforms. This is because although time-domain +OFDM signals also follow Gaussian distributions, the discrete +nature of the digital constellations utilized in such systems is +visible in the frequency domain, a feature which can therefore +be exploited to detect the presence of communications. In +contrast, in LPD systems, signals must be Gaussian in both +the time and the frequency domains. +The security of LPD communication has been analyzed +from an information-theoretic perspective [4, 5], but under +the assumption that perfect channel state information (CSI) is +available at both transmitter and receiver [4, 5], which in turn +implies the exchange of reference signals, thus increasing the +risk of detection by an adversary. This fundamental weakness +of existing LDP methods calls for the design of noncoherent +Y. Katsuki and N. Ishikawa are with the Faculty of Engineering, Yokohama +National University, 240-8501 Kanagawa, Japan (e-mail: ishikawa-naoki- +fr@ynu.ac.jp). G. Abreu is with the School of Computer Science and En- +gineering, Jacobs University Bremen, 28759 Bremen, Germany. K. Ishibashi +is with the Advanced Wireless and Communication Research Center, The +University of Electro-Communications, 182-8585 Tokyo, Japana. This work +was supported in part by the Japan Science and Technology Agency, Strategic +International Collaborative Research Program (JST SICORP), Japan, under +Grant JPMJSC20C1. +LPD schemes, which on the other hand is challenging under +the optimal LDP Gaussian signaling, and therefore remains an +open issue hindering the practicality of the technology [2]. +Against this background, a noncoherent detection scheme +for LDP systems employing Gaussian signals is proposed in +this letter, as an enabling technology for LPD communications. +The proposed scheme builds on principles of differential +encoding [6] to construct complex Gaussian reference and +data signals, such that thanks to the differential structure, no +periodic reference signals are required to track CSI. The pro- +posed scheme has advantages over the representative Gaussian +signaling scheme of [7, 8] decoded via the semi-blind detector +[9], which can be considered the state-of-the-art in the area. +The contributions of the article can be summarized as follows. +• We design a new noncoherent detection scheme for +LPD communications. The proposed scheme inserts a +reference matrix at the beginning of the transmission +frame and generates differentially-encoded symbols, all +of which follow the ideal Gaussian distribution in the +frequency domain, resulting in Gaussian-distributed time- +domain signals. Since Gaussian-distributed signals make +CSI estimation difficult due to their maximally-entropic +feature, the issue is circumvented by the design of a +robust noncoherent detector. +• We demonstrate that the proposed scheme achieves +better BER performance than state-of-the-art of LPD +methods employing Gaussian signaling [7, 8] with +semi-blind detector [9]. In addition, our analysis indicates +that the detection complexity is reduced by a linear factor +while maintaining the same security level as the ideal +Gaussian signaling. +II. SYSTEM MODEL +Consider a multiple-input multiple-output (MIMO) system +in which a legitimate transmitter, Alice, equipped with M +transmit antennas, communicates with a legitimate receiver, +Bob, equipped with N receive antennas, in the presence of +an illegitimate adversary, Willie, which tries to detect Alice’s +communications. The narrow-band received signals at Bob is +modeled as1 +Y(i) = H(i)S(i) + V(i) ∈ CN×T , +(1) +for 0 < i ≤ W, where i is the transmission index, W is the +frame length, T is the number of time slots, S(i) ∈ CM×T is a +1We remark that this system model is readily applicable to MIMO-OFDM +scenarios [6]. +©2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any +copyrighted component of this work in other works. DOI: 10.1109/LWC.2022.3233619 +arXiv:2301.02954v1 [eess.SP] 8 Jan 2023 + +AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS +2 +space-time codeword, and the elements of H(i) ∈ CN×M and +V(i) ∈ CN×T follow CN(0, 1) and CN(0, σ2 +v), respectively, +with the signal-to-noise ratio (SNR) is defined as 1/σ2 +v. +Let B be the number of information bits conveyed by +the codeword S(i) ∈ S, where S ≜ {S1, · · · , S2B} is the +codebook. The bit error rate (BER) associated with signals as +in (1) depends on the so-called coding gain [10] +G ≜ +min +p̸=q∈{1,··· ,2B} +� +(Sp − Sq)H(Sp − Sq) +� 1 +N , +(2) +which represents the minimum Euclidean distance between +different space-time codewords in the codebook. +We emphasize that the main interest of LPD communication +with M ≪ N is in uplink scenario, since for massive MIMO +systems, downlink physical layer security schemes such as +the one proposed in [11] can generate Gaussian signals. As +a consequence, LPD communications in downlink will be +considered out of the scope of this letter. +III. REFERENCE STATE-OF-THE-ART SCHEMES +Before we introduce our proposed method, let us briefly +revise relevant state-of-the-art techniques, in particular the +conventional Gaussian signaling scheme of [7, 8] and the con- +ventional semi-blind detector of [9]. Combining both schemes +is a straightforward task, and results in an exclusive Gaussian +signaling scheme that can work for MIMO-OFDM scenarios. +Hence, we regard the combination as a performance baseline +reference to our contribution. +A. Conventional Gaussian Signaling Scheme of [7, 8] +Chaos MIMO (C-MIMO) [7] is a MIMO modulation +scheme designed with basis on chaos theory2, in which +B = MT information bits b = [b1 b2 · · · bB] ∈ BB are +mapped onto complex-valued Gaussian symbols, such that the +associated the M × T space-time codeword is in a form [8] +S(i) ≜ +1 +√ +M +� +���� +s1 +sM+1 +· · · +sMT −M+1 +s2 +sM+2 +· · · +sMT −M+2 +... +... +... +... +sM +s2M +· · · +sMT +� +����∈ CM×T , +(3) +where each symbol sk is generated by the Box-Muller’s +transform [8] +sk = +� +− log +� +c(x) +k +� � +cos +� +2πc(y) +k +� ++ j sin +� +2πc(y) +k +�� +(4) +and j denotes the imaginary number. +Here, c(x) +k +and c(y) +k +are uniform pseudo-random numbers +generated by a shared key, the input bit sequence, and the +Bernoulli shift map transition, which are given by [8] +c(x) +k += arccos (cos (37π (Re[ck] + Im[ck]))) /π, +(5a) +c(y) +k += arcsin (sin (43π (Re[ck]−Im[ck]))) /π+1/2, (5b) +2In our simulations, C-MIMO achieved the best coding gain among physical +layer encryption schemes. Thus, we consider it to be a representative Gaussian +signaling scheme. +where we have +ck = Re[zNs+b(k+B/2) mod B] + Im[zNs+b(k+B/2+1) mod B]. +(6) +In (6), the chaos sequences are defined by [8] +Re[zl] = 2 · Re[zl−1] mod (1 − 10−16), +(7a) +Im[zl] = 2 · Im[zl−1] mod (1 − 10−16). +(7b) +for l = 1, 2, · · · , Ns, Ns + 1, where the transition factor is +typically set to a large number such as Ns = 100 [8], and +both sequences are initialized by Re[z0] = Γ(Re[ck−1], bk−1), +Im[z0] = Γ(Im[ck−1], bk mod B), where [8] +Γ(a, b) ≜ +� +� +� +a +if (b = 0), +1 − a +if (b = 1 and a > 1/2), +a + 1/2 +if (b = 1 and a ≤ 1/2), +(8) +where c0 ∈ C is a pre-shared key that satisfies 0 < Re[c0] < 1 +and 0 < Im[c0] < 1. +We remark that due to the Box-Muller transform described +by (4), the resultant symbol sk follows CN(0, 1).3 +B. Conventional Coherent and Semi-Blind Detection of [9] +Algorithm 1 Conventional Semi-blind Detection [9]. +Input: ¯Y = +� +Y(1) Y(2) · · · +Y(W) +� +∈ CN×W T , ˆH(0) +Output: ˆS(l) +1: Set the iteration index l = 0. +2: repeat +3: +Given ˆH(l), perform ML detection for each sub-matrix +of ¯Y and obtain a set of estimates +ˆS(l) = +� +ˆS(1) ˆS(2) · · · +ˆS(W) +� +∈ CM×W T +4: +Update the channel matrix by ˆH(l+1) = ¯Y +� +ˆS(l)�+ +. +5: +Set l = l + 1. +6: until l < Imax. +Under the assumption that a perfect estimate of +ˆH is +available at the receiver, maximum likelihood (ML) detection +can be carried out via +ˆS(i) = arg min +S +∥Y(i) − ˆHS∥2 +F, +(9) +where an optimal S is searched over the set of C-MIMO +codewords. +In practice, however, frequent transmission of reference +signals is required to improve the accuracy of CSI estimation +by Bob, which in turn also increases the probability that +transmissions are detected by Willie. To alleviate this problem, +the semi-blind detection scheme of [9] may be exploited. +In such a scheme, a reference signal IM is transmited at +the first instance (i.e., when i = 0), which is used to +obtain a rough initial channel estimate ˆH(0). Then, over the +subsequent transmission of W blocks, the received signals are +concatenated and more accurate CSI is obtained iteratively, as +summarized in Algorithm 1, where we use the pseudo-inverse +matrix A+ = AH � +AAH�−1. +3The open-source implementation of C-MIMO is available at https://github. +com/ishikawalab/wiphy/blob/master/wiphy/examples/okamoto2012chaos.py + +AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS +3 +IV. PROPOSED NONCOHERENT GAUSSIAN SIGNALING +(NGS) FOR LPD COMMUNICATIONS +In this section, we introduce the proposed NGS scheme +satisfying the requirements of LPD communications. +A. Gaussian Signal Transmission +Based on a shared secret seed, Alice and Bob generate a +Gaussian reference matrix G ∈ CM×KM, where K is a rep- +etition number, and each element of G follows CN(0, 1/M), +i.e., E[∥G∥2 +F] = KM. Similarly, Alice and Bob generate +a Gaussian projection matrix E(i) ∈ CM×T , where each +element of E(i) follows CN(0, 1/M), i.e., E[∥E(i)∥2 +F] = T, +where the index i indicates that the projection matrix varies +over time. +After the above Gaussian matrices are prepared, at the index +i = 0, the Gaussian reference matrix is transmitted. Then, +for 0 < i ≤ W, B bits of information are mapped onto +a unitary matrix X(i) ∈ CM×M, which is selected from a +codebook of 2B matrices {X1, · · · , X2B}. Here, we use the +classic differential encoding in the form +˜S(i) = +� IM +if i = 0, or +˜S(i − 1)X(i) +if i > 0. +(10) +Finally, the space-time codeword at i is generated by +S(i) = +� G +if i = 0, or +˜S(i)E(i) +if i > 0. +(11) +The important question here is what type of unitary matrix +X(i) would lead each element of (11) to follow an independent +complex Gaussian distribution. Since any point on the unit cir- +cle does not change the statistical property, one can expect that +any unitary matrix that has one unit-norm nonzero element in +each column results in the Gaussian distribution. Differential +schemes that satisfy such property include the diagonal unitary +coding (DUC) [12] and differential spatial modulation [13] +methods, but since the DUC approach maximizes the coding +gain due to its optimized structure, it will be adopted here in +our NGS scheme. +In the DUC scheme [12], the B input bits are mapped +onto the integers b = 0, 1, · · · , 2B − 1, and the corresponding +unitary matrix is generated as +Xb = diag +� +exp +� +j 2πb +2B u1 +� +, · · · , exp +� +j 2πb +2B uM +�� +, (12) +where the factors 0 < u1 ≤ · · · uM ≤ 2B/2 ∈ Z are designed +so as to maximize the diversity product given by4 +min +b∈{1,··· ,2B−1} +����� +M +� +m=1 +sin +�πbum +2B +������ +1 +M +. +(13) +Notice that the encoding scheme described by (12) becomes +identical to the classic differential phase-shift keying modu- +lation if M = 1 and u1 = 1. In addition, although the ideal +transmission rate is R = B/T, since the Gaussian reference +matrix GM×KM is inserted, the effective transmission rate is +4The +optimized +factors +are +available +online +at +https://github.com/ +ishikawalab/wiphy/blob/master/wiphy/code/duc.py. +Reff = +BW +MK + WT = ηR, +(14) +where we have implicitly defined the transmission efficiency +η ≜ (1 + MK/(WT))−1. +As given, the efficiency decreases as K increases. The ref- +erence insertion ratio is calculated as 1−η. In the performance +comparisons, we will set the ratio to 5%, but we remark that +the performance of the proposed scheme remains constant +in high-speed mobile scenarios even with lower reference +insertion ratios such as 1% and 0.1%. +B. Noncoherent Detection of Gaussian Signaling +The noncoherent detection of the Gaussian signaling scheme +proposed in Subsection IV-A is not a straightforward task. +We extend the noncoherent detection scheme proposed in [6] +to support time-varying Gaussian reference and projection +matrices. Specifically, for the data blocks 0 < i ≤ W, the +proposed noncoherent ML detector is described by +ˆX(i) = arg min +X +∥Y(i) − ˆY(i − 1)XE(i)∥2 +F, +(15) +where the matrices ˆY(i) are given by +ˆY(0) ≜ Y(0)G+ = H(0)˜S(0) + V(0)G+, +(16) +at i = 0, where each element of V(0) ∈ CN×KM follows +CN(0, σ2 +v), and +ˆY(i) ≜ βY(i)EH(i)+ˆY(i−1) ˆX(i)(IM−βE(i)EH(i)), (17) +for i > 0, respectively, with the parameter β ≜ 1 − α +containing a forgetting factor 0 < α < 1 that determines the +inter-dependence between Y(i) and ˆY(i − 1). +Notice that due to (15), even if Willie successfully detects +the LPD communications, he cannot decrypt data symbols +because the projection of E(i) induces phase ambiguity, which +implies that the proposed NGS also works as a physical layer +encryption scheme. +V. THEORETICAL ANALYSIS +In this section, we compare the proposed NGS LPD com- +munication scheme against the conventional C-MIMO in terms +of security and complexity. System configurations of the +conventional C-MIMO and the proposed NGS schemes are +summarized in Table I. +A. Security Analysis +In LPD communications, the achievable security level has +been evaluated by the Willie’s minimum detection error prob- +ability [3–5]. Let the case when Alice does not transmit +symbols to Bob be referred to as the null-hypothesis H0, +with likelihood function p0(y), and the alternative case when +Alice does transmit to Bob be referred to as the alternative +hypothesis H1, with likelihood function p1(y). The received +signal at Willie under both hypotheses can be expressed as [3] +� H0 : y = v, or +H1 : y = s + v, +(18) + +AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS +4 +TABLE I +SYSTEM CONFIGURATIONS OF THE CONVENTIONAL C-MIMO AND THE PROPOSED NGS SCHEMES. +Reference Signal (i = 0) +Data Symbols (0 < i ≤ W) +ML Detector +Conventional C-MIMO +Gaussian reference matrix G ∈ CM×KM +Okamoto’s C-MIMO encoding method [8] +Semi-blind detector [9] +(see Section IV-A) +(see Section III-A) +(see Algorithm 1) +Proposed NGS +Gaussian reference matrix G ∈ CM×KM +DUC & Gaussian projection matrix E(i) +Noncoherent detector +(see Section IV-A) +(see Section IV-A) +(see Section IV-B) +Fig. 1. Lower bounds of Willie’s minimum detection error probability ξ∗. +where s and v denote the transmit signal and additive white +Gaussian noise, respectively. +Assuming that all transmit signals are Gaussian, Willie’s +detection error probability can be expressed as [3] +ξ∗ ≥ ξ∗ +min ≜ 1 − +� +D(p1∥p0)/2, +(19) +where the lower-bounding quantity ξ∗ +min was derived in [3, 4] +and D(p1∥p0) denotes the Kullback-Leiber (KL) divergence +between the likelihoods of observing y under the null and the +alternative hypotheses, respectively. +In Fig. 1, we evaluate the lower bounds of Willie’s minimum +detection error probability ξ∗ +min, where the conventional C- +MIMO and the proposed NGS were considered. As expected, +both schemes exhibit the same lower bound since the resulting +constellation in both successfully follow the ideal Gaussian +distribution. Additionally, it is observed that as M increases, +ξ∗ +min becomes larger because less power is allocated to each +antenna, which is preferable in LPD communications. In short, +it can be said that the proposed scheme has the same security +level of conventional C-MIMO. +We remark that the repetition number K has no effect on +ξ∗ +min since it does not influence the statistical property of +transmit signals, which can be inferred from [3–5]. +B. Complexity Analysis +Next, the conventional C-MIMO and the proposed NGS are +evaluated in terms of encoding and decoding complexities. +Encoding complexity: +In both the C-MIMO and NGS +schemes a total of BW [bits] are transmitted within the frame +length W. However, in order to generate a codebook, C- +MIMO requires 2BW complex-valued random variables, since +C-MIMO directly maps information bits onto complex-valued +Gaussian symbols. +In contrast, NGS requires only BW complex-valued ran- +dom variables since the codebook is constructed as in (12), +before being transformed by the Gaussian projection matrix. +Thus, the number of Gaussian variables to be generated can +be significantly reduced in the proposed NGS LPD scheme as +the number of bits B increases. +Decoding complexity: The complexity orders of the con- +ventional semi-blind detector Cc (see Algorithm 1) and of the +proposed detector Cp (see Subsection IV-B) are, respectively +Cc =2BWImax(4MNT +� �� � +ˆHS ++ 4NT +� �� � +∥·∥2 +F +) ++ MW(Imax − 1) +� +8MT + 4N +W + M 2 +W +� +� +�� +� +¯Y[ˆS(l)] ++ +=O +� +2BWImax(4MNT + 4NT) +� +, +(20) +Cp =2BW(4MNT +� �� � +ˆY(i−1)X ++ 4NT +� �� � +∥·∥2 +F ++ 4MT +� �� � +XE(i) +) ++ +MW(8MN + 4NT + 4MT) +� +�� +� +βY(i)EH(i)+ˆY(i−1) ˆX(i)(IM−βE(i)EH(i)) +=O +� +2BW(4MNT + 4NT + 4MT) +� +, +(21) +where only the significant numbers of real-valued floating- +point multiplication operations are counted. +It follows that the ratio between both is given by +Cp +Cc += O +� +1 +Imax +�1 + +1 +M + 1 +N +1 + +1 +M +�� +≈ O +� +1 +Imax +� +, +(22) +which means that the complexity order of the conventional +semi-blind detector is approximately Imax higher than that of +the proposed detector. +VI. NUMERICAL PERFORMANCE COMPARISONS +In this section, the conventional C-MIMO decoded via the +semi-blind detector and the proposed NGS LPD scheme are +empirically evaluated in terms of their bit error rates (BERs) +and coding gains. Again, the system configurations of both +schemes are summarized in Table I. In our simulations, we +set the minimum SNR to −20 dB since transmission power in +LPD communications systems is typically small, and adopt +the forgetting factor α = 0.8. In addition, although the +proposed NGS supports time-varying channels, we limit the +comparisons to quasi-static Rayleigh fading channels because +the conventional C-MIMO with the semi-blind detector only +supports such channel condition. +Note that also in the conventional C-MIMO scheme, a +Gaussian reference matrix G ∈ CM×KM is transmitted first +for the purpose of channel estimation, since all signals in LPD + +AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS +5 +systems must be complex Gaussian. The semi-blind detection +algorithm is subsequently performed using the rough initial +channel estimate ˆH(0) = ˆY(0) given by (16). +First, a comparison between the BERs of the conventional +C-MIMO [7] scheme with semi-blind detection [9] and the +proposed NGS LPD method is offered in Fig. 2, for the case +where M = 2, N = 64, T = 1, B = 2, and overhead factor +K = 1 or K = 3. As a lower bound, the BER of the NGS +scheme with perfect CSI is also included. The results show +that the proposed NGS LPD system achieves the best BER in +all cases, with a gain of 16 dB over conventional C-MIMO +observed for K = 1, and a performance rapidly approaching +the lower bound for larger K. +In order to better investigate the reason behind the large +BER advantage of NGS LPD seen in Fig. 2, we compare in +Fig. 3 the coding gains of NGS LPD and alternative schemes5, +with N = 1 and T = 1, and M varying from 1 to 8. +The results of Fig. 3 show that thanks to the optimized +DUC constellation, the proposed NGS LPD scheme achieves +coding gains equal or close to those of conventional spatial +multiplexing, while generating Gaussian signals. In compari- +son, the coding gains of conventional C-MIMO was found to +be close or equal to these of random Gaussian constellations. +VII. CONCLUSION +We proposed a solution to an open issue regarding the +noncoherent detection of Gaussian-distributed signals, which +is especially important for LPD communications. To solve +the issue, we relied on newly proposed Gaussian projection +scheme, applied over optimized DUC constellations. Refer- +ence signals were also designed so as to follow the Gaussian +distribution. Thus, the proposed scheme communicates only +using Gaussian signals, which satisfies the common require- +ment of LPD communication. With the use of OFDM, the +Gaussian signals in the frequency domain result in Gaussian- +distributed time-domain signals. An analysis was provided, +which clarified that the proposed NGS generates perfectly +Gaussian symbols at a fraction of the complexity of C-MIMO +schemes. Simulation results also revealed large BER gains +over the latter alternative, due to the higher coding gains +afforded by the new method. Since the proposed NGS requires +shorter reference signals, it is expected to be suitable for low- +overhead LPD communications in high-mobility scenarios. +REFERENCES +[1] B. A. Bash, D. Goeckel, D. Towsley, and S. Guha, “Hiding information +in noise: Fundamental limits of covert wireless communication,” IEEE +Communications Magazine, vol. 53, no. 12, pp. 26–31, 2015. +[2] S. Yan, X. Zhou, J. Hu, and S. V. Hanly, “Low probability +of detection communication: Opportunities and challenges,” IEEE +Wireless Communications, vol. 26, no. 5, pp. 19–25, 2019. +[3] S. +Yan, +Y. +Cong, +S. +V. +Hanly, +and +X. +Zhou, +“Gaussian +signalling for covert communications,” IEEE Transactions on Wireless +Communications, vol. 18, no. 7, pp. 3542–3553, 2019. +[4] A. Bendary, A. Abdelaziz, and C. E. Koksal, “Achieving positive covert +capacity over MIMO AWGN channels,” IEEE Journal on Selected +Areas in Information Theory, vol. 2, no. 1, pp. 149–162, 2021. +[5] T. X. Zheng, H. M. Wang, D. W. K. Ng, and J. Yuan, “Multi- +antenna covert communications in random wireless networks,” IEEE +5Note that although spatial multiplexing does not generate Gaussian signals, +it was added to the comparison only as a reference to coding gain. +Fig. 2. Comparison of BERs achieved with Gaussian reference signals and +Gaussian projection matrix, with M = 2, N = 64, T = 1 and B = 2. +Fig. 3. Comparison of coding gains achieved with Gaussian projection matrix, +with B = M, N = 1 and T = 1. +Transactions on Wireless Communications, vol. 18, no. 3, pp. 1974– +1987, 2019. +[6] N. Ishikawa, R. Rajashekar, C. Xu, S. Sugiura, and L. Hanzo, +“Differential space-time coding dispensing with channel estimation +approaches +the +performance +of +its +coherent +counterpart +in +the +open-loop massive MIMO-OFDM downlink,” IEEE Transactions on +Communications, vol. 66, no. 12, pp. 6190–6204, 2018. +[7] E. Okamoto, “A chaos MIMO transmission scheme for secure com- +munications on physical layer,” in IEEE 73rd Vehicular Technology +Conference, Budapest, Hungary, 15-18 May, 2011. +[8] E. Okamoto and N. Horiike, “Performance improvement of chaos +MIMO scheme using advanced stochastic characteristics,” IEICE +Communications Express, vol. 5, no. 10, pp. 371–377, 2016. +[9] S. Chen, S. Sugiura, and L. Hanzo, “Semi-blind joint channel estimation +and data detection for space-time shift keying systems,” IEEE Signal +Processing Letters, vol. 17, no. 12, pp. 993–996, 2010. +[10] L. Hanzo, O. Alamri, M. El-Hajjar, and N. Wu, Near-capacity +multi-functional MIMO systems. +John Wiley & Sons, Ltd, May 2009. +[11] T. R. Dean and A. J. Goldsmith, “Physical-layer cryptography through +massive MIMO,” IEEE Transactions on Information Theory, vol. 63, +no. 8, pp. 5419–5436, 2017. +[12] B. Hochwald and W. Sweldens, “Differential unitary space-time +modulation,” IEEE Transactions on Communications, vol. 48, no. 12, +pp. 2041–2052, 2000. +[13] Y. Bian, X. Cheng, M. Wen, L. Yang, H. V. Poor, and B. Jiao, “Differ- +ential spatial modulation,” IEEE Transactions on Vehicular Technology, +vol. 64, no. 7, pp. 3262–3268, 2015. + diff --git a/YNE1T4oBgHgl3EQfJgP2/content/tmp_files/load_file.txt b/YNE1T4oBgHgl3EQfJgP2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67a391c09cf754e2083dbd9de1181605fcb892c4 --- /dev/null +++ b/YNE1T4oBgHgl3EQfJgP2/content/tmp_files/load_file.txt @@ -0,0 +1,291 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf,len=290 +page_content='AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS 1 A New Noncoherent Gaussian Signaling Scheme for Low Probability of Detection Communications Yuma Katsuki, Graduate Student Member, IEEE, Giuseppe Thadeu Freitas de Abreu, Senior Member, IEEE, Koji Ishibashi, Senior Member, IEEE, and Naoki Ishikawa, Senior Member, IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Abstract—We propose a novel, Gaussian signaling mechanism for low probability of detection (LPD) communication systems with either single or multiple antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The new scheme is de- signed to allow the noncoherent detection of Gaussian-distributed signals, enabling LPD communications using signals that follow the complex Gaussian distribution in the time and frequency domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' It is demonstrated via simulations that the proposed scheme achieves better performance than a comparable conven- tional scheme over the entire SNR region, with the advantage becoming more significant in scenarios with lower overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Index Terms—Multiple-input multiple-output (MIMO), low probability of detection (LPD), Gaussian signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' INTRODUCTION L OW probability of detection (LPD) systems are a new type of covert wireless communication technology [1] that is expected to play a key role in applications such as stealth IoT and military networks [2], which require high levels of security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In the LPD communication systems, a legitimate transmitter attempts to communicate with a legitimate receiver without being detected by an illegitimate adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Low-power zero-mean complex-valued Gaussian signals are ideal for LPD, as it has been recently shown [3] that such signals minimize the probability of detection by illegitimate adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' This is unlike most current wireless systems, many of which rely on orthogonal frequency division multiplexing (OFDM) waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' This is because although time-domain OFDM signals also follow Gaussian distributions, the discrete nature of the digital constellations utilized in such systems is visible in the frequency domain, a feature which can therefore be exploited to detect the presence of communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In contrast, in LPD systems, signals must be Gaussian in both the time and the frequency domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The security of LPD communication has been analyzed from an information-theoretic perspective [4, 5], but under the assumption that perfect channel state information (CSI) is available at both transmitter and receiver [4, 5], which in turn implies the exchange of reference signals, thus increasing the risk of detection by an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' This fundamental weakness of existing LDP methods calls for the design of noncoherent Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Katsuki and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Ishikawa are with the Faculty of Engineering, Yokohama National University, 240-8501 Kanagawa, Japan (e-mail: ishikawa-naoki- fr@ynu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='jp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Abreu is with the School of Computer Science and En- gineering, Jacobs University Bremen, 28759 Bremen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Ishibashi is with the Advanced Wireless and Communication Research Center, The University of Electro-Communications, 182-8585 Tokyo, Japana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' This work was supported in part by the Japan Science and Technology Agency, Strategic International Collaborative Research Program (JST SICORP), Japan, under Grant JPMJSC20C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' LPD schemes, which on the other hand is challenging under the optimal LDP Gaussian signaling, and therefore remains an open issue hindering the practicality of the technology [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Against this background, a noncoherent detection scheme for LDP systems employing Gaussian signals is proposed in this letter, as an enabling technology for LPD communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The proposed scheme builds on principles of differential encoding [6] to construct complex Gaussian reference and data signals, such that thanks to the differential structure, no periodic reference signals are required to track CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The pro- posed scheme has advantages over the representative Gaussian signaling scheme of [7, 8] decoded via the semi-blind detector [9], which can be considered the state-of-the-art in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The contributions of the article can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' We design a new noncoherent detection scheme for LPD communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The proposed scheme inserts a reference matrix at the beginning of the transmission frame and generates differentially-encoded symbols, all of which follow the ideal Gaussian distribution in the frequency domain, resulting in Gaussian-distributed time- domain signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Since Gaussian-distributed signals make CSI estimation difficult due to their maximally-entropic feature, the issue is circumvented by the design of a robust noncoherent detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' We demonstrate that the proposed scheme achieves better BER performance than state-of-the-art of LPD methods employing Gaussian signaling [7, 8] with semi-blind detector [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In addition, our analysis indicates that the detection complexity is reduced by a linear factor while maintaining the same security level as the ideal Gaussian signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' SYSTEM MODEL Consider a multiple-input multiple-output (MIMO) system in which a legitimate transmitter, Alice, equipped with M transmit antennas, communicates with a legitimate receiver, Bob, equipped with N receive antennas, in the presence of an illegitimate adversary, Willie, which tries to detect Alice’s communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The narrow-band received signals at Bob is modeled as1 Y(i) = H(i)S(i) + V(i) ∈ CN×T , (1) for 0 < i ≤ W, where i is the transmission index, W is the frame length, T is the number of time slots, S(i) ∈ CM×T is a 1We remark that this system model is readily applicable to MIMO-OFDM scenarios [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' ©2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='1109/LWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='3233619 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='02954v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='SP] 8 Jan 2023 AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS 2 space-time codeword, and the elements of H(i) ∈ CN×M and V(i) ∈ CN×T follow CN(0, 1) and CN(0, σ2 v), respectively, with the signal-to-noise ratio (SNR) is defined as 1/σ2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Let B be the number of information bits conveyed by the codeword S(i) ∈ S, where S ≜ {S1, · · · , S2B} is the codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The bit error rate (BER) associated with signals as in (1) depends on the so-called coding gain [10] G ≜ min p̸=q∈{1,··· ,2B} � (Sp − Sq)H(Sp − Sq) � 1 N , (2) which represents the minimum Euclidean distance between different space-time codewords in the codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' We emphasize that the main interest of LPD communication with M ≪ N is in uplink scenario, since for massive MIMO systems, downlink physical layer security schemes such as the one proposed in [11] can generate Gaussian signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' As a consequence, LPD communications in downlink will be considered out of the scope of this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' REFERENCE STATE-OF-THE-ART SCHEMES Before we introduce our proposed method, let us briefly revise relevant state-of-the-art techniques, in particular the conventional Gaussian signaling scheme of [7, 8] and the con- ventional semi-blind detector of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Combining both schemes is a straightforward task, and results in an exclusive Gaussian signaling scheme that can work for MIMO-OFDM scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Hence, we regard the combination as a performance baseline reference to our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Conventional Gaussian Signaling Scheme of [7, 8] Chaos MIMO (C-MIMO) [7] is a MIMO modulation scheme designed with basis on chaos theory2, in which B = MT information bits b = [b1 b2 · · · bB] ∈ BB are mapped onto complex-valued Gaussian symbols, such that the associated the M × T space-time codeword is in a form [8] S(i) ≜ 1 √ M � ���� s1 sM+1 · · sMT −M+1 s2 sM+2 · · sMT −M+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' sM s2M · · sMT � ����∈ CM×T , (3) where each symbol sk is generated by the Box-Muller’s transform [8] sk = � − log � c(x) k � � cos � 2πc(y) k � + j sin � 2πc(y) k �� (4) and j denotes the imaginary number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Here, c(x) k and c(y) k are uniform pseudo-random numbers generated by a shared key, the input bit sequence, and the Bernoulli shift map transition, which are given by [8] c(x) k = arccos (cos (37π (Re[ck] + Im[ck]))) /π, (5a) c(y) k = arcsin (sin (43π (Re[ck]−Im[ck]))) /π+1/2, (5b) 2In our simulations, C-MIMO achieved the best coding gain among physical layer encryption schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Thus, we consider it to be a representative Gaussian signaling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' where we have ck = Re[zNs+b(k+B/2) mod B] + Im[zNs+b(k+B/2+1) mod B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (6) In (6), the chaos sequences are defined by [8] Re[zl] = 2 · Re[zl−1] mod (1 − 10−16), (7a) Im[zl] = 2 · Im[zl−1] mod (1 − 10−16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (7b) for l = 1, 2, · · · , Ns, Ns + 1, where the transition factor is typically set to a large number such as Ns = 100 [8], and both sequences are initialized by Re[z0] = Γ(Re[ck−1], bk−1), Im[z0] = Γ(Im[ck−1], bk mod B), where [8] Γ(a, b) ≜ � � � a if (b = 0), 1 − a if (b = 1 and a > 1/2), a + 1/2 if (b = 1 and a ≤ 1/2), (8) where c0 ∈ C is a pre-shared key that satisfies 0 < Re[c0] < 1 and 0 < Im[c0] < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' We remark that due to the Box-Muller transform described by (4), the resultant symbol sk follows CN(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Conventional Coherent and Semi-Blind Detection of [9] Algorithm 1 Conventional Semi-blind Detection [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Input: ¯Y = � Y(1) Y(2) · · · Y(W) � ∈ CN×W T , ˆH(0) Output: ˆS(l) 1: Set the iteration index l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 2: repeat 3: Given ˆH(l), perform ML detection for each sub-matrix of ¯Y and obtain a set of estimates ˆS(l) = � ˆS(1) ˆS(2) · · · ˆS(W) � ∈ CM×W T 4: Update the channel matrix by ˆH(l+1) = ¯Y � ˆS(l)�+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 5: Set l = l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 6: until l < Imax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Under the assumption that a perfect estimate of ˆH is available at the receiver, maximum likelihood (ML) detection can be carried out via ˆS(i) = arg min S ∥Y(i) − ˆHS∥2 F, (9) where an optimal S is searched over the set of C-MIMO codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In practice, however, frequent transmission of reference signals is required to improve the accuracy of CSI estimation by Bob, which in turn also increases the probability that transmissions are detected by Willie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' To alleviate this problem, the semi-blind detection scheme of [9] may be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In such a scheme, a reference signal IM is transmited at the first instance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=', when i = 0), which is used to obtain a rough initial channel estimate ˆH(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Then, over the subsequent transmission of W blocks, the received signals are concatenated and more accurate CSI is obtained iteratively, as summarized in Algorithm 1, where we use the pseudo-inverse matrix A+ = AH � AAH�−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 3The open-source implementation of C-MIMO is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' com/ishikawalab/wiphy/blob/master/wiphy/examples/okamoto2012chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='py AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS 3 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' PROPOSED NONCOHERENT GAUSSIAN SIGNALING (NGS) FOR LPD COMMUNICATIONS In this section, we introduce the proposed NGS scheme satisfying the requirements of LPD communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Gaussian Signal Transmission Based on a shared secret seed, Alice and Bob generate a Gaussian reference matrix G ∈ CM×KM, where K is a rep- etition number, and each element of G follows CN(0, 1/M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=', E[∥G∥2 F] = KM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Similarly, Alice and Bob generate a Gaussian projection matrix E(i) ∈ CM×T , where each element of E(i) follows CN(0, 1/M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=', E[∥E(i)∥2 F] = T, where the index i indicates that the projection matrix varies over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' After the above Gaussian matrices are prepared, at the index i = 0, the Gaussian reference matrix is transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Then, for 0 < i ≤ W, B bits of information are mapped onto a unitary matrix X(i) ∈ CM×M, which is selected from a codebook of 2B matrices {X1, · · · , X2B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Here, we use the classic differential encoding in the form ˜S(i) = � IM if i = 0, or ˜S(i − 1)X(i) if i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (10) Finally, the space-time codeword at i is generated by S(i) = � G if i = 0, or ˜S(i)E(i) if i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (11) The important question here is what type of unitary matrix X(i) would lead each element of (11) to follow an independent complex Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Since any point on the unit cir- cle does not change the statistical property, one can expect that any unitary matrix that has one unit-norm nonzero element in each column results in the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Differential schemes that satisfy such property include the diagonal unitary coding (DUC) [12] and differential spatial modulation [13] methods, but since the DUC approach maximizes the coding gain due to its optimized structure, it will be adopted here in our NGS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In the DUC scheme [12], the B input bits are mapped onto the integers b = 0, 1, · · · , 2B − 1, and the corresponding unitary matrix is generated as Xb = diag � exp � j 2πb 2B u1 � , · · · , exp � j 2πb 2B uM �� , (12) where the factors 0 < u1 ≤ · · · uM ≤ 2B/2 ∈ Z are designed so as to maximize the diversity product given by4 min b∈{1,··· ,2B−1} ����� M � m=1 sin �πbum 2B ������ 1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (13) Notice that the encoding scheme described by (12) becomes identical to the classic differential phase-shift keying modu- lation if M = 1 and u1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In addition, although the ideal transmission rate is R = B/T, since the Gaussian reference matrix GM×KM is inserted, the effective transmission rate is 4The optimized factors are available online at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='com/ ishikawalab/wiphy/blob/master/wiphy/code/duc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Reff = BW MK + WT = ηR, (14) where we have implicitly defined the transmission efficiency η ≜ (1 + MK/(WT))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' As given, the efficiency decreases as K increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The ref- erence insertion ratio is calculated as 1−η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In the performance comparisons, we will set the ratio to 5%, but we remark that the performance of the proposed scheme remains constant in high-speed mobile scenarios even with lower reference insertion ratios such as 1% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Noncoherent Detection of Gaussian Signaling The noncoherent detection of the Gaussian signaling scheme proposed in Subsection IV-A is not a straightforward task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' We extend the noncoherent detection scheme proposed in [6] to support time-varying Gaussian reference and projection matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' for the data blocks 0 < i ≤ W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' the proposed noncoherent ML detector is described by ˆX(i) = arg min X ∥Y(i) − ˆY(i − 1)XE(i)∥2 F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (15) where the matrices ˆY(i) are given by ˆY(0) ≜ Y(0)G+ = H(0)˜S(0) + V(0)G+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (16) at i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' where each element of V(0) ∈ CN×KM follows CN(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' σ2 v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' and ˆY(i) ≜ βY(i)EH(i)+ˆY(i−1) ˆX(i)(IM−βE(i)EH(i)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (17) for i > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' with the parameter β ≜ 1 − α containing a forgetting factor 0 < α < 1 that determines the inter-dependence between Y(i) and ˆY(i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Notice that due to (15), even if Willie successfully detects the LPD communications, he cannot decrypt data symbols because the projection of E(i) induces phase ambiguity, which implies that the proposed NGS also works as a physical layer encryption scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' THEORETICAL ANALYSIS In this section, we compare the proposed NGS LPD com- munication scheme against the conventional C-MIMO in terms of security and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' System configurations of the conventional C-MIMO and the proposed NGS schemes are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Security Analysis In LPD communications, the achievable security level has been evaluated by the Willie’s minimum detection error prob- ability [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Let the case when Alice does not transmit symbols to Bob be referred to as the null-hypothesis H0, with likelihood function p0(y), and the alternative case when Alice does transmit to Bob be referred to as the alternative hypothesis H1, with likelihood function p1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The received signal at Willie under both hypotheses can be expressed as [3] � H0 : y = v, or H1 : y = s + v, (18) AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS 4 TABLE I SYSTEM CONFIGURATIONS OF THE CONVENTIONAL C-MIMO AND THE PROPOSED NGS SCHEMES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Reference Signal (i = 0) Data Symbols (0 < i ≤ W) ML Detector Conventional C-MIMO Gaussian reference matrix G ∈ CM×KM Okamoto’s C-MIMO encoding method [8] Semi-blind detector [9] (see Section IV-A) (see Section III-A) (see Algorithm 1) Proposed NGS Gaussian reference matrix G ∈ CM×KM DUC & Gaussian projection matrix E(i) Noncoherent detector (see Section IV-A) (see Section IV-A) (see Section IV-B) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Lower bounds of Willie’s minimum detection error probability ξ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' where s and v denote the transmit signal and additive white Gaussian noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Assuming that all transmit signals are Gaussian, Willie’s detection error probability can be expressed as [3] ξ∗ ≥ ξ∗ min ≜ 1 − � D(p1∥p0)/2, (19) where the lower-bounding quantity ξ∗ min was derived in [3, 4] and D(p1∥p0) denotes the Kullback-Leiber (KL) divergence between the likelihoods of observing y under the null and the alternative hypotheses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 1, we evaluate the lower bounds of Willie’s minimum detection error probability ξ∗ min, where the conventional C- MIMO and the proposed NGS were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' As expected, both schemes exhibit the same lower bound since the resulting constellation in both successfully follow the ideal Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Additionally, it is observed that as M increases, ξ∗ min becomes larger because less power is allocated to each antenna, which is preferable in LPD communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In short, it can be said that the proposed scheme has the same security level of conventional C-MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' We remark that the repetition number K has no effect on ξ∗ min since it does not influence the statistical property of transmit signals, which can be inferred from [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Complexity Analysis Next, the conventional C-MIMO and the proposed NGS are evaluated in terms of encoding and decoding complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Encoding complexity: In both the C-MIMO and NGS schemes a total of BW [bits] are transmitted within the frame length W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' However, in order to generate a codebook, C- MIMO requires 2BW complex-valued random variables, since C-MIMO directly maps information bits onto complex-valued Gaussian symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In contrast, NGS requires only BW complex-valued ran- dom variables since the codebook is constructed as in (12), before being transformed by the Gaussian projection matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Thus, the number of Gaussian variables to be generated can be significantly reduced in the proposed NGS LPD scheme as the number of bits B increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Decoding complexity: The complexity orders of the con- ventional semi-blind detector Cc (see Algorithm 1) and of the proposed detector Cp (see Subsection IV-B) are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' respectively Cc =2BWImax(4MNT � �� � ˆHS + 4NT � �� � ∥·∥2 F ) + MW(Imax − 1) � 8MT + 4N W + M 2 W � � �� � ¯Y[ˆS(l)] + =O � 2BWImax(4MNT + 4NT) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (20) Cp =2BW(4MNT � �� � ˆY(i−1)X + 4NT � �� � ∥·∥2 F + 4MT � �� � XE(i) ) + MW(8MN + 4NT + 4MT) � �� � βY(i)EH(i)+ˆY(i−1) ˆX(i)(IM−βE(i)EH(i)) =O � 2BW(4MNT + 4NT + 4MT) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' (21) where only the significant numbers of real-valued floating- point multiplication operations are counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' It follows that the ratio between both is given by Cp Cc = O � 1 Imax �1 + 1 M + 1 N 1 + 1 M �� ≈ O � 1 Imax � , (22) which means that the complexity order of the conventional semi-blind detector is approximately Imax higher than that of the proposed detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' NUMERICAL PERFORMANCE COMPARISONS In this section, the conventional C-MIMO decoded via the semi-blind detector and the proposed NGS LPD scheme are empirically evaluated in terms of their bit error rates (BERs) and coding gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Again, the system configurations of both schemes are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In our simulations, we set the minimum SNR to −20 dB since transmission power in LPD communications systems is typically small, and adopt the forgetting factor α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In addition, although the proposed NGS supports time-varying channels, we limit the comparisons to quasi-static Rayleigh fading channels because the conventional C-MIMO with the semi-blind detector only supports such channel condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Note that also in the conventional C-MIMO scheme, a Gaussian reference matrix G ∈ CM×KM is transmitted first for the purpose of channel estimation, since all signals in LPD AUTHOR’S FINAL VERSION ACCEPTED FOR PUBLICATION IN IEEE WIRELESS COMMUNICATIONS LETTERS 5 systems must be complex Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The semi-blind detection algorithm is subsequently performed using the rough initial channel estimate ˆH(0) = ˆY(0) given by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' First, a comparison between the BERs of the conventional C-MIMO [7] scheme with semi-blind detection [9] and the proposed NGS LPD method is offered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 2, for the case where M = 2, N = 64, T = 1, B = 2, and overhead factor K = 1 or K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' As a lower bound, the BER of the NGS scheme with perfect CSI is also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The results show that the proposed NGS LPD system achieves the best BER in all cases, with a gain of 16 dB over conventional C-MIMO observed for K = 1, and a performance rapidly approaching the lower bound for larger K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In order to better investigate the reason behind the large BER advantage of NGS LPD seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 2, we compare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 3 the coding gains of NGS LPD and alternative schemes5, with N = 1 and T = 1, and M varying from 1 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' The results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' 3 show that thanks to the optimized DUC constellation, the proposed NGS LPD scheme achieves coding gains equal or close to those of conventional spatial multiplexing, while generating Gaussian signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' In compari- son, the coding gains of conventional C-MIMO was found to be close or equal to these of random Gaussian constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' CONCLUSION We proposed a solution to an open issue regarding the noncoherent detection of Gaussian-distributed signals, which is especially important for LPD communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' To solve the issue, we relied on newly proposed Gaussian projection scheme, applied over optimized DUC constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Refer- ence signals were also designed so as to follow the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Thus, the proposed scheme communicates only using Gaussian signals, which satisfies the common require- ment of LPD communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' With the use of OFDM, the Gaussian signals in the frequency domain result in Gaussian- distributed time-domain signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' An analysis was provided, which clarified that the proposed NGS generates perfectly Gaussian symbols at a fraction of the complexity of C-MIMO schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Simulation results also revealed large BER gains over the latter alternative, due to the higher coding gains afforded by the new method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Since the proposed NGS requires shorter reference signals, it is expected to be suitable for low- overhead LPD communications in high-mobility scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Bash, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfJgP2/content/2301.02954v1.pdf'} +page_content=' Goeckel, D.' metadata={'source': 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Owens, Haeran Cho and Matteo Barigozzi +Abstract The package fnets for the R language implements the suite of methodologies proposed by +Barigozzi et al. (2022) for the network estimation and forecasting of high-dimensional time series under +a factor-adjusted vector autoregressive model, which permits strong spatial and temporal correlations +in the data. Additionally, we provide tools for visualising the networks underlying the time series data +after adjusting for the presence of factors. The package also offers data-driven methods for selecting +tuning parameters including the number of factors, vector autoregressive order and thresholds for +estimating the edge sets of the networks of interest in time series analysis. We demonstrate various +features of fnets on simulated datasets as well as real data on electricity prices. +Introduction +Vector autoregressive (VAR) models are popularly adopted for modelling time series datasets collected +in many disciplines including economics (Koop, 2013), finance (Barigozzi and Brownlees, 2019), +neuroscience (Kirch et al., 2015) and systems biology (Shojaie and Michailidis, 2010), to name a few. +By fitting a VAR model to the data, we can infer dynamic interdependence between the variables and +forecast future values. In particular, estimating the non-zero elements of the VAR parameter matrices +recovers directed edges between the components of vector time series in a Granger causality network. +Besides, by estimating the precision matrix (inverse of the covariance matrix) of the VAR innovations, +we can define a network representing their contemporaneous dependencies by means of partial +correlations. Finally, the inverse of the long-run covariance matrix of the data simultaneously captures +lead-lag and contemporaneous co-movements of the variables. For the network interpretation of VAR +modelling, see e.g. Dahlhaus (2000), Eichler (2007), Billio et al. (2012) and Barigozzi and Brownlees +(2019). +Fitting VAR models to the data quickly becomes a high-dimensional problem as the number of +parameters grows quadratically with the dimensionality of the data. There exists a mature literature +on ℓ1-regularisation methods for estimating VAR models in high dimensions under suitable sparsity +assumptions on the VAR parameters (Basu and Michailidis, 2015; Han et al., 2015; Kock and Callot, +2015; Medeiros and Mendes, 2016; Nicholson et al., 2020; Liu and Zhang, 2021). Consistency of such +methods is derived under the assumption that the spectral density matrix of the data has bounded +eigenvalues. However, in many applications, the datasets exhibit strong serial and cross-sectional +correlations which leads to the violation of this assumption. As a motivating example, we introduce a +dataset of node-specific prices in the PJM (Pennsylvania, New Jersey and Maryland) power pool area +in the United States, see Energy price data for further details. Figure 1 demonstrates that the leading +eigenvalue of the long-run covariance matrix (i.e. spectral density matrix at frequency 0) increases +linearly as the dimension of the data increases, which implies the presence of latent common factors in +the panel data (Forni et al., 2000). Additionally, the left panel of Figure 2 shows the inadequacy of +fitting a VAR model to such data under the sparsity assumption via ℓ1-regularisation methods, unless +the presence of strong correlations is accounted for by a factor-adjustment step as in the right panel. +Barigozzi et al. (2022) propose the FNETS methodology for factor-adjusted VAR modelling of +high-dimensional, second-order stationary time series. Under their proposed model, the data is +decomposed into two latent components such that the factor-driven component accounts for pervasive +leading, lagging or contemporaneous co-movements of the variables, while the remaining idiosyncratic +dynamic dependence between the variables is modelled by a sparse VAR process. Then, FNETS +provides tools for inferring the networks underlying the latent VAR process and forecasting. +In this paper, we present an R package named fnets which implements the FNETS methodology. +It provides a range of user-friendly tools for estimating and visualising the networks representing the +interconnectedness of time series variables, and for producing forecasts. In addition, fnets thoroughly +addresses the problem of selecting tuning parameters ranging from the number of factors and the VAR +order, to regularisation and thresholding parameters adopted for producing sparse and interpretable +networks. As such, a simple call of the main routine of fnets requires the input data only, and it +outputs an object of S3 class fnets which is supported by a plot method for network visualisation +and a predict method for time series forecasting. +There exist several R packages for fitting VAR models and their extensions to high-dimensional +1 +arXiv:2301.11675v1 [stat.CO] 27 Jan 2023 + +1 +3 +5 +7 +9 +11 +14 +17 +20 +23 +26 +29 +32 +35 +38 +41 +44 +47 +2 +4 +6 +8 +10 +12 +14 +1 +3 +5 +7 +9 +11 +14 +17 +20 +23 +26 +29 +32 +35 +38 +41 +44 +47 +2 +4 +6 +8 +10 +12 +14 +First +Second +First +Second +Figure 1: Box plots of the two largest eigenvalues (y-axis) of the long-run covariance matrix estimated +from the energy price data collected between 01/01/2021 and 19/07/2021 (n = 200), see Data example +for further details. Cross-sections of the data are randomly sampled 100 times for each given dimension +p ∈ {2, . . . , 50} (x-axis) to produce the box plots. +Figure 2: Granger causal networks defined in (5) obtained from fitting a VAR(1) model to the energy +price data analysed in Figure 1, without (left) and with (right) the factor adjustment step outlined +in FNETS: Network estimation. Edge weights (proportional to the size of coefficient estimates) are +visualised by the width of each edge, and the nodes are coloured according to their groupings, see +Data example for further details. +time series by means of Lasso-type estimation techniques, see lsvar (Bai, 2021), sparsevar (Vazzoler, +2021), nets (Brownlees, 2020), mgm (Haslbeck and Waldorp, 2020), graphicalVAR (Epskamp et al., +2018), bigVAR (Nicholson et al., 2017), and bigtime (Wilms et al., 2021). The package fnets is clearly +distinguished from, and complements, the above list by handling strong cross-sectional and serial +correlations in the data via factor-adjustment step. In addition, the FNETS methodology operates under +the most general approach to high-dimensional time series factor modelling termed the Generalised +Dynamic Factor (GDFM), first proposed in Forni et al. (2000) and further investigated in Forni et al. +(2015). Accordingly, fnets is the first R package to provide tools for high-dimensional panel data +analysis under the GDFM, such as fast computation of spectral density and autocovariance matrices +via the Fast Fourier Transform. +FNETS methodology +In this section, we introduce the factor-adjusted VAR model and describe the FNETS methodology +proposed in Barigozzi et al. (2022) for network estimation and forecasting of high-dimensional time +series. We limit ourselves to describing the key steps of FNETS and refer to the above paper for its +comprehensive treatment, both methodologically and theoretically. +2 + +Factor-adjusted VAR model +A zero-mean, p-variate process ξt follows a VAR(d) model if it satisfies +ξt = +d +∑ +ℓ=1 +Aℓξt−ℓ + Γ1/2εt, +(1) +where Aℓ ∈ Rp×p, 1 ≤ ℓ ≤ d, determine how future values of the series depend on their past. For the +p-variate random vector εt = (ε1t, . . . , εpt)⊤, we assume that εit are independently and identically +distributed (i.i.d.) for all i and t with IE(εit) = 0 and Var(εit) = 1. Then, the positive definite matrix +Γ ∈ Rp×p is the covariance matrix of the innovations Γ1/2εt. +In the literature on factor modelling of high-dimensional time series, the factor-driven component +exhibits strong cross-sectional and/or serial correlations by ‘loading’ finite-dimensional vectors of +factors linearly. Among many time series factor models, the GDFM (Forni et al., 2000) provides +the most general approach where the p-variate factor-driven component χt admits the following +representation +χt = B(L)ut = +∞ +∑ +ℓ=0 +Bℓut−ℓ with ut = (u1t, . . . , uqt)⊤ and Bℓ ∈ Rp×q, +(2) +for some fixed q, where L stands for the lag operator. The q-variate random vector ut contains the +common factors which are loaded across the variables and time by the filter B(L) = ∑∞ +ℓ=0 BℓLℓ, and +it is assumed that ujt are i.i.d. with IE(ujt) = 0 and Var(ujt) = 1. The model (2) reduces to a static +factor model (Bai, 2003; Stock and Watson, 2002; Fan et al., 2013) when B(L) admits a decomposition +B(L) = M(1)(L)M(2)(L) with M(k)(L) = ∑mk +ℓ=0 M(k) +ℓ Lℓ for k = 1, 2, where M(1) ∈ Rp×q and M(2) ∈ +Rq×q. Then, we can write +χt = +m1 +∑ +ℓ=0 +M(1) +ℓ ft−ℓ = ΛFt where Ft = (f⊤ +t , . . . , f⊤ +t−m1)⊤ and ft = +m2 +∑ +ℓ=0 +M(2) +ℓ ut−ℓ, +(3) +with r = q(m1 + 1) as the dimension of static factors Ft. Throughout, we refer to the models (2) and (3) +as unrestricted and restricted to highlight that the latter imposes more restrictions on the model. +Barigozzi et al. (2022) propose a factor-adjusted VAR model under which we observe a zero-mean, +second-order stationary process Xt = (X1t, . . . , Xpt)⊤ for t = 1, . . . , n, that permits a decomposition +into the sum of the unobserved components ξt and χt, i.e. +Xt = ξt + χt. +(4) +We assume that IE(εitujt′) = 0 for all i, j, t and t′ as is commonly assumed in the literature, such that +IE(ξitχi′t′) = 0 for all 1 ≤ i, i′ ≤ p and t, t′ ∈ Z. +Networks +Under (4), it is of interest to infer three types of networks representing the interconnectedness of +Xt after factor adjustment. Let V = {1, . . . , p} denote the set of vertices representing the p cross- +sections. Then, the VAR parameter matrices, Aℓ = [Aℓ,ii′, 1 ≤ i, i′ ≤ p], encode the directed network +N G = (V, EG) representing Granger causal linkages, where the set of edges are given by +EG = +�(i, i′) ∈ V × V : Aℓ,ii′ ̸= 0 for some 1 ≤ ℓ ≤ d +� +. +(5) +Here, the presence of an edge (i, i′) ∈ EG indicates that ξi′,t−ℓ Granger causes ξit at some lag 1 ≤ ℓ ≤ d +(Dahlhaus, 2000). +The second network contains undirected edges representing contemporaneous cross-sectional +dependence in VAR innovations Γ1/2εt, denoted by N C = (V, EC). We have (i, i′) ∈ EC if and only if +the partial correlation between the i-th and i′-th elements of Γ1/2εt is non-zero. Specifically, letting +Γ−1 = ∆ = [δii′, 1 ≤ i, i′ ≤ p], the set of edges is given by +EC = +� +(i, i′) ∈ V × V : i ̸= i′ and − +δii′ +√δii · δi′i′ ̸= 0 +� +. +(6) +Finally, we can summarise the aforementioned lead-lag and contemporaneous relations between +the variables in a single, undirected network N L = (V, EL) by means of the long-run partial cor- +relations of ξt. Let Ω = [ωii′, 1 ≤ i, i′ ≤ p] denote the long-run partial covariance matrix of ξt, i.e. +3 + +the inverse of the zero-frequency spectral density of ξt which is given by Ω = 2πA⊤(1)∆A(1) with +A(z) = I − ∑d +ℓ=1 Aℓzℓ. Then, the edge set of N L is given by +EL = +� +(i, i′) ∈ V × V : i ̸= i′ and − +ωii′ +√ωii · ωi′i′ ̸= 0 +� +. +(7) +FNETS: Network estimation +We describe the three-step methodology for estimating the networks N G, N C and N L. Throughout, +we assume that the VAR order d is known, and discuss its selection in Tuning parameter selection. +Step 1: Factor adjustment +The autocovariance (ACV) matrices of ξt, denoted by Γξ(ℓ) = IE(ξt−ℓξ⊤ +t ) for ℓ ≥ 0 and Γξ(ℓ) = +(Γξ(−ℓ))⊤ for ℓ < 0, play a key role in network estimation. Since ξt is not directly observed, we +propose to adjust for the presence of the factor-driven χt via dynamic PCA, for the estimation of Γξ(ℓ). +Here, we assume that the number of factors, either q under the more general model in (2) or r under +the restricted model in (3), are known and defer the discussion on their choice to Tuning parameter +selection. With the sample ACV matrices of Xt, +�Γx(ℓ) = 1 +n +n +∑ +t=ℓ+1 +Xt−ℓX⊤ +t when ℓ ≥ 0 +and +�Γx(ℓ) = (�Γx(−ℓ))⊤ when ℓ < 0, +we estimate the spectral density of Xt by +�Σx(ωk) = 1 +2π +m +∑ +ℓ=−m +K +� ℓ +m +� +�Γx(ℓ) exp(−ιℓωk), +where K(·) denotes a kernel, m the kernel bandwidth (for its choice, see Tuning parameter selection) +and ωk = 2πk/(2m + 1) the Fourier frequencies. We adopt the Bartlett kernel as K(·) which ensures +positive semi-definiteness of �Σx(ω) and also �Γξ(ℓ) estimating Γξ(ℓ) obtained as described below. +Performing PCA on �Σx(ωk) at each ωk, we obtain the estimator of the spectral density matrix of +χt as �Σχ(ωk) = ∑ +q +j=1 �µx,j(ωk)�ex,j(ωk)(�ex,j(ωk))∗, where �µx,j(ωk) denotes the j-th largest eigenvalue +of �Σx(ωk), �ex,j(ωk) its associated eigenvector, and for any vector a ∈ Cn, we denote its transposed +complex conjugate by a∗. Then taking the inverse Fourier transform of �Σχ(ωk), −m ≤ k ≤ m, leads to +an estimator of Γχ(ℓ), the ACV matrix of χt, as +�Γχ(ℓ) = +2π +2m + 1 +m +∑ +k=−m +�Σχ(ωk) exp(ιℓωk) +for − m ≤ ℓ ≤ m. +Finally, we estimate the ACV of ξt by +�Γξ(ℓ) = �Γx(ℓ) − �Γχ(ℓ). +(8) +When we assume the restricted factor model in (3), the factor-adjustment step is simplified as it +suffices to perform PCA in the time domain. Denoting the eigenvector of the sample covariance matrix +�Γx(0) associated with its j-th largest eigenvalue by �ex,j, we obtain �Γχ(ℓ) = �Ex�E⊤x �Γx(ℓ)�Ex�E⊤x , where +�Ex = [�ex,j, 1 ≤ j ≤ r]. Then, we obtain �Γξ(ℓ) as in (8). +Step 2: Estimation of N G +Recall from (5) that N G representing Granger causal linkages, has its edge set determined by the +VAR transition matrices Aℓ, 1 ≤ ℓ ≤ d. By the Yule-Walker equation, we have β0 = [A1, . . . , Ad]⊤ = +G(d)−1g(d), where +G(d) = +� +���� +Γξ(0) +Γξ(−1) +. . . +Γξ(−d + 1) +Γξ(1) +Γξ(0) +. . . +Γξ(−d + 2) +... +Γξ(d − 1) +Γξ(d − 2) +. . . +Γξ(0) +� +���� +and +g(d) = +� +���� +Γξ(1) +Γξ(2) +... +Γξ(d) +� +���� . +(9) +4 + +We propose to estimate β0 as a regularised Yule-Walker estimator based on �G(d) and �g(d), each of +which is obtained by replacing Γξ(ℓ) with �Γξ(ℓ) (see (8)) in the definition of G(d) and g(d). +For any matrix A = [aij] ∈ Rm×n, let |A|1 = ∑m +i=1 ∑n +j=1 |aij|, |A|∞ = max1≤i≤m max1≤j≤n |aij| and +tr(A) = ∑m +i=1 aii when m = n. We consider two estimators of β0. Firstly, we adopt a Lasso-type +estimator which solves an ℓ1-regularised M-estimation problem +�βlas = arg min +β∈Rpd×p +tr +� +β⊤ �G(d)β − 2β⊤�g(d) +� ++ λ|β|1 +(10) +with a tuning parameter λ > 0. In the implementation, we solve (10) via the fast iterative shrinkage- +thresholding algorithm (FISTA, Beck and Teboulle (2009)). Alternatively, we adopt a constrained +ℓ1-minimisation approach closely related to the Dantzig selector (Candes and Tao, 2007, DS): +�βDS = arg min +β∈Rpd×p +|β|1 +subject to +��� �G(d)β − �g(d) +��� +∞ ≤ λ +(11) +for some tuning parameter λ > 0. We divide (11) into p sub-problems and obtain each column of �βDS +via the simplex algorithm (using the function lp in lpSolve). +Barigozzi et al. (2022) establish the consistency of both �βlas and �βDS but, as is typically the case for +ℓ1-regularisation methods, they do not achieve exact recovery of the support of β0. Hence we propose +to estimate the edge set of N G by thresholding the elements of �β with some threshold t > 0, where +either �β = �βlas or �β = �βDS, i.e. +�β(t) = +��βij · I{|�βij|>t}, 1 ≤ i ≤ pd, 1 ≤ j ≤ p +� +. +(12) +We discuss cross validation and information criterion methods for selecting λ, and a data-driven +choice of t, in Tuning parameter selection. +Step 3: Estimation of N C and N L +From the definitions of N C and N L given in (6) and (7), their edge sets are obtained by estimating +∆ = Γ−1 and Ω = 2πA⊤(1)∆A(1). Given �β = [�A1, . . . , �Ad]⊤, some estimator of the VAR parameter +matrices obtained as in either (10) or (11), a natural estimator of Γ arises from the Yule-Walker equation +Γ = Γξ(0) − ∑d +ℓ=1 AℓΓξ(ℓ) = Γξ(0) − (β0)⊤g, as �Γ = �Γξ(0) − �β⊤�g. In high dimensions, it is not +feasible or recommended to directly invert �Γ to estimate ∆. Therefore, we adopt a constrained +ℓ1-minimisation method motivated by the CLIME methodology of Cai et al. (2011). +Specifically, the CLIME estimator of ∆ is obtained by first solving +ˇ∆ = arg minM∈Rp×p|M|1 +subject to +����ΓM − I +��� +∞ ≤ η, +(13) +and applying a symmetrisation step to ˇ∆ = [ ˇδii′, 1 ≤ i, j ≤ p] as +�∆ = [�δii′, 1 ≤ i, i′ ≤ p] with �δii′ = ˇδii′ · I{| ˇδii′ |≤| ˇδi′i|} + ˇδi′i · I{| ˇδi′i|<| ˇδii′ |}. +(14) +for some tuning parameter η > 0. Cai et al. (2016) propose ACLIME, which improves the CLIME +estimator by selecting the parameter η in (14) adaptively. It first produces the estimators of the diagonal +entries δii, 1 ≤ i ≤ p, as in (14) with η1 = 2 +� +log(p)/n as the tuning parameter. Then these estimates +are used for adaptive tuning parameter selection in the second step. We provide the full description +of the ACLIME estimator along with the details of its implementation in ACLIME estimator of the +Appendix. +Given the estimators � +A(1) = I − ∑d +ℓ=1 �Aℓ and �∆, we estimate Ω by �Ω = 2π � +A⊤(1)�∆ � +A(1). In +Barigozzi et al. (2022), �∆ and �Ω are shown to be consistent in ℓ∞- and ℓ1-norms under suitable sparsity +assumptions. However, an additional thresholding step as in (12) is required to guarantee consistency +in estimating the support of ∆ and Ω and consequently the edge sets of N C and N L. We discuss +data-driven selection of these thresholds and η in Tuning parameter selection. +FNETS: Forecasting +Following the estimation procedure, FNETS performs forecasting by estimating the best linear predic- +tor of Xn+a given Xt, t ≤ n, for a fixed integer a ≥ 1. This is achieved by separately producing the best +linear predictors of χn+a and ξn+a as described below, and then combining them. +5 + +Forecasting the factor-driven component +For given a ≥ 0, the best linear predictor of χn+a given Xt, t ≤ n, under (2) is +χn+a|n = +∞ +∑ +ℓ=0 +Bℓ+aun−ℓ. +Forni et al. (2015) show that the model (2) admits a low-rank VAR representation with ut as the +innovations under mild conditions, and Forni et al. (2017) propose the estimators of Bℓ and ut based +on this representation which make use of the estimators of the ACV of χt obtained as described in +Step 1. Then, a natural estimator of χn+a|n is +�χunr +n+a|n = +K +∑ +ℓ=0 +�Bℓ+a�un−ℓ +(15) +for some truncation lag K. We refer to �χunr +n+a|n as the unrestricted estimator of χn+a|n as it is obtained +without imposing any restrictions on the factor model (2). +When χt admits the static representation in (3), we can show that χn+a|n = Γχ(−a)EχM−1 +χ E⊤ +χ χn, +where Mχ ∈ Rr×r is a diagonal matrix with the r eigenvalues of Γχ(0) on its diagonal and Eχ ∈ Rp×r +the matrix of the corresponding eigenvectors. This suggests an estimator +�χres +n+a|n = �Γχ(−a)�Eχ � +M +−1 +χ �E⊤ +χ Xn, +(16) +where � +Mχ and �Eχ are obtained from the eigendecomposition of �Γχ(0). We refer to �χres +n+a|n as the +restricted estimator of χn+a|n. As a by-product, we obtain the in-sample estimators of χt, t ≤ n, as +�χt|n = �χt, with either of the two estimators in (15) and (16). +Forecasting the latent VAR process +Once the VAR parameters are estimated either as in (10) or (11), we produce an estimator of ξn+a|n = +∑d +ℓ=1 Aℓξn+a−ℓ, the best linear predictor of ξn+a given Xt, t ≤ n, as +�ξn+a|n = +max(1,a)−1 +∑ +ℓ=1 +�Aℓ�ξn+a−ℓ|n + +d +∑ +ℓ=max(1,a) +�Aℓ�ξn+a−ℓ. +(17) +Here, �ξn+1−ℓ = Xn+1−ℓ − �χn+1−ℓ denotes the in-sample estimator of ξn+1−ℓ, which may be obtained +with either of the two (in-sample) estimators of the factor-driven component in (15) and (16). +Tuning parameter selection +Factor numbers q and r +The estimation and forecasting tools of the FNETS methodology require the selection of the number of +factors, i.e. q under the unrestricted factor model in (2), and r under the restricted, static factor model +in (3). Under (2), there exists a large gap which diverges with p, between the q leading eigenvalues of +the spectral density matrix of Xt and the remainder. We provide two methods for selecting the factor +number q, which make use of the postulated eigengap using �µx,j(ωk), 1 ≤ j ≤ p, the eigenvalues of +the spectral density matrix estimator of Xt at a given Fourier frequency ωk, −m ≤ k ≤ m. +Hallin and Liška (2007) propose an information criterion for selecting the number of factors under +the model (2) and further, a methodology for tuning the multiplicative constant in the penalty. Define +IC(b, c) = log +� +� 1 +p +p +∑ +j=b+1 +1 +2m + 1 +m +∑ +k=−m +�µx,j(ωk) +� +� + b · c · pen(n, p), +(18) +where pen(n, p) = min(p, m2, √ +n/m)−1/2 by default (for other choices of the information criterion, +see Appendix A) and c > 0 a constant. Provided that pen(n, p) → 0 sufficiently slowly, for an +arbitrary value of c, the factor number q is consistently estimated by the minimiser of IC(b, c) over +b ∈ {0, . . . , ¯q}, with some fixed ¯q as the maximum allowable number of factors. However, this is not +the case in finite sample, and Hallin and Liška (2007) propose to simultaneously select q and c. First, +6 + +we identify �q(nl, pl, c) = arg min0≤b≤ ¯q IC(nl, pl, b, c) where IC(nl, pl, b, c) is constructed analogously +to IC(b, c), except that it only involves the sub-sample {Xit, 1 ≤ i ≤ pl, 1 ≤ t ≤ nl}, for sequences +0 < n1 < . . . < nL = n and 0 < p1 < . . . < pL = p. Then, denoting the sample variance of +�q(nl, pl, c), 1 ≤ l ≤ L, by S(c), we select �q = �q(n, p, �c) with �c corresponding to the second interval +of stability with S(c) = 0 for the mapping c �→ S(c) as c increases from 0 to some cmax (the first +stable interval is where ¯q is selected with a very small value of c). Figure 3 plots �q(n, p, c) and S(c) +for varying values of c obtained from a dataset simulated in Data generation. In the implementation +of this methodology, we set nl = n − (L − l)⌊n/20⌋ and pl = ⌊3p/4 + lp/40⌋ with L = 10, and +¯q = min(50, ⌊ +� +min(n − 1, p)⌋). +IC 1 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +3 +4 +5 +6 +7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q +Sc +IC 2 +0.0 +0.5 +1.0 +1.5 +2.0 +1 +2 +3 +4 +5 +6 +7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q +Sc +IC 3 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +3 +4 +5 +6 +7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q +Sc +IC 4 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +3 +4 +5 +6 +7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q +Sc +IC 5 +0.0 +0.5 +1.0 +1.5 +2.0 +1 +2 +3 +4 +5 +6 +7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q +Sc +IC 6 +0.0 +0.5 +1.0 +1.5 +2.0 +2 +3 +4 +5 +6 +7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +q +Sc +Figure 3: Plots of c against �q(n, p, c) (in circle, y-axis on the left) and S(c) (in triangle, y-axis on the +right) with the six IC (see Appendix A) implemented in the function factor.number of fnets, on a +dataset simulated as in Data generation (with n = 500, p = 50 and q = 2). With the default choice of +IC in (18) (IC5), we obtain �q = �q(n, p, �c) = 2 correctly estimating q = 2. +Alternatively, we can adopt the ratio-based estimator �q = arg min1≤b≤ ¯q ER(b) proposed in +Avarucci et al. (2022), where +ER(b) = +� +m +∑ +k=−m +�µx,b+1(ωk) +�−1 � +m +∑ +k=−m +�µx,b(ωk) +� +. +(19) +These methods are readily modified to select the number of factors r under the restricted factor +model in (3), by replacing (2m + 1)−1 ∑m +k=−m �µx,j(ωk) with �µx,j, the eigenvalues of the sample covari- +ance matrix �Γx(0). We refer to Bai and Ng (2002) and Alessi et al. (2010) for the discussion of the +information criterion-based method in this setting, and Ahn and Horenstein (2013) for that of the +eigenvalue ratio-based method. +Threshold t +Motivated by Liu et al. (2021), we propose a method for data-driven selection of the threshold t, which +is applied to the estimators of Aℓ, 1 ≤ ℓ ≤ d (see (12)), ∆ or Ω for estimating the edge sets of N G, N C +or N L, respectively. +Let B = [bij] ∈ Rm×n denote a matrix for which a threshold is to be selected, i.e. B may be either +�β = [�A1, . . . , �Ad]⊤, �∆0 (�∆ with diagonals set to zero) or �Ω0 ( �Ω with diagonals set to zero) obtained +from Steps 2 and 3 of FNETS. We work with �∆0 and �Ω0 since we do not threshold the diagonal entries +of �∆ and �Ω. As such estimators are shown to achieve consistency in ℓν-norm for some ν > 0, we expect +there exists a large gap between the entries of B corresponding to true positives and false positives. +Further, it is expected that the number of edges reduces at a faster rate when increasing the threshold +from 0 towards this (unknown) gap, compared to when increasing the threshold from the gap to |B|∞. +Therefore, we propose to identify this gap by casting the problem as that of locating a single change +point in the trend of the ratio of edges to non-edges, +Ratiok = +|B(tk)|0 +max(N − |B(tk)|0, 1), +k = 1, . . . , M. +Here, B(t) = [bij · I{|bij|>t}], |B(t)|0 = ∑m1 +i=1 ∑m2 +j=1 I{|bij|>t} and {tk, 1 ≤ k ≤ M : 0 = t1 < t2 < · · · < +7 + +tM = |B|∞} denotes a sequence of candidate threshold values. We recommend using an exponentially +growing sequence for {tk}M +k=1 since the size of the false positive entries tends to be very small. The +quantity N in the denominator of Ratiok is set as N = p2d when B = �β, and N = p(p − 1) when +B = �∆0 or B = �Ω0. Then, from the difference quotient +Diffk = Ratiok − Ratiok−1 +tk − tk−1 +, +k = 2, . . . , M, +we compute the cumulative sum (CUSUM) statistic +CUSUMk = +� +k(M − k) +M +����� +1 +k +k +∑ +l=2 +Diffl − +1 +M − k +M +∑ +l=k+1 +Diffl +����� , +k = 2, . . . , M − 1, +and select tada = tk∗ with k∗ = arg max2≤k≤M−1CUSUMk. For illustration, Figure 4 plots Ratiok and +CUSUMk against candidate thresholds for the dataset simulated in Data generation. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.00 +0.02 +0.04 +0.06 +threshold +ratio +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +threshold +abs(cusum) +Figure 4: Ratiok (left) and CUSUMk (right) plotted against tk when B = �βlas obtained from the data +simulated in Data generation with n = 500 and p = 50, as a Lasso estimator of the VAR parameter +matrix, with the selected tada denoted by the vertical lines. +VAR order d, λ and η +Step 2 and Step 3 of the network estimation methodology of FNETS involve the selection of the tuning +parameters λ and η (see (10), (11) and (13)) and the VAR order d. While there exist a variety of methods +available for VAR order selection in fixed dimensions (Lütkepohl, 2005, Chapter 4), data-driven +selection of d in high dimensions remains largely unaddressed with few exceptions (Nicholson et al., +2020; Krampe and Margaritella, 2021; Zheng, 2022). We suggest two methods for jointly selecting λ +and d for Step 2. The first method is also applicable for selecting η in Step 3. +Cross validation +We describe the use of cross validation (CV) for simultaneously selecting d and λ in Step 2. The data +is partitioned into L folds, Il = {n◦ +l + 1, . . . , n◦ +l+1} with n◦ +l = min(l⌈n/L⌉, n), 1 ≤ l ≤ L, and each +fold is split into a training set Itrain +l += {n◦ +l + 1, . . . , ⌈(n◦ +l + n◦ +l+1)/2⌉} and a test set Itest +l += Il \ Itrain +l +. +On each fold, β0 is estimated from {Xt, t ∈ Itrain +l +} as either the Lasso (10) or the Dantzig selector (11) +estimators with λ as the tuning parameter and some b as the VAR order, say �βtrain +l +(λ, b), using which +we compute the CV measure +CV(λ, b) = +L +∑ +l=1 +tr +� +�Γtest +ξ,l (0) − ( �βtrain +l +(λ, b))⊤�gtest +l +(b)− +(�gtest +l +(b))⊤ �βtrain +l +(λ, b) + ( �βtrain +l +(λ, b))⊤ �Gtest +l +(b) �βtrain +l +(λ, b) +� +, +where �Γtest +ξ,l (ℓ), �Gtest +l +(b) and �gtest +l +(b) are generated analogously as �Γξ(ℓ), �G(b) and �g(b), respectively, +from the test set {Xt, t ∈ Itest +l +}. Although we do not directly observe ξt, the measure CV(λ, b) gives +an approximation of the prediction error. Then, we select (�λ, �d) = arg minλ∈Λ,1≤b≤ ¯d CV(λ, b), where +Λ is a grid of values for λ, and ¯d ≥ 1 is a pre-determined upper bound on the VAR order. A similar +8 + +approach is taken for the selection of η with a Burg matrix divergence-based CV measure: +CV(η) = +L +∑ +l=1 +tr +� +�∆train +l +(η)�Γtest +l +� +− log +����∆train +l +(η)�Γtest +l +��� − p. +Here, �∆train +l +(η) denotes the estimator of ∆ with η as the tuning parameter from {Xt, t ∈ Itrain +l +}, and +�Γtest +l +the estimator of Γ from {Xt, t ∈ Itest +l +}, see Step 3 for the descriptions of the estimators. For both +CV procedures, we set L = 1 in the numerical results reported in Simulations. +Extended Bayesian information criterion +Alternatively, to select the pair (λ, d) in Step 2, we propose to use the extended Bayesian information cri- +terion (eBIC) of Chen and Chen (2008), originally proposed for variable selection in high-dimensional +linear regression. Let �β(λ, b, tada) denote the thresholded version of �β(λ, b) as in (12) with the threshold +tada chosen as described in Threshold t. Then, letting s(λ, b) = | �β(λ, b, tada)|0, we define +eBICα(λ, b) = n +2 log (L(λ, b)) + s(λ, b) log(n) + 2α log +� bp2 +s(λ, b) +� +, +where +(20) +L(λ, b) = tr +� +�G(b) − ( �β(λ, b))⊤�g(b) − (�g(b))⊤ �β(λ, b) + ( �β(λ, b))⊤ �G(b) �β(λ, b) +� +. +Then, we select (�λ, �d) = arg minλ∈Λ,1≤b≤ ¯d eBICα(λ, b). The constant α ∈ (0, 1) determines the degree +of penalisation which may be chosen from the relationship between n and p. Preliminary simulations +suggest that α = 0 is a suitable choice for the dimensions (n, p) considered in our numerical studies. +Other tuning parameters +Motivated by theoretical results reported in Barigozzi et al. (2022), we select the kernel bandwidth +for Step 1 of FNETS as m = ⌊4(n/ log(n))1/3⌋. In forecasting the factor-driven component as in (15), +we set the truncation lag at K = 20, as it is expected that the elements of Bℓ decay as ℓ increases for +stationary time series. +Package overview +fnets is available from the Comprehensive R Archive Network (CRAN). The main function, fnets, +implements the FNETS methodology for the input data and returns an object of S3 class fnets. +fnets.var implements Step 2 of the FNETS methodology estimating the VAR parameters only, and is +applicable directly for VAR modelling of high-dimensional time series. fnets.factor.model performs +factor modelling under either of the two models (2) and (3), and returns an object of class fm. We +provide predict methods for the objects of classes fnets and fm, and a plot method for the objects of +the fnets class. We recommend that the input time series for the above functions to be transformed to +stationarity (if necessary) after a unit root test. In this section, we demonstrate how to use the functions +included with the package. +Data generation +For illustration, we generate an example dataset of n = 500 and p = 50 following the model (4). fnets +provides functions for this purpose. For given n and p, the function sim.var generates the VAR(1) +process following (1) with d = 1, Γ as supplied to the function (Γ = I by default), and A1 generated +as described in Simulations. The function sim.unrestricted generates the factor-driven component +under the unrestricted factor model in (2) with q dynamic factors (q = 2 by default) and the filter B(L) +generated as in model (C1) of Simulations. +set.seed(111) +n <- 500 +p <- 50 +x <- sim.var(n, p)$data + sim.unrestricted(n, p)$data +Throughout this section, we use the thus-generated dataset in demonstrating fnets unless specified +otherwise. There also exists sim.restricted which generates the factor-driven component under the +restricted factor model in (3). For all data generation functions, the default is to use the standard normal +9 + +distribution for generating ut and εt, while supplying the argument heavy = TRUE, the innovations are +generated from √ +3/5 · t5, the t-distribution with 5 degrees of freedom scaled to have unit variance. +Calling fnets with default parameters +The function fnets can be called with the p × n data matrix x as the only input, which sets all other +arguments to their default choices. +out <- fnets(x) +By default, it performs the factor-adjustment under the unrestricted model in (2) with q estimated +by minimising the IC in (18). The VAR parameter matrix is estimated via the Lasso estimator in (10) +with d = 1 as the VAR order and the tuning parameters λ and η chosen via CV, and no thresholding is +performed. This returns an object of class fnets whose entries are described in Table 1. +Table 1: Entries of S3 objects of class fnets +Name +Description +Type +q +Factor number +integer +spec +Spectral density matrices for Xt, χt and ξt (when fm.restricted = FALSE) +list +acv +Autocovariance matrices for Xt, χt and ξt +list +loadings +Estimates of Bℓ, 0 ≤ ℓ ≤ K (when fm.restricted = FALSE) +array +or Λ (when fm.restricted = TRUE) +factors +Estimates of {ut} (when fm.restricted = FALSE) +array +or {Ft} (when fm.restricted = TRUE) +idio.var +Estimates of Aℓ, 1 ≤ ℓ ≤ d and Γ, and d and λ used +list +lrpc +Estimates of ∆, Ω, (long-run) partial correlations and η used +list +mean.x +Sample mean vector +vector +var.method +Estimation method for Aℓ +string +do.lrpc +Whether to estimate the long-run partial correlations (input parameter) +Boolean +kern.bw +Kernel bandwidth (when fm.restricted = FALSE, input parameter) +double +Calling fnets with optional parameters +We can also specify the arguments of fnets to control how Steps 1–3 of FNETS are to be performed. +The full model call is as follows: +out <- fnets(x, center = TRUE, fm.restricted = FALSE, +q = c("ic", "er"), ic.op = NULL, kern.bw = NULL, +common.args = list(factor.var.order = NULL, max.var.order = NULL, trunc.lags = 20, +n.perm = 10), var.order = 1, var.method = c("lasso", "ds"), +var.args = list(n.iter = NULL, n.cores = min(parallel::detectCores() - 1, 3)), +do.threshold = FALSE, do.lrpc = TRUE, lrpc.adaptive = FALSE, +tuning.args = list(tuning = c("cv", "bic"), n.folds = 1, penalty = NULL, +path.length = 10, do.plot = FALSE) +) +Here, we discuss a selection of input arguments. The center argument will de-mean the input. +fm.restricted determines whether to perform the factor-adjustment under the restricted factor model +in (3) or not. If the number of factors is known, we can specify q with a non-negative integer. Otherwise, +it can be set as "ic" or "er" which selects the factor number estimator to be used between (18) and (19). +When q = "ic", setting the argument ic.op as an integer between 1 and 6 specifies the choice of the +IC (see Appendix A) where the default is ic.op = 5. kern.bw takes a positive integer which specifies +the bandwidth to be used in Step 1 of FNETS. The list common.args specifies arguments for estimating +Bℓ and ut under (2), and relates to the low-rank VAR representation of χt under the unrestricted +factor model. var.order specifies a vector of positive integers to be considered in VAR order selection. +var.method determines the method for VAR parameter estimation, which can be either "lasso" (for +the estimator in (10)) or "ds" (for that in (11)). The list var.args takes additional parameters for Step 2 +of FNETS, such as the number of gradient descent steps (n.iter, when var.method = "lasso") or the +number of cores to use for parallel computing (n.cores, when var.method = "ds"). do.threshold +selects whether to threshold the estimators of Aℓ, 1 ≤ ℓ ≤ d, ∆ and Ω. It is possible to perform +Steps 1–2 of FNETS only without estimating ∆ and Ω by setting do.lrpc = FALSE. If do.lrpc = TRUE, +lrpc.adaptive specifies whether to use the non-adaptive estimator in (13) or the ACLIME estimator. +10 + +Granger causal network +Long-run partial correlation heatmap +-1.0 +-0.5 +0.0 +0.5 +1.0 +Figure 5: Estimated networks for data simulated as in Data generation. Left: Granger causal net- +work N G. A directed arrow from node i to node i′ indicates that variable i Granger causes node i′, and +the edge weights proportional to the size of estimated coefficients are visualised by the edge width. +Right: Long-run partial correlation network N L where the edge weights (i.e. partial correlations) are +visualised by the colour. +The list tuning.args supplies arguments to the CV or eBIC procedures, including the number of folds +L (n.folds), the eBIC parameter α (penalty, see (20)) and the length of the grid of values for λ and/or +η (path.length). Finally, it is possible to set only a subset of the arguments of var.args, common.args, +and tuning.args whereby the unspecified arguments are set to their default values. +The factor adjustment (Step 1) and VAR parameter estimation (Step 2) functionalities can be +accessed individually by calling fnets.factor.model and fnets.var, respectively. The latter is equiv- +alent to calling fnets with q = 0 and do.lrpc = FALSE. The former returns an object of class fm which +contains the entries of the fnets object in Table 1 that relate to the factor-driven component only. +Network visualisation +Using the plot method available for the objects of class fnets, we can visualise the Granger network +N G induced by the estimated VAR parameter matrices, see the left panel of Figure 5. +plot(out, type = "granger", display = "network") +Setting the argument type to "pc" or "lrpc", we can visualise N C given by the partial correlations of +VAR innovations or N L given by the long-run partial correlations of ξt. This displays an igraph object +from igraph (Csardi et al., 2006). We can instead visualise the networks as a heat map, with the edge +weights colour-coded by setting display = "heatmap". We plot N L as a heat map in the right panel +of Figure 5 using the following command. +plot(out, type = "lrpc", display = "heatmap") +Forecasting +The fnets objects are supported by the predict method with which we can perform h-step ahead +forecasting of the input data. For example, we can produce a one-step ahead forecast of Xn+1 as +pr <- predict(out, x, h = 1, fc.restricted = TRUE) +pr$forecast +The argument fc.restricted specifies whether to use the estimator �χres +n+h|n in (16) generated +under a restricted factor model (3), or �χunr +n+h|n in (15) generated without such a restriction. Internally, +predict.fnets calls common.predict and idio.predict to sequentially produce forecasts and in- +sample estimators of the factor-driven component and the VAR process, and the outputs are reported +together with the h-step ahead forecast of the input data, see Table 2. +11 + +Table 2: Entries of the output from predict.fnets +Name +Description +Type +forecast +The h-step ahead forecast of Xt +list +common.predict +Output of common.pred containing +list +$is +p × n matrix containing the in-sample estimator of χt +$fc +p × h matrix containing the h-step ahead forecasts of χt +$h +Input parameter +$r +Factor number (only produced when fc.restricted = TRUE) +idio.predict +Output of idio.pred containing is, fc and h +list +mean.x +Sample mean vector +vector +Factor number estimation +It is of independent interest to estimate the number of factors (if any) in the input dataset. The function +factor.number provides access to the two methods for selecting q described in Factor numbers q and +r. The code +fn <- factor.number(x, fm.restricted = FALSE, do.plot = TRUE) +calls the information criterion-based factor number estimation method in (18), and setting do.plot = +TRUE returns Figure 3 which visualises the results. Alternatively, we call the eigenvalue ratio-based +method in (19) as +fn <- factor.number(x, method = "er", fm.restricted = FALSE) +In this case, setting do.plot = TRUE produces a plot of ER(b) against the candidate factor number +b ∈ {1, . . . , ¯q}. +Visualisation of the tuning parameter selection procedure +We provide tools for visualising the tuning parameter selection results adopted in Steps 2 and 3 of +FNETS (see VAR order d, λ and η). These tools are accessible from both fnets and fnets.var by +setting tuning.args = list(do.plot = TRUE), e.g. +set.seed(111) +n <- 500 +p <- 10 +x <- sim.var(n, p)$data +out <- fnets(x, q = 0, var.order = 1:3, +tuning.args = list(tuning = "cv", do.plot = TRUE)) +This generates the two plots reported in Figure 6 which visualise the CV errors computed as described +in Cross validation and, in particular, the left plot shows that the VAR order is correctly selected by this +approach. When tuning.args = list(tuning = "bic"), the results from the eBIC method described +in Extended Bayesian information criterion adopted in Step 2, is similarly visualised in place of the +left panel of Figure 6. +Simulations +Barigozzi et al. (2022) provide comprehensive simulation results on the estimation and forecasting +performance of FNETS in comparison with competing methodologies. Therefore in this paper, we focus +on assessing the performance of the methods for selecting tuning parameters such as the threshold and +VAR order, which are implemented in fnets and discussed in Tuning parameter selection. Additionally +in Appendix B, we compare the adaptive and the non-adaptive estimators in estimating ∆ and also +investigate how their performance is carried over to estimating Ω. +Settings +We consider the following data generating processes for the factor-driven component χt: +12 + +5e-04 +5e-03 +5e-02 +5e-01 +7.8 +8.0 +8.2 +8.4 +8.6 +8.8 +CV for VAR parameter estimation +1 +2 +3 +0.01 +0.02 +0.05 +0.10 +0.20 +0.50 +1.00 +0 +5 +10 +15 +20 +25 +CV for (LR)PC matrix estimation +Figure 6: Plots of CV(λ, b) against λ with b ∈ {1, 2, 3} (left) and CV(η) against η (right). Vertical lines +denote where the minimum CV measure is attained with respect to λ and η, respectively. +(C1) Taken from Forni et al. (2017), χit is generated as a sum of AR processes χit = ∑ +q +j=1 aij(1 − +αijL)−1ujt with q = 2, where ujt ∼iid N (0, 1), aij ∼iid U[−1, 1] and αij ∼iid U[−0.8, 0.8] with +U[a, b] denoting a uniform distribution. Then, χt does not admit a static representation in (3). +(C2) χt = 0, i.e. the VAR process is directly observed as Xt = ξt. +For generating a VAR(d) process ξt, we first generate a directed Erd˝os-Rényi random graph N = +(V, E) on V = {1, . . . , p} with the link probability 1/p, and set entries of Ad such that Ad,ii′ = 0.275 +when (i, i′) ∈ E and Ad,ii′ = 0 otherwise. Also, we set Aℓ = O for ℓ < d. The VAR innovations are +generated as below. +(E1) Gaussian with the covariance matrix Γ = ∆−1 = I. +(E2) Gaussian with the covariance matrix Γ = ∆−1 such that δii = 1, δi,i+1 = δi+1,i = 0.6, δi,i+2 = +δi+2,i = 0.3, and δii′ = 0 for |i − i′| ≥ 3. +For each setting, we generate 100 realisations. +Results: Threshold selection +We assess the performance of the adaptive threshold. We generate χt as in (C1) and fix d = 1 for gener- +ating ξt and further, treat d as known. We consider (n, p) ∈ {(200, 50), (200, 100), (500, 100), (500, 200)}. +Then we estimate Ω using the thresholded Lasso estimator of A1 (see (10) and (12)) with two choices +of thresholds, t = tada generated as described in Threshold t and t = 0. To assess the performance +of �Ω = [ �ωii′] in recovering of the support of Ω = [ωii′], i.e. {(i, i′) : ωii′ ̸= 0}, we plot the receiver +operating characteristic (ROC) curves of true positive rate (TPR) against false positive rate (FPR), +where +TPR = |{(i, i′) : �ωii′ ̸= 0 and ωii′ ̸= 0}| +|{(i, i′) : ωii′ ̸= 0}| +and +FPR = |{(i, i′) : �ωii′ ̸= 0 and ωii′ = 0}| +|{(i, i′) : ωii′ = 0}| +. +Figure 7 plots the ROC curves averaged over 100 realisations when t = tada and t = 0. When ∆ = I +under (E1), we see little improvement from adopting tada as the support recovery performance is +already good even without thresholding. However, when ∆ ̸= I under (E2), the adaptive threshold +leads to improved support recovery especially when the sample size is large. Tables 3 and 4 in +Appendix C additionally report the errors in estimating A1 and Ω with and without thresholding, +where we see little change is brought by thresholding. In summary, we conclude that the estimators +already perform reasonably well without thresholding, and the adaptive threshold tada brings marginal +improvement in support recovery which is of interest in network estimation. +13 + +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +model +E1 +E2 +method +adaptive threshold +threshold=0 +n = 200, p = 50 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 200, p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 500, p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 500, p = 200 +Figure 7: ROC curves of TPR against FPR for �β(t) (12) (with �β = �βlas) when t = tada and t = 0 in +recovering the support of Ω, averaged over 100 realisations. Vertical lines indicate FPR = 0.05 +Results: VAR order selection +We compare the performance of the CV and eBIC methods proposed in VAR order d, λ and η for +selecting the order of the VAR process. Here, we consider the case when χt = 0 (setting (C2)) and when +ξt is generated under (E1) with d ∈ {1, 3}. We set (n, p) ∈ {(200, 10), (200, 20), (500, 10), (500, 20)} +where the range of p is in line with the simulation studies conducted in the relevant literature (see e.g. +Zheng (2022)). We consider {1, 2, 3, 4} as the candidate VAR orders. Figure 8 and Table 5 in Appendix +C show that CV works reasonably well regardless of d ∈ {1, 3}, with slightly better performance +observed together with the DS estimator. On the other hand, eBIC tends to over-estimate the VAR +order when d = 1 while under-estimating it when d = 3, and hence is less reliable compared to the CV +method. +0 +25 +50 +75 +100 +0 +1 +2 +3 +n = 200, p = 10, d = 1 +0 +25 +50 +75 +100 +-2 +-1 +0 +1 +method +BIC, DS +BIC, Lasso +CV, DS +CV, Lasso +n = 200, p = 10, d = 3 +0 +25 +50 +75 +100 +0 +1 +2 +3 +n = 200, p = 20, d = 1 +0 +25 +50 +75 +100 +-2 +-1 +0 +1 +n = 200, p = 20, d = 3 +0 +25 +50 +75 +100 +0 +1 +2 +3 +n = 500, p = 10, d = 1 +0 +25 +50 +75 +100 +-2 +-1 +0 +1 +n = 500, p = 10, d = 3 +0 +25 +50 +75 +100 +0 +1 +2 +3 +n = 500, p = 20, d = 1 +0 +25 +50 +75 +100 +-2 +-1 +0 +1 +n = 500, p = 20, d = 3 +Figure 8: Box plots of �d − d over 100 realisations when the VAR order is selected by the CV and eBIC +methods in combination with the Lasso (10) and the DS (11) estimators. +14 + +Data example +Energy price data +Electricity is more difficult to store than physical commodities, which results in high volatility and +seasonality in spot prices (Han et al., 2022). Global market deregulation has increased the volume +of electricity trading, which promotes the development of better forecasting and risk management +methods. We analyse a dataset of node-specific prices in the PJM (Pennsylvania, New Jersey and +Maryland) power pool area in the United States, accessed using dataminer2.pjm.com. There are +four node types in the panel, which are Zone, Aggregate, Hub and Extra High Voltage (EHV); for +their definitions, see Table 8 and for the names and types of p = 50 nodes, see Table 9, all found in +Appendix D. The series we model is the sum of the real time congestion price and marginal loss price +or, equivalently, the difference between the spot price at a given location and the overall system price, +where the latter can be thought of as an observed factor in the local spot price. These are obtained as +hourly prices and then averaged over each day as per Maciejowska and Weron (2013). We remove +any short-term seasonality by subtracting a separate mean for each day of the week, then stabilise the +variance by applying the inverse hyperbolic sine transformation (Uniejewski et al., 2017). +Network estimation +We select the data collected from 01/01/2021 to 19/07/2021 (n = 200). The information criterion +in (18) returns a single factor (�q = 1), and �d = 1 is selected by CV. See Figure 9 for the heat maps +visualising the three networks N G, N C and N L described in Networks, which are produced by fnets. +N G +N C +N L +PJM +AECO +BGE +DPL +JCPL +METED +PECO +PEPCO +PPL +PENELEC +PSEG +BRANDONSH +BRUNSWICK +COOKSTOWN +DOVER +DPL NORTH +DPL SOUTH +EASTON +ECRRF +EPHRATA +FAIRLAWN +HOMERCIT +HOMERCIT UNIT1 +HOMERCIT UNIT2 +HOMERCIT UNIT3 +KITTATNY 230 +MANITOU +MONTVILLE +PENNTECH +PPL_ALLUGI +SENECA +SOUTHRIV 230 +SUNBURY LBRG +TRAYNOR +UGI +VINELAND +WELLSBORO +EASTERN HUB +WEST INT HUB +WESTERN HUB +ALBURTIS +BRANCHBURG +BRIGHTON +BURCHESHILL +CALVERTC +CHALKPT +CONASTONE +CONEMAUGH +DEANS +ELROY +PJM +AECO +BGE +DPL +JCPL +METED +PECO +PEPCO +PPL +PENELEC +PSEG +BRANDONSH +BRUNSWICK +COOKSTOWN +DOVER +DPL NORTH +DPL SOUTH +EASTON +ECRRF +EPHRATA +FAIRLAWN +HOMERCIT +HOMERCIT UNIT1 +HOMERCIT UNIT2 +HOMERCIT UNIT3 +KITTATNY 230 +MANITOU +MONTVILLE +PENNTECH +PPL_ALLUGI +SENECA +SOUTHRIV 230 +SUNBURY LBRG +TRAYNOR +UGI +VINELAND +WELLSBORO +EASTERN HUB +WEST INT HUB +WESTERN HUB +ALBURTIS +BRANCHBURG +BRIGHTON +BURCHESHILL +CALVERTC +CHALKPT +CONASTONE +CONEMAUGH +DEANS +ELROY +PJM +AECO +BGE +DPL +JCPL +METED +PECO +PEPCO +PPL +PENELEC +PSEG +BRANDONSH +BRUNSWICK +COOKSTOWN +DOVER +DPL NORTH +DPL SOUTH +EASTON +ECRRF +EPHRATA +FAIRLAWN +HOMERCIT +HOMERCIT UNIT1 +HOMERCIT UNIT2 +HOMERCIT UNIT3 +KITTATNY 230 +MANITOU +MONTVILLE +PENNTECH +PPL_ALLUGI +SENECA +SOUTHRIV 230 +SUNBURY LBRG +TRAYNOR +UGI +VINELAND +WELLSBORO +EASTERN HUB +WEST INT HUB +WESTERN HUB +ALBURTIS +BRANCHBURG +BRIGHTON +BURCHESHILL +CALVERTC +CHALKPT +CONASTONE +CONEMAUGH +DEANS +ELROY +PJM +AECO +BGE +DPL +JCPL +METED +PECO +PEPCO +PPL +PENELEC +PSEG +BRANDONSH +BRUNSWICK +COOKSTOWN +DOVER +DPL NORTH +DPL SOUTH +EASTON +ECRRF +EPHRATA +FAIRLAWN +HOMERCIT +HOMERCIT UNIT1 +HOMERCIT UNIT2 +HOMERCIT UNIT3 +KITTATNY 230 +MANITOU +MONTVILLE +PENNTECH +PPL_ALLUGI +SENECA +SOUTHRIV 230 +SUNBURY LBRG +TRAYNOR +UGI +VINELAND +WELLSBORO +EASTERN HUB +WEST INT HUB +WESTERN HUB +ALBURTIS +BRANCHBURG +BRIGHTON +BURCHESHILL +CALVERTC +CHALKPT +CONASTONE +CONEMAUGH +DEANS +ELROY +PJM +AECO +BGE +DPL +JCPL +METED +PECO +PEPCO +PPL +PENELEC +PSEG +BRANDONSH +BRUNSWICK +COOKSTOWN +DOVER +DPL NORTH +DPL SOUTH +EASTON +ECRRF +EPHRATA +FAIRLAWN +HOMERCIT +HOMERCIT UNIT1 +HOMERCIT UNIT2 +HOMERCIT UNIT3 +KITTATNY 230 +MANITOU +MONTVILLE +PENNTECH +PPL_ALLUGI +SENECA +SOUTHRIV 230 +SUNBURY LBRG +TRAYNOR +UGI +VINELAND +WELLSBORO +EASTERN HUB +WEST INT HUB +WESTERN HUB +ALBURTIS +BRANCHBURG +BRIGHTON +BURCHESHILL +CALVERTC +CHALKPT +CONASTONE +CONEMAUGH +DEANS +ELROY +PJM +AECO +BGE +DPL +JCPL +METED +PECO +PEPCO +PPL +PENELEC +PSEG +BRANDONSH +BRUNSWICK +COOKSTOWN +DOVER +DPL NORTH +DPL SOUTH +EASTON +ECRRF +EPHRATA +FAIRLAWN +HOMERCIT +HOMERCIT UNIT1 +HOMERCIT UNIT2 +HOMERCIT UNIT3 +KITTATNY 230 +MANITOU +MONTVILLE +PENNTECH +PPL_ALLUGI +SENECA +SOUTHRIV 230 +SUNBURY LBRG +TRAYNOR +UGI +VINELAND +WELLSBORO +EASTERN HUB +WEST INT HUB +WESTERN HUB +ALBURTIS +BRANCHBURG +BRIGHTON +BURCHESHILL +CALVERTC +CHALKPT +CONASTONE +CONEMAUGH +DEANS +ELROY +Figure 9: Heat maps of the three networks underlying the energy price data collected over the period +01/01/2021–19/07/2021. Left: N G obtained with the Lasso estimator (10) combined with the adaptive +threshold tada. Middle: N C obtained with the ACLIME estimator of ∆. Right: N L obtained by +combining the estimators of VAR parameters and ∆. The heat maps in the left column are in the scale +of [−0.6, 0.6] while those in the middle and right columns are in the scale of [−1, 1], with red hues +denoting large positive values and blue hues large negative values. In the axis labels, Zone-type nodes +are coloured in blue, Aggregate-types in green, Hub-types in red, and EHV-types in black. +The non-zero entries of the VAR parameter matrix estimates tend to take positive values, indicating +that high energy prices are persistent and spill over to other nodes. Considering the node types, Zone- +type nodes (blue) tend not to have causal links from other nodes in N G, but do have causal links +outwards to nodes of other types, which reflects the behaviour of the electrical transmission system. +Aggregate-types (green) have strong causal links from other nodes, while EHV-types (black) have links +inward from Zone-types but not from Aggregrate-types. This carries forward to N L where we observe +that EHV-type nodes do not have long-run dependence with nodes belonging to Aggregate-types. +Summary +We introduce the R package fnets which implements the FNETS methodology proposed by Barigozzi +et al. (2022) for network estimation and forecasting of high-dimensional time series exhibiting strong +correlations. It further implements data-driven methods for selecting tuning parameters, and provides +tools for high-dimensional time series factor modelling under the GDFM which are of independent +interest. 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[p8, 14] +17 + +Dom Owens +School of Mathematics, University of Bristol +Supported by EPSRC Centre for Doctoral Training (EP/S023569/1) +dom.owens@bristol.ac.uk +Haeran Cho +School of Mathematics, University of Bristol +Supported by the Leverhulme Trust (RPG-2019-390) +haeran.cho@bristol.ac.uk +Matteo Barigozzi +Department of Economics, Università di Bologna +Supported by MIUR (PRIN 2017, Grant 2017TA7TYC) +matteo.barigozzi@unibo.it +18 + +Appendix A: Information criteria for factor number selection +Here we list information criteria for factor number estimation which are implemented in fnets and +accessible by the functions fnets, fnets.factor.model and factor.number by setting the argument +ic.op at an integer belonging to {1, . . . , 6}. When fm.restricted = FALSE, we have +IC1: +� +1 +p ∑ +p +j=b+1 +1 +2m+1 ∑m +k=−m �µx,j(ωk) +� ++ b · c · (m−2 + √ +m/n + p−1) · log(min(p, m2, √ +n/m)), +IC2: +� +1 +p ∑ +p +j=b+1 +1 +2m+1 ∑m +k=−m �µx,j(ωk) +� ++ b · c · (min(p, m2, √ +n/m))−1/2, +IC3: +� +1 +p ∑ +p +j=b+1 +1 +2m+1 ∑m +k=−m �µx,j(ωk) +� ++ b · c · (min(p, m2, √ +n/m))−1 · log(min(p, m2, √ +n/m)), +IC4: log +� +1 +p ∑ +p +j=b+1 +1 +2m+1 ∑m +k=−m �µx,j(ωk) +� ++ b · c · (m−2 + √ +m/n + p−1) · log(min(p, m2, √ +n/m)), +IC5: log +� +1 +p ∑ +p +j=b+1 +1 +2m+1 ∑m +k=−m �µx,j(ωk) +� ++ b · c · (min(p, m2, √ +n/m))−1/2, +IC6: log +� +1 +p ∑ +p +j=b+1 +1 +2m+1 ∑m +k=−m �µx,j(ωk) +� ++ b · c · (min(p, m2, √ +n/m))−1 · log(min(p, m2, √ +n/m)) . +When fm.restricted = TRUE, we use one of +IC1: +� +1 +p ∑ +p +j=b+1 �µx,j +� ++ b · c · (n + p)/(np) · log(np/(n + p)), +IC2: +� +1 +p ∑ +p +j=b+1 �µx,j +� ++ b · c · (n + p)/(np) · log(np/(n + p)), +IC3: +� +1 +p ∑ +p +j=b+1 �µx,j +� ++ b · c · log(min(n, p))/(min(n, p)), +IC4: log +� +1 +p ∑ +p +j=b+1 �µx,j +� ++ b · c · (n + p)/(np) · log(np/(n + p)), +IC5: log +� +1 +p ∑ +p +j=b+1 �µx,j +� ++ b · c · (n + p)/(np) · log(np/(n + p)), +IC6: log +� +1 +p ∑ +p +j=b+1 �µx,j +� ++ b · c · log(min(n, p))/(min(n, p)). +Whether fm.restricted = FALSE or not, the default choice is ic.op = 5. +Appendix B: ACLIME estimator +We provide a detailed description of the adaptive extension of the CLIME estimator of ∆ in (13), +extending the methodology proposed in Cai et al. (2016) for precision matrix estimation in the +independent setting. Let �Γ∗ = �Γ + n−1I and η1 = 2 +� +log(p)/n . +Step 1: Let ˇ∆(1) = [ ˇδ(1) +ii′ ] be the solution to +ˇ∆(1) +·i′ = arg minm∈Rp|m|1 +subject to +(21) +���(�Γ∗m − ei′)i +��� ≤ η1(�γii ∨ �γi′i′)mi′ ∀ 1 ≤ i ≤ p and mi′ > 0, +for i′ = 1, . . . , p. Then we obtain truncated estimates +�δ(1) +ii += ˇδ(1) +ii +· I{|�γii|≤√ +n/ log(p)} + +� +log(p) +n +· I{|�γii|>√ +n/ log(p)}. +Step 2: We obtain +ˇ∆(2) +·i′ = arg minm∈Rp|m|1 +subject to +���(�Γ∗m − ei′)i +��� ≤ η2 +� +�γii�δ(1) +i′i′ +∀ 1 ≤ i ≤ p, +where η2 > 0 is a tuning parameter. Since ˇ∆(2) is not guaranteed to be symmetric, the final +estimator is obtained after a symmetrisation step: +�∆ada = [�δii′, 1 ≤ i, i′ ≤ p] with �δ(2) +ii′ = ˇδ(2) +ii′ · I{| ˇδ(2) +ii′ |≤| ˇδ(2) +i′i |} + ˇδ(2) +i′i · I{| ˇδ(2) +i′i |<| ˇδ(2) +ii′ |}. +(22) +19 + +The constraints in (21) incorporate the parameter in the right-hand side. To use linear programming +software to solve this, we formulate the constraints for each 1 ≤ i′ ≤ p as +∀1 ≤ i ≤ p, +((�Γ∗ − Qi′)m − ei′)i ≤ 0, +∀1 ≤ i ≤ p, +−((�Γ∗ + Qi′)m − ei′)i ≤ 0, +mi′ > 0. +where Qi′ has entries qii′ = η1(�γii ∨ �γi′i′) in column i′ and 0 elsewhere. +Appendix C: Additional simulation results +Threshold selection +Tables 3 and 4 report the errors in estimating A1 and Ω when the threshold t = tada or t = 0 is applied +to the estimator of A1 obtained by either the Lasso (10) or the DS (11) estimators. With a matrix γ as +an estimand we measure the estimation error of its estimator �γ using the following (scaled) matrix +norms: +LF = ∥�γ − γ∥F +∥γ∥F +and +L2 = ∥�γ − γ∥ +∥γ∥ +. +Table 3: Errors in estimating A1 with t ∈ {0, tada} in combination with the Lasso (10) and the DS (11) +estimators, measured by LF and L2, averaged over 100 realisations (with standard errors reported in +brackets). We also report the average TPR when FPR = 0.05 and the corresponding standard error. +See Results: Threshold selection in the main text for further information. +t = 0 +t = tada +�βlas +�βDS +�βlas +�βDS +Model +n +p +TPR +LF +L2 +TPR +LF +L2 +TPR +LF +L2 +TPR +LF +L2 +(E1) +200 +50 +0.9681 +0.6234 +0.7204 +0.8991 +0.4299 +0.3747 +0.9413 +0.6226 +0.7204 +0.6932 +0.4487 +0.3960 +(0.050) +(0.081) +(0.118) +(0.096) +(0.280) +(0.225) +(0.112) +(0.088) +(0.121) +(0.216) +(0.256) +(0.206) +200 +100 +0.9398 +0.6696 +0.8113 +0.8810 +0.5772 +0.4362 +0.8832 +0.6710 +0.8132 +0.6491 +0.6025 +0.4642 +(0.091) +(0.096) +(0.096) +(0.094) +(0.449) +(0.271) +(0.182) +(0.108) +(0.100) +(0.246) +(0.418) +(0.250) +500 +100 +0.9990 +0.4648 +0.6682 +0.9304 +0.2740 +0.2604 +0.9971 +0.4608 +0.6645 +0.7237 +0.2806 +0.2699 +(0.003) +(0.054) +(0.094) +(0.065) +(0.158) +(0.138) +(0.010) +(0.056) +(0.095) +(0.199) +(0.133) +(0.111) +500 +200 +0.9986 +0.5068 +0.7729 +0.9167 +0.3680 +0.3882 +0.9964 +0.5023 +0.7637 +0.7095 +0.3889 +0.4014 +(0.003) +(0.058) +(0.081) +(0.076) +(0.196) +(0.134) +(0.006) +(0.061) +(0.082) +(0.256) +(0.187) +(0.126) +(E2) +200 +50 +0.9595 +0.6375 +0.7075 +0.8828 +0.4673 +0.4280 +0.9442 +0.6356 +0.7079 +0.6720 +0.4835 +0.4433 +(0.053) +(0.077) +(0.094) +(0.107) +(0.324) +(0.255) +(0.064) +(0.079) +(0.096) +(0.212) +(0.303) +(0.241) +200 +100 +0.9624 +0.6200 +0.6909 +0.8093 +0.4519 +0.4090 +0.9435 +0.6175 +0.6913 +0.5903 +0.4765 +0.4324 +(0.072) +(0.079) +(0.089) +(0.100) +(0.385) +(0.251) +(0.093) +(0.082) +(0.090) +(0.182) +(0.371) +(0.243) +500 +100 +0.9970 +0.4657 +0.5533 +0.9304 +0.3434 +0.3621 +0.9958 +0.4638 +0.5525 +0.8384 +0.3370 +0.3634 +(0.006) +(0.056) +(0.076) +(0.089) +(0.158) +(0.153) +(0.008) +(0.058) +(0.077) +(0.182) +(0.140) +(0.144) +500 +200 +0.9981 +0.4702 +0.5658 +0.9205 +0.3684 +0.3740 +0.9945 +0.4686 +0.5665 +0.8154 +0.3663 +0.3803 +(0.003) +(0.065) +(0.091) +(0.088) +(0.182) +(0.162) +(0.014) +(0.068) +(0.093) +(0.205) +(0.159) +(0.145) +Table 4: Errors in estimating Ω with t ∈ {0, tada} applied to the estimator of A1 in combination with +the Lasso (10) and the DS (11) estimators, measured by LF and L2, averaged over 100 realisations +(with standard errors reported in brackets). We also report the average TPR when FPR = 0.05 and the +corresponding standard error. See Results: Threshold selection in the main text for further information. +t = 0 +t = tada +�βlas +�βDS +�βlas +�βDS +Model +n +p +TPR +LF +L2 +TPR +LF +L2 +TPR +LF +L2 +TPR +LF +L2 +(E1) +200 +50 +0.8714 +0.4143 +0.5553 +0.8622 +0.4217 +0.5691 +0.8685 +0.4145 +0.5559 +0.8640 +0.4217 +0.5695 +(0.108) +(0.048) +(0.066) +(0.119) +(0.054) +(0.070) +(0.118) +(0.049) +(0.067) +(0.121) +(0.055) +(0.070) +200 +100 +0.8827 +0.4320 +0.5890 +0.8961 +0.4379 +0.5949 +0.8684 +0.4326 +0.5892 +0.8867 +0.4386 +0.5960 +(0.084) +(0.050) +(0.072) +(0.080) +(0.046) +(0.065) +(0.139) +(0.052) +(0.074) +(0.120) +(0.048) +(0.066) +500 +100 +0.9909 +0.3311 +0.4916 +0.9886 +0.3391 +0.4989 +0.9928 +0.3303 +0.4901 +0.9901 +0.3380 +0.4975 +(0.016) +(0.031) +(0.069) +(0.021) +(0.036) +(0.065) +(0.015) +(0.032) +(0.069) +(0.018) +(0.037) +(0.066) +500 +200 +0.9942 +0.3520 +0.5287 +0.9916 +0.3511 +0.5400 +0.9954 +0.3512 +0.5273 +0.9672 +0.3528 +0.5399 +(0.009) +(0.038) +(0.054) +(0.018) +(0.045) +(0.065) +(0.008) +(0.039) +(0.055) +(0.129) +(0.055) +(0.072) +(E2) +200 +50 +0.4074 +0.7831 +0.8353 +0.4027 +0.7942 +0.8335 +0.4063 +0.7832 +0.8353 +0.4045 +0.7943 +0.8336 +(0.073) +(0.089) +(0.072) +(0.087) +(0.079) +(0.034) +(0.072) +(0.089) +(0.072) +(0.089) +(0.079) +(0.034) +200 +100 +0.4178 +0.8406 +0.8690 +0.3541 +0.9119 +0.8879 +0.4486 +0.8407 +0.8690 +0.4038 +0.9120 +0.8880 +(0.091) +(0.108) +(0.036) +(0.107) +(0.126) +(0.045) +(0.091) +(0.108) +(0.036) +(0.123) +(0.126) +(0.045) +500 +100 +0.5405 +0.8267 +0.8118 +0.5632 +0.7910 +0.7953 +0.5406 +0.8267 +0.8117 +0.5628 +0.7910 +0.7951 +(0.111) +(0.125) +(0.047) +(0.122) +(0.166) +(0.062) +(0.111) +(0.125) +(0.047) +(0.123) +(0.166) +(0.062) +500 +200 +0.5951 +0.8713 +0.8519 +0.6487 +0.8184 +0.8259 +0.6918 +0.8713 +0.8519 +0.7101 +0.8184 +0.8258 +(0.175) +(0.165) +(0.088) +(0.159) +(0.182) +(0.090) +(0.148) +(0.165) +(0.088) +(0.122) +(0.182) +(0.090) +20 + +VAR order selection +Table 5 reports the results of VAR order estimation over 100 realisations. +Table 5: Distribution of �d − d over 100 realisations when the VAR order is selected by the CV and +eBIC methods in combination with the Lasso (10) and the DS (11) estimators, see Results: VAR order +selection in the main text for further information. +CV +eBIC +�βlas +�βDS +�βlas +�βDS +d +n +p +0 +1 +2 +3 +0 +1 +2 +3 +0 +1 +2 +3 +0 +1 +2 +3 +1 +200 +10 +81 +10 +4 +5 +91 +6 +2 +1 +64 +17 +11 +8 +64 +12 +16 +8 +200 +20 +94 +6 +0 +0 +94 +5 +1 +0 +68 +10 +9 +13 +75 +10 +7 +8 +500 +10 +94 +5 +1 +0 +86 +7 +4 +3 +65 +17 +11 +7 +65 +18 +9 +8 +500 +20 +97 +2 +0 +1 +98 +1 +1 +0 +70 +15 +8 +7 +64 +14 +10 +12 +-2 +-1 +0 +1 +-2 +-1 +0 +1 +-2 +-1 +0 +1 +-2 +-1 +0 +1 +3 +200 +10 +0 +0 +77 +23 +0 +0 +78 +22 +27 +3 +49 +21 +30 +6 +49 +15 +200 +20 +0 +0 +97 +3 +0 +0 +85 +15 +32 +1 +48 +19 +31 +2 +58 +9 +500 +10 +0 +0 +76 +24 +0 +0 +83 +17 +30 +4 +43 +23 +29 +2 +40 +29 +500 +20 +0 +0 +74 +26 +0 +0 +97 +3 +29 +3 +45 +23 +25 +4 +53 +18 +CLIME vs. ACLIME estimators +We compare the performance of the adaptive and non-adaptive estimators for the VAR innovation +precision matrix ∆ and its impact on the estimation of Ω, the inverse of the long-run covariance matrix +of the data (see Step 3). We generate χt as in (C1), fix d = 1 and treat it as known and consider +(n, p) ∈ {(200, 50), (200, 100), (500, 100), (500, 200)}. +In Tables 6 and 7, we report the errors of ∆ and Ω. We consider both the Lasso (10) and DS (11) +estimators of VAR parameters, and CLIME and ACLIME estimators for ∆, which lead to four different +estimators for ∆ and Ω, respectively. Overall, we observe that with increasing n, the performance of +all estimators improve according to all metrics regardless of the scenarios (E1) or (E2), while increasing +p has an adverse effect. The two methods perform similarly in setting (E1) when ∆ = I. There is +marginal improvement for adopting the ACLIME estimator noticeable under (E2), particularly in +TPR. Figures 10 and 11 shows the ROC curves for the support recovery of ∆ and Ω when the Lasso +estimator is used. +Table 6: Errors in estimating ∆ using CLIME and ACLIME estimators, measured by LF and L2, +averaged over 100 realisations (with standard errors reported in brackets). We also report the average +TPR when FPR = 0.05 and the corresponding standard errors. +CLIME +ACLIME +�βlas +�βDS +�βlas +�βDS +Model +n +p +TPR +LF +L2 +TPR +LF +L2 +TPR +LF +L2 +TPR +LF +L2 +(E1) +200 +50 +1.000 +0.215 +0.489 +1.000 +0.220 +0.497 +1.000 +0.207 +0.472 +1.000 +0.209 +0.469 +(0.000) +(0.047) +(0.223) +(0.000) +(0.047) +(0.182) +(0.002) +(0.043) +(0.173) +(0.000) +(0.041) +(0.116) +200 +100 +1.000 +0.235 +0.513 +1.000 +0.241 +0.521 +1.000 +0.223 +0.507 +1.000 +0.228 +0.518 +(0.000) +(0.036) +(0.089) +(0.000) +(0.036) +(0.107) +(0.000) +(0.033) +(0.084) +(0.000) +(0.034) +(0.099) +500 +100 +1.000 +0.181 +0.458 +1.000 +0.183 +0.466 +1.000 +0.176 +0.452 +1.000 +0.178 +0.458 +(0.000) +(0.022) +(0.062) +(0.000) +(0.029) +(0.087) +(0.000) +(0.022) +(0.052) +(0.000) +(0.028) +(0.069) +500 +200 +1.000 +0.198 +0.510 +1.000 +0.193 +0.492 +1.000 +0.187 +0.505 +1.000 +0.182 +0.489 +(0.000) +(0.027) +(0.066) +(0.000) +(0.035) +(0.065) +(0.000) +(0.026) +(0.056) +(0.000) +(0.033) +(0.057) +(E2) +200 +50 +0.659 +0.422 +0.816 +0.662 +0.391 +0.608 +0.682 +0.397 +0.706 +0.687 +0.380 +0.600 +(0.058) +(0.101) +(0.654) +(0.057) +(0.031) +(0.144) +(0.055) +(0.056) +(0.351) +(0.054) +(0.030) +(0.176) +200 +100 +0.639 +0.417 +0.695 +0.637 +0.420 +0.720 +0.669 +0.404 +0.663 +0.668 +0.405 +0.684 +(0.044) +(0.039) +(0.205) +(0.042) +(0.043) +(0.249) +(0.041) +(0.037) +(0.162) +(0.039) +(0.037) +(0.193) +500 +100 +0.730 +0.372 +0.764 +0.726 +0.499 +1.708 +0.735 +0.358 +0.650 +0.734 +0.361 +0.718 +(0.035) +(0.097) +(0.828) +(0.039) +(1.101) +(7.586) +(0.032) +(0.038) +(0.322) +(0.031) +(0.056) +(0.517) +500 +200 +0.729 +0.370 +0.711 +0.728 +0.362 +0.736 +0.737 +0.363 +0.647 +0.737 +0.354 +0.673 +(0.028) +(0.035) +(0.355) +(0.028) +(0.035) +(0.384) +(0.023) +(0.026) +(0.239) +(0.024) +(0.028) +(0.279) +21 + +Table 7: Errors in estimating Ω using CLIME and ACLIME estimators of ∆, measured by LF and L2, +averaged over 100 realisations (with standard errors reported in brackets). We also report the average +TPR when FPR = 0.05 and the corresponding standard errors. +CLIME +ACLIME +�βlas +�βDS +�βlas +�βDS +Model +n +p +TPR +LF +L2 +TPR +LF +L2 +TPR +LF +L2 +TPR +LF +L2 +(E1) +200 +50 +0.871 +0.415 +0.557 +0.862 +0.422 +0.571 +0.867 +0.411 +0.558 +0.856 +0.417 +0.570 +(0.108) +(0.050) +(0.070) +(0.119) +(0.055) +(0.080) +(0.106) +(0.051) +(0.088) +(0.114) +(0.053) +(0.083) +200 +100 +0.883 +0.432 +0.589 +0.896 +0.438 +0.595 +0.868 +0.423 +0.583 +0.883 +0.429 +0.587 +(0.084) +(0.050) +(0.072) +(0.080) +(0.046) +(0.065) +(0.088) +(0.048) +(0.077) +(0.085) +(0.045) +(0.061) +500 +100 +0.991 +0.331 +0.492 +0.989 +0.339 +0.499 +0.991 +0.328 +0.490 +0.989 +0.337 +0.498 +(0.016) +(0.031) +(0.069) +(0.021) +(0.036) +(0.065) +(0.015) +(0.033) +(0.070) +(0.019) +(0.036) +(0.067) +500 +200 +0.994 +0.352 +0.529 +0.992 +0.351 +0.540 +0.994 +0.344 +0.525 +0.990 +0.342 +0.537 +(0.009) +(0.038) +(0.054) +(0.018) +(0.045) +(0.065) +(0.009) +(0.038) +(0.056) +(0.014) +(0.044) +(0.068) +(E2) +200 +50 +0.509 +0.532 +0.724 +0.510 +0.514 +0.664 +0.504 +0.518 +0.679 +0.507 +0.506 +0.658 +(0.078) +(0.071) +(0.243) +(0.068) +(0.043) +(0.137) +(0.071) +(0.055) +(0.162) +(0.063) +(0.043) +(0.141) +200 +100 +0.511 +0.541 +0.683 +0.513 +0.542 +0.695 +0.509 +0.531 +0.674 +0.504 +0.531 +0.679 +(0.059) +(0.047) +(0.082) +(0.065) +(0.051) +(0.093) +(0.062) +(0.045) +(0.084) +(0.061) +(0.046) +(0.084) +500 +100 +0.640 +0.450 +0.655 +0.624 +0.544 +1.099 +0.642 +0.441 +0.597 +0.637 +0.440 +0.617 +(0.066) +(0.072) +(0.402) +(0.079) +(0.866) +(3.714) +(0.059) +(0.036) +(0.118) +(0.060) +(0.047) +(0.204) +500 +200 +0.670 +0.461 +0.630 +0.658 +0.450 +0.630 +0.677 +0.456 +0.612 +0.661 +0.445 +0.605 +(0.045) +(0.041) +(0.116) +(0.043) +(0.040) +(0.117) +(0.041) +(0.036) +(0.075) +(0.037) +(0.037) +(0.082) +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +model +E1 +E2 +method +ACLIME +CLIME +n = 200, p = 50 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 200, p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 500, p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 500, p = 200 +Figure 10: ROC curves of TPR against FPR for �∆ with CLIME and ACLIME estimators in recovering +the support of ∆, averaged over 100 realisations. Vertical lines indicate FPR = 0.05. +22 + +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +model +E1 +E2 +method +ACLIME +CLIME +n = 200, p = 50 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 200, p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 500, p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +n = 500, p = 200 +Figure 11: ROC curves of TPR against FPR for �Ω with CLIME and ACLIME estimators in recovering +the support of Ω, averaged over 100 realisations. Vertical lines indicate FPR = 0.05. +23 + +Appendix D: Dataset information +Table 8 defines the four node types in the panel. Table 9 describes the dataset analysed in Data example. +Table 8: Node type definitions for energy price data. +Name +Definition +Zone +A transmission owner’s area within the PJM Region. +Aggregate +A group of more than one individual bus into a pricing node (pnode) +that is considered as a whole in the Energy Market and other various systems +and Markets within PJM. +Hub +A group of more than one individual bus into a regional pricing node (pnode) +developed to produce a stable price signal in the Energy Market +and other various systems and Markets within PJM. +Extra High Voltage (EHV) +Nodes at 345kV and above on the PJM system. +24 + +Table 9: Names, IDs and Types for the 50 power nodes in the energy price dataset. +Name +Node ID +Node Type +PJM +1 +ZONE +AECO +51291 +ZONE +BGE +51292 +ZONE +DPL +51293 +ZONE +JCPL +51295 +ZONE +METED +51296 +ZONE +PECO +51297 +ZONE +PEPCO +51298 +ZONE +PPL +51299 +ZONE +PENELEC +51300 +ZONE +PSEG +51301 +ZONE +BRANDONSH +51205 +AGGREGATE +BRUNSWICK +51206 +AGGREGATE +COOKSTOWN +51211 +AGGREGATE +DOVER +51214 +AGGREGATE +DPL NORTH +51215 +AGGREGATE +DPL SOUTH +51216 +AGGREGATE +EASTON +51218 +AGGREGATE +ECRRF +51219 +AGGREGATE +EPHRATA +51220 +AGGREGATE +FAIRLAWN +51221 +AGGREGATE +HOMERCIT +51229 +AGGREGATE +HOMERCIT UNIT1 +51230 +AGGREGATE +HOMERCIT UNIT2 +51231 +AGGREGATE +HOMERCIT UNIT3 +51232 +AGGREGATE +KITTATNY 230 +51238 +AGGREGATE +MANITOU +51239 +AGGREGATE +MONTVILLE +51241 +AGGREGATE +PENNTECH +51246 +AGGREGATE +PPL_ALLUGI +51252 +AGGREGATE +SENECA +51255 +AGGREGATE +SOUTHRIV 230 +51261 +AGGREGATE +SUNBURY LBRG +51270 +AGGREGATE +TRAYNOR +51277 +AGGREGATE +UGI +51279 +AGGREGATE +VINELAND +51280 +AGGREGATE +WELLSBORO +51285 +AGGREGATE +EASTERN HUB +51217 +HUB +WEST INT HUB +51287 +HUB +WESTERN HUB +51288 +HUB +ALBURTIS +52443 +EHV +BRANCHBURG +52444 +EHV +BRIGHTON +52445 +EHV +BURCHESHILL +52446 +EHV +CALVERTC +52447 +EHV +CHALKPT +52448 +EHV +CONASTONE +52449 +EHV +CONEMAUGH +52450 +EHV +DEANS +52451 +EHV +ELROY +52452 +EHV +25 + diff --git a/YtFJT4oBgHgl3EQf6y0f/content/tmp_files/load_file.txt b/YtFJT4oBgHgl3EQf6y0f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1978729d7409af20a7af95ec525b49dbc30cf52c --- /dev/null +++ b/YtFJT4oBgHgl3EQf6y0f/content/tmp_files/load_file.txt @@ -0,0 +1,2608 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf,len=2607 +page_content='fnets: An R Package for Network Estimation and Forecasting via Factor-Adjusted VAR Modelling by Dom Owens, Haeran Cho and Matteo Barigozzi Abstract The package fnets for the R language implements the suite of methodologies proposed by Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022) for the network estimation and forecasting of high-dimensional time series under a factor-adjusted vector autoregressive model, which permits strong spatial and temporal correlations in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Additionally, we provide tools for visualising the networks underlying the time series data after adjusting for the presence of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The package also offers data-driven methods for selecting tuning parameters including the number of factors, vector autoregressive order and thresholds for estimating the edge sets of the networks of interest in time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We demonstrate various features of fnets on simulated datasets as well as real data on electricity prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Introduction Vector autoregressive (VAR) models are popularly adopted for modelling time series datasets collected in many disciplines including economics (Koop, 2013), finance (Barigozzi and Brownlees, 2019), neuroscience (Kirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2015) and systems biology (Shojaie and Michailidis, 2010), to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' By fitting a VAR model to the data, we can infer dynamic interdependence between the variables and forecast future values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In particular, estimating the non-zero elements of the VAR parameter matrices recovers directed edges between the components of vector time series in a Granger causality network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Besides, by estimating the precision matrix (inverse of the covariance matrix) of the VAR innovations, we can define a network representing their contemporaneous dependencies by means of partial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Finally, the inverse of the long-run covariance matrix of the data simultaneously captures lead-lag and contemporaneous co-movements of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For the network interpretation of VAR modelling, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Dahlhaus (2000), Eichler (2007), Billio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2012) and Barigozzi and Brownlees (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Fitting VAR models to the data quickly becomes a high-dimensional problem as the number of parameters grows quadratically with the dimensionality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' There exists a mature literature on ℓ1-regularisation methods for estimating VAR models in high dimensions under suitable sparsity assumptions on the VAR parameters (Basu and Michailidis, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Kock and Callot, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Medeiros and Mendes, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Liu and Zhang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Consistency of such methods is derived under the assumption that the spectral density matrix of the data has bounded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' However, in many applications, the datasets exhibit strong serial and cross-sectional correlations which leads to the violation of this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' As a motivating example, we introduce a dataset of node-specific prices in the PJM (Pennsylvania, New Jersey and Maryland) power pool area in the United States, see Energy price data for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Figure 1 demonstrates that the leading eigenvalue of the long-run covariance matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' spectral density matrix at frequency 0) increases linearly as the dimension of the data increases, which implies the presence of latent common factors in the panel data (Forni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Additionally, the left panel of Figure 2 shows the inadequacy of fitting a VAR model to such data under the sparsity assumption via ℓ1-regularisation methods, unless the presence of strong correlations is accounted for by a factor-adjustment step as in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022) propose the FNETS methodology for factor-adjusted VAR modelling of high-dimensional, second-order stationary time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Under their proposed model, the data is decomposed into two latent components such that the factor-driven component accounts for pervasive leading, lagging or contemporaneous co-movements of the variables, while the remaining idiosyncratic dynamic dependence between the variables is modelled by a sparse VAR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, FNETS provides tools for inferring the networks underlying the latent VAR process and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In this paper, we present an R package named fnets which implements the FNETS methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' It provides a range of user-friendly tools for estimating and visualising the networks representing the interconnectedness of time series variables, and for producing forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In addition, fnets thoroughly addresses the problem of selecting tuning parameters ranging from the number of factors and the VAR order, to regularisation and thresholding parameters adopted for producing sparse and interpretable networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' As such, a simple call of the main routine of fnets requires the input data only, and it outputs an object of S3 class fnets which is supported by a plot method for network visualisation and a predict method for time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' There exist several R packages for fitting VAR models and their extensions to high-dimensional 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='11675v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='CO] 27 Jan 2023 1 3 5 7 9 11 14 17 20 23 26 29 32 35 38 41 44 47 2 4 6 8 10 12 14 1 3 5 7 9 11 14 17 20 23 26 29 32 35 38 41 44 47 2 4 6 8 10 12 14 First Second First Second Figure 1: Box plots of the two largest eigenvalues (y-axis) of the long-run covariance matrix estimated from the energy price data collected between 01/01/2021 and 19/07/2021 (n = 200), see Data example for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Cross-sections of the data are randomly sampled 100 times for each given dimension p ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , 50} (x-axis) to produce the box plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Figure 2: Granger causal networks defined in (5) obtained from fitting a VAR(1) model to the energy price data analysed in Figure 1, without (left) and with (right) the factor adjustment step outlined in FNETS: Network estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Edge weights (proportional to the size of coefficient estimates) are visualised by the width of each edge, and the nodes are coloured according to their groupings, see Data example for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' time series by means of Lasso-type estimation techniques, see lsvar (Bai, 2021), sparsevar (Vazzoler, 2021), nets (Brownlees, 2020), mgm (Haslbeck and Waldorp, 2020), graphicalVAR (Epskamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2018), bigVAR (Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2017), and bigtime (Wilms et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The package fnets is clearly distinguished from, and complements, the above list by handling strong cross-sectional and serial correlations in the data via factor-adjustment step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In addition, the FNETS methodology operates under the most general approach to high-dimensional time series factor modelling termed the Generalised Dynamic Factor (GDFM), first proposed in Forni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2000) and further investigated in Forni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Accordingly, fnets is the first R package to provide tools for high-dimensional panel data analysis under the GDFM, such as fast computation of spectral density and autocovariance matrices via the Fast Fourier Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' FNETS methodology In this section, we introduce the factor-adjusted VAR model and describe the FNETS methodology proposed in Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022) for network estimation and forecasting of high-dimensional time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We limit ourselves to describing the key steps of FNETS and refer to the above paper for its comprehensive treatment, both methodologically and theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 2 Factor-adjusted VAR model A zero-mean, p-variate process ξt follows a VAR(d) model if it satisfies ξt = d ∑ ℓ=1 Aℓξt−ℓ + Γ1/2εt, (1) where Aℓ ∈ Rp×p, 1 ≤ ℓ ≤ d, determine how future values of the series depend on their past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For the p-variate random vector εt = (ε1t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , εpt)⊤, we assume that εit are independently and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=') for all i and t with IE(εit) = 0 and Var(εit) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, the positive definite matrix Γ ∈ Rp×p is the covariance matrix of the innovations Γ1/2εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In the literature on factor modelling of high-dimensional time series, the factor-driven component exhibits strong cross-sectional and/or serial correlations by ‘loading’ finite-dimensional vectors of factors linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Among many time series factor models, the GDFM (Forni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2000) provides the most general approach where the p-variate factor-driven component χt admits the following representation χt = B(L)ut = ∞ ∑ ℓ=0 Bℓut−ℓ with ut = (u1t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , uqt)⊤ and Bℓ ∈ Rp×q, (2) for some fixed q, where L stands for the lag operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The q-variate random vector ut contains the common factors which are loaded across the variables and time by the filter B(L) = ∑∞ ℓ=0 BℓLℓ, and it is assumed that ujt are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' with IE(ujt) = 0 and Var(ujt) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The model (2) reduces to a static factor model (Bai, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Stock and Watson, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2013) when B(L) admits a decomposition B(L) = M(1)(L)M(2)(L) with M(k)(L) = ∑mk ℓ=0 M(k) ℓ Lℓ for k = 1, 2, where M(1) ∈ Rp×q and M(2) ∈ Rq×q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, we can write χt = m1 ∑ ℓ=0 M(1) ℓ ft−ℓ = ΛFt where Ft = (f⊤ t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , f⊤ t−m1)⊤ and ft = m2 ∑ ℓ=0 M(2) ℓ ut−ℓ, (3) with r = q(m1 + 1) as the dimension of static factors Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Throughout, we refer to the models (2) and (3) as unrestricted and restricted to highlight that the latter imposes more restrictions on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022) propose a factor-adjusted VAR model under which we observe a zero-mean, second-order stationary process Xt = (X1t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , Xpt)⊤ for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , n, that permits a decomposition into the sum of the unobserved components ξt and χt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Xt = ξt + χt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (4) We assume that IE(εitujt′) = 0 for all i, j, t and t′ as is commonly assumed in the literature, such that IE(ξitχi′t′) = 0 for all 1 ≤ i, i′ ≤ p and t, t′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Networks Under (4), it is of interest to infer three types of networks representing the interconnectedness of Xt after factor adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Let V = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , p} denote the set of vertices representing the p cross- sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, the VAR parameter matrices, Aℓ = [Aℓ,ii′, 1 ≤ i, i′ ≤ p], encode the directed network N G = (V, EG) representing Granger causal linkages, where the set of edges are given by EG = �(i, i′) ∈ V × V : Aℓ,ii′ ̸= 0 for some 1 ≤ ℓ ≤ d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (5) Here, the presence of an edge (i, i′) ∈ EG indicates that ξi′,t−ℓ Granger causes ξit at some lag 1 ≤ ℓ ≤ d (Dahlhaus, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The second network contains undirected edges representing contemporaneous cross-sectional dependence in VAR innovations Γ1/2εt, denoted by N C = (V, EC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We have (i, i′) ∈ EC if and only if the partial correlation between the i-th and i′-th elements of Γ1/2εt is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Specifically, letting Γ−1 = ∆ = [δii′, 1 ≤ i, i′ ≤ p], the set of edges is given by EC = � (i, i′) ∈ V × V : i ̸= i′ and − δii′ √δii · δi′i′ ̸= 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (6) Finally, we can summarise the aforementioned lead-lag and contemporaneous relations between the variables in a single, undirected network N L = (V, EL) by means of the long-run partial cor- relations of ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Let Ω = [ωii′, 1 ≤ i, i′ ≤ p] denote the long-run partial covariance matrix of ξt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 3 the inverse of the zero-frequency spectral density of ξt which is given by Ω = 2πA⊤(1)∆A(1) with A(z) = I − ∑d ℓ=1 Aℓzℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, the edge set of N L is given by EL = � (i, i′) ∈ V × V : i ̸= i′ and − ωii′ √ωii · ωi′i′ ̸= 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (7) FNETS: Network estimation We describe the three-step methodology for estimating the networks N G, N C and N L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Throughout, we assume that the VAR order d is known, and discuss its selection in Tuning parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Step 1: Factor adjustment The autocovariance (ACV) matrices of ξt, denoted by Γξ(ℓ) = IE(ξt−ℓξ⊤ t ) for ℓ ≥ 0 and Γξ(ℓ) = (Γξ(−ℓ))⊤ for ℓ < 0, play a key role in network estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Since ξt is not directly observed, we propose to adjust for the presence of the factor-driven χt via dynamic PCA, for the estimation of Γξ(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Here, we assume that the number of factors, either q under the more general model in (2) or r under the restricted model in (3), are known and defer the discussion on their choice to Tuning parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' With the sample ACV matrices of Xt, �Γx(ℓ) = 1 n n ∑ t=ℓ+1 Xt−ℓX⊤ t when ℓ ≥ 0 and �Γx(ℓ) = (�Γx(−ℓ))⊤ when ℓ < 0, we estimate the spectral density of Xt by �Σx(ωk) = 1 2π m ∑ ℓ=−m K � ℓ m � �Γx(ℓ) exp(−ιℓωk), where K(·) denotes a kernel, m the kernel bandwidth (for its choice, see Tuning parameter selection) and ωk = 2πk/(2m + 1) the Fourier frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We adopt the Bartlett kernel as K(·) which ensures positive semi-definiteness of �Σx(ω) and also �Γξ(ℓ) estimating Γξ(ℓ) obtained as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Performing PCA on �Σx(ωk) at each ωk, we obtain the estimator of the spectral density matrix of χt as �Σχ(ωk) = ∑ q j=1 �µx,j(ωk)�ex,j(ωk)(�ex,j(ωk))∗, where �µx,j(ωk) denotes the j-th largest eigenvalue of �Σx(ωk), �ex,j(ωk) its associated eigenvector, and for any vector a ∈ Cn, we denote its transposed complex conjugate by a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then taking the inverse Fourier transform of �Σχ(ωk), −m ≤ k ≤ m, leads to an estimator of Γχ(ℓ), the ACV matrix of χt, as �Γχ(ℓ) = 2π 2m + 1 m ∑ k=−m �Σχ(ωk) exp(ιℓωk) for − m ≤ ℓ ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Finally, we estimate the ACV of ξt by �Γξ(ℓ) = �Γx(ℓ) − �Γχ(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (8) When we assume the restricted factor model in (3), the factor-adjustment step is simplified as it suffices to perform PCA in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Denoting the eigenvector of the sample covariance matrix �Γx(0) associated with its j-th largest eigenvalue by �ex,j, we obtain �Γχ(ℓ) = �Ex�E⊤x �Γx(ℓ)�Ex�E⊤x , where �Ex = [�ex,j, 1 ≤ j ≤ r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, we obtain �Γξ(ℓ) as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Step 2: Estimation of N G Recall from (5) that N G representing Granger causal linkages, has its edge set determined by the VAR transition matrices Aℓ, 1 ≤ ℓ ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' By the Yule-Walker equation, we have β0 = [A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , Ad]⊤ = G(d)−1g(d), where G(d) = � ���� Γξ(0) Γξ(−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Γξ(−d + 1) Γξ(1) Γξ(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Γξ(−d + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Γξ(d − 1) Γξ(d − 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Γξ(0) � ���� and g(d) = � ���� Γξ(1) Γξ(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Γξ(d) � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (9) 4 We propose to estimate β0 as a regularised Yule-Walker estimator based on �G(d) and �g(d), each of which is obtained by replacing Γξ(ℓ) with �Γξ(ℓ) (see (8)) in the definition of G(d) and g(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For any matrix A = [aij] ∈ Rm×n, let |A|1 = ∑m i=1 ∑n j=1 |aij|, |A|∞ = max1≤i≤m max1≤j≤n |aij| and tr(A) = ∑m i=1 aii when m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We consider two estimators of β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Firstly, we adopt a Lasso-type estimator which solves an ℓ1-regularised M-estimation problem �βlas = arg min β∈Rpd×p tr � β⊤ �G(d)β − 2β⊤�g(d) � + λ|β|1 (10) with a tuning parameter λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In the implementation, we solve (10) via the fast iterative shrinkage- thresholding algorithm (FISTA, Beck and Teboulle (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Alternatively, we adopt a constrained ℓ1-minimisation approach closely related to the Dantzig selector (Candes and Tao, 2007, DS): �βDS = arg min β∈Rpd×p |β|1 subject to ��� �G(d)β − �g(d) ��� ∞ ≤ λ (11) for some tuning parameter λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We divide (11) into p sub-problems and obtain each column of �βDS via the simplex algorithm (using the function lp in lpSolve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022) establish the consistency of both �βlas and �βDS but, as is typically the case for ℓ1-regularisation methods, they do not achieve exact recovery of the support of β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Hence we propose to estimate the edge set of N G by thresholding the elements of �β with some threshold t > 0, where either �β = �βlas or �β = �βDS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' �β(t) = ��βij · I{|�βij|>t}, 1 ≤ i ≤ pd, 1 ≤ j ≤ p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (12) We discuss cross validation and information criterion methods for selecting λ, and a data-driven choice of t, in Tuning parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Step 3: Estimation of N C and N L From the definitions of N C and N L given in (6) and (7), their edge sets are obtained by estimating ∆ = Γ−1 and Ω = 2πA⊤(1)∆A(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Given �β = [�A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , �Ad]⊤, some estimator of the VAR parameter matrices obtained as in either (10) or (11), a natural estimator of Γ arises from the Yule-Walker equation Γ = Γξ(0) − ∑d ℓ=1 AℓΓξ(ℓ) = Γξ(0) − (β0)⊤g, as �Γ = �Γξ(0) − �β⊤�g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In high dimensions, it is not feasible or recommended to directly invert �Γ to estimate ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Therefore, we adopt a constrained ℓ1-minimisation method motivated by the CLIME methodology of Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Specifically, the CLIME estimator of ∆ is obtained by first solving ˇ∆ = arg minM∈Rp×p|M|1 subject to ����ΓM − I ��� ∞ ≤ η, (13) and applying a symmetrisation step to ˇ∆ = [ ˇδii′, 1 ≤ i, j ≤ p] as �∆ = [�δii′, 1 ≤ i, i′ ≤ p] with �δii′ = ˇδii′ · I{| ˇδii′ |≤| ˇδi′i|} + ˇδi′i · I{| ˇδi′i|<| ˇδii′ |}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (14) for some tuning parameter η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2016) propose ACLIME, which improves the CLIME estimator by selecting the parameter η in (14) adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' It first produces the estimators of the diagonal entries δii, 1 ≤ i ≤ p, as in (14) with η1 = 2 � log(p)/n as the tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then these estimates are used for adaptive tuning parameter selection in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We provide the full description of the ACLIME estimator along with the details of its implementation in ACLIME estimator of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Given the estimators � A(1) = I − ∑d ℓ=1 �Aℓ and �∆, we estimate Ω by �Ω = 2π � A⊤(1)�∆ � A(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022), �∆ and �Ω are shown to be consistent in ℓ∞- and ℓ1-norms under suitable sparsity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' However, an additional thresholding step as in (12) is required to guarantee consistency in estimating the support of ∆ and Ω and consequently the edge sets of N C and N L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We discuss data-driven selection of these thresholds and η in Tuning parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' FNETS: Forecasting Following the estimation procedure, FNETS performs forecasting by estimating the best linear predic- tor of Xn+a given Xt, t ≤ n, for a fixed integer a ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' This is achieved by separately producing the best linear predictors of χn+a and ξn+a as described below, and then combining them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 5 Forecasting the factor-driven component For given a ≥ 0, the best linear predictor of χn+a given Xt, t ≤ n, under (2) is χn+a|n = ∞ ∑ ℓ=0 Bℓ+aun−ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Forni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2015) show that the model (2) admits a low-rank VAR representation with ut as the innovations under mild conditions, and Forni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2017) propose the estimators of Bℓ and ut based on this representation which make use of the estimators of the ACV of χt obtained as described in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, a natural estimator of χn+a|n is �χunr n+a|n = K ∑ ℓ=0 �Bℓ+a�un−ℓ (15) for some truncation lag K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We refer to �χunr n+a|n as the unrestricted estimator of χn+a|n as it is obtained without imposing any restrictions on the factor model (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' When χt admits the static representation in (3), we can show that χn+a|n = Γχ(−a)EχM−1 χ E⊤ χ χn, where Mχ ∈ Rr×r is a diagonal matrix with the r eigenvalues of Γχ(0) on its diagonal and Eχ ∈ Rp×r the matrix of the corresponding eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' This suggests an estimator �χres n+a|n = �Γχ(−a)�Eχ � M −1 χ �E⊤ χ Xn, (16) where � Mχ and �Eχ are obtained from the eigendecomposition of �Γχ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We refer to �χres n+a|n as the restricted estimator of χn+a|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' As a by-product, we obtain the in-sample estimators of χt, t ≤ n, as �χt|n = �χt, with either of the two estimators in (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Forecasting the latent VAR process Once the VAR parameters are estimated either as in (10) or (11), we produce an estimator of ξn+a|n = ∑d ℓ=1 Aℓξn+a−ℓ, the best linear predictor of ξn+a given Xt, t ≤ n, as �ξn+a|n = max(1,a)−1 ∑ ℓ=1 �Aℓ�ξn+a−ℓ|n + d ∑ ℓ=max(1,a) �Aℓ�ξn+a−ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (17) Here, �ξn+1−ℓ = Xn+1−ℓ − �χn+1−ℓ denotes the in-sample estimator of ξn+1−ℓ, which may be obtained with either of the two (in-sample) estimators of the factor-driven component in (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Tuning parameter selection Factor numbers q and r The estimation and forecasting tools of the FNETS methodology require the selection of the number of factors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' q under the unrestricted factor model in (2), and r under the restricted, static factor model in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Under (2), there exists a large gap which diverges with p, between the q leading eigenvalues of the spectral density matrix of Xt and the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We provide two methods for selecting the factor number q, which make use of the postulated eigengap using �µx,j(ωk), 1 ≤ j ≤ p, the eigenvalues of the spectral density matrix estimator of Xt at a given Fourier frequency ωk, −m ≤ k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Hallin and Liška (2007) propose an information criterion for selecting the number of factors under the model (2) and further, a methodology for tuning the multiplicative constant in the penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Define IC(b, c) = log � � 1 p p ∑ j=b+1 1 2m + 1 m ∑ k=−m �µx,j(ωk) � � + b · c · pen(n, p), (18) where pen(n, p) = min(p, m2, √ n/m)−1/2 by default (for other choices of the information criterion, see Appendix A) and c > 0 a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Provided that pen(n, p) → 0 sufficiently slowly, for an arbitrary value of c, the factor number q is consistently estimated by the minimiser of IC(b, c) over b ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , ¯q}, with some fixed ¯q as the maximum allowable number of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' However, this is not the case in finite sample, and Hallin and Liška (2007) propose to simultaneously select q and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' First, 6 we identify �q(nl, pl, c) = arg min0≤b≤ ¯q IC(nl, pl, b, c) where IC(nl, pl, b, c) is constructed analogously to IC(b, c), except that it only involves the sub-sample {Xit, 1 ≤ i ≤ pl, 1 ≤ t ≤ nl}, for sequences 0 < n1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' < nL = n and 0 < p1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' < pL = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, denoting the sample variance of �q(nl, pl, c), 1 ≤ l ≤ L, by S(c), we select �q = �q(n, p, �c) with �c corresponding to the second interval of stability with S(c) = 0 for the mapping c �→ S(c) as c increases from 0 to some cmax (the first stable interval is where ¯q is selected with a very small value of c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Figure 3 plots �q(n, p, c) and S(c) for varying values of c obtained from a dataset simulated in Data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In the implementation of this methodology, we set nl = n − (L − l)⌊n/20⌋ and pl = ⌊3p/4 + lp/40⌋ with L = 10, and ¯q = min(50, ⌊ � min(n − 1, p)⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 q Sc IC 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 q Sc Figure 3: Plots of c against �q(n, p, c) (in circle, y-axis on the left) and S(c) (in triangle, y-axis on the right) with the six IC (see Appendix A) implemented in the function factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='number of fnets, on a dataset simulated as in Data generation (with n = 500, p = 50 and q = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' With the default choice of IC in (18) (IC5), we obtain �q = �q(n, p, �c) = 2 correctly estimating q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Alternatively, we can adopt the ratio-based estimator �q = arg min1≤b≤ ¯q ER(b) proposed in Avarucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022), where ER(b) = � m ∑ k=−m �µx,b+1(ωk) �−1 � m ∑ k=−m �µx,b(ωk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (19) These methods are readily modified to select the number of factors r under the restricted factor model in (3), by replacing (2m + 1)−1 ∑m k=−m �µx,j(ωk) with �µx,j, the eigenvalues of the sample covari- ance matrix �Γx(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We refer to Bai and Ng (2002) and Alessi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2010) for the discussion of the information criterion-based method in this setting, and Ahn and Horenstein (2013) for that of the eigenvalue ratio-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Threshold t Motivated by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2021), we propose a method for data-driven selection of the threshold t, which is applied to the estimators of Aℓ, 1 ≤ ℓ ≤ d (see (12)), ∆ or Ω for estimating the edge sets of N G, N C or N L, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Let B = [bij] ∈ Rm×n denote a matrix for which a threshold is to be selected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' B may be either �β = [�A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , �Ad]⊤, �∆0 (�∆ with diagonals set to zero) or �Ω0 ( �Ω with diagonals set to zero) obtained from Steps 2 and 3 of FNETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We work with �∆0 and �Ω0 since we do not threshold the diagonal entries of �∆ and �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' As such estimators are shown to achieve consistency in ℓν-norm for some ν > 0, we expect there exists a large gap between the entries of B corresponding to true positives and false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Further, it is expected that the number of edges reduces at a faster rate when increasing the threshold from 0 towards this (unknown) gap, compared to when increasing the threshold from the gap to |B|∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Therefore, we propose to identify this gap by casting the problem as that of locating a single change point in the trend of the ratio of edges to non-edges, Ratiok = |B(tk)|0 max(N − |B(tk)|0, 1), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Here, B(t) = [bij · I{|bij|>t}], |B(t)|0 = ∑m1 i=1 ∑m2 j=1 I{|bij|>t} and {tk, 1 ≤ k ≤ M : 0 = t1 < t2 < · · · < 7 tM = |B|∞} denotes a sequence of candidate threshold values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We recommend using an exponentially growing sequence for {tk}M k=1 since the size of the false positive entries tends to be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The quantity N in the denominator of Ratiok is set as N = p2d when B = �β, and N = p(p − 1) when B = �∆0 or B = �Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, from the difference quotient Diffk = Ratiok − Ratiok−1 tk − tk−1 , k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , M, we compute the cumulative sum (CUSUM) statistic CUSUMk = � k(M − k) M ����� 1 k k ∑ l=2 Diffl − 1 M − k M ∑ l=k+1 Diffl ����� , k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , M − 1, and select tada = tk∗ with k∗ = arg max2≤k≤M−1CUSUMk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For illustration, Figure 4 plots Ratiok and CUSUMk against candidate thresholds for the dataset simulated in Data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='06 threshold ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 threshold abs(cusum) Figure 4: Ratiok (left) and CUSUMk (right) plotted against tk when B = �βlas obtained from the data simulated in Data generation with n = 500 and p = 50, as a Lasso estimator of the VAR parameter matrix, with the selected tada denoted by the vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' VAR order d, λ and η Step 2 and Step 3 of the network estimation methodology of FNETS involve the selection of the tuning parameters λ and η (see (10), (11) and (13)) and the VAR order d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' While there exist a variety of methods available for VAR order selection in fixed dimensions (Lütkepohl, 2005, Chapter 4), data-driven selection of d in high dimensions remains largely unaddressed with few exceptions (Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Krampe and Margaritella, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Zheng, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We suggest two methods for jointly selecting λ and d for Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The first method is also applicable for selecting η in Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Cross validation We describe the use of cross validation (CV) for simultaneously selecting d and λ in Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The data is partitioned into L folds, Il = {n◦ l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , n◦ l+1} with n◦ l = min(l⌈n/L⌉, n), 1 ≤ l ≤ L, and each fold is split into a training set Itrain l = {n◦ l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , ⌈(n◦ l + n◦ l+1)/2⌉} and a test set Itest l = Il \\ Itrain l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' On each fold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' β0 is estimated from {Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' t ∈ Itrain l } as either the Lasso (10) or the Dantzig selector (11) estimators with λ as the tuning parameter and some b as the VAR order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' say �βtrain l (λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' using which we compute the CV measure CV(λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' b) = L ∑ l=1 tr � �Γtest ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='l (0) − ( �βtrain l (λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' b))⊤�gtest l (b)− (�gtest l (b))⊤ �βtrain l (λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' b) + ( �βtrain l (λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' b))⊤ �Gtest l (b) �βtrain l (λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' b) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' where �Γtest ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='l (ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' �Gtest l (b) and �gtest l (b) are generated analogously as �Γξ(ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' �G(b) and �g(b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' from the test set {Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' t ∈ Itest l }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Although we do not directly observe ξt, the measure CV(λ, b) gives an approximation of the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, we select (�λ, �d) = arg minλ∈Λ,1≤b≤ ¯d CV(λ, b), where Λ is a grid of values for λ, and ¯d ≥ 1 is a pre-determined upper bound on the VAR order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' A similar 8 approach is taken for the selection of η with a Burg matrix divergence-based CV measure: CV(η) = L ∑ l=1 tr � �∆train l (η)�Γtest l � − log ����∆train l (η)�Γtest l ��� − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Here, �∆train l (η) denotes the estimator of ∆ with η as the tuning parameter from {Xt, t ∈ Itrain l }, and �Γtest l the estimator of Γ from {Xt, t ∈ Itest l }, see Step 3 for the descriptions of the estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For both CV procedures, we set L = 1 in the numerical results reported in Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Extended Bayesian information criterion Alternatively, to select the pair (λ, d) in Step 2, we propose to use the extended Bayesian information cri- terion (eBIC) of Chen and Chen (2008), originally proposed for variable selection in high-dimensional linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Let �β(λ, b, tada) denote the thresholded version of �β(λ, b) as in (12) with the threshold tada chosen as described in Threshold t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, letting s(λ, b) = | �β(λ, b, tada)|0, we define eBICα(λ, b) = n 2 log (L(λ, b)) + s(λ, b) log(n) + 2α log � bp2 s(λ, b) � , where (20) L(λ, b) = tr � �G(b) − ( �β(λ, b))⊤�g(b) − (�g(b))⊤ �β(λ, b) + ( �β(λ, b))⊤ �G(b) �β(λ, b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, we select (�λ, �d) = arg minλ∈Λ,1≤b≤ ¯d eBICα(λ, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The constant α ∈ (0, 1) determines the degree of penalisation which may be chosen from the relationship between n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Preliminary simulations suggest that α = 0 is a suitable choice for the dimensions (n, p) considered in our numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Other tuning parameters Motivated by theoretical results reported in Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022), we select the kernel bandwidth for Step 1 of FNETS as m = ⌊4(n/ log(n))1/3⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In forecasting the factor-driven component as in (15), we set the truncation lag at K = 20, as it is expected that the elements of Bℓ decay as ℓ increases for stationary time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Package overview fnets is available from the Comprehensive R Archive Network (CRAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The main function, fnets, implements the FNETS methodology for the input data and returns an object of S3 class fnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' fnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var implements Step 2 of the FNETS methodology estimating the VAR parameters only, and is applicable directly for VAR modelling of high-dimensional time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' fnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='model performs factor modelling under either of the two models (2) and (3), and returns an object of class fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We provide predict methods for the objects of classes fnets and fm, and a plot method for the objects of the fnets class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We recommend that the input time series for the above functions to be transformed to stationarity (if necessary) after a unit root test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In this section, we demonstrate how to use the functions included with the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Data generation For illustration, we generate an example dataset of n = 500 and p = 50 following the model (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' fnets provides functions for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For given n and p, the function sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var generates the VAR(1) process following (1) with d = 1, Γ as supplied to the function (Γ = I by default), and A1 generated as described in Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The function sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='unrestricted generates the factor-driven component under the unrestricted factor model in (2) with q dynamic factors (q = 2 by default) and the filter B(L) generated as in model (C1) of Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='seed(111) n <- 500 p <- 50 x <- sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var(n, p)$data + sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='unrestricted(n, p)$data Throughout this section, we use the thus-generated dataset in demonstrating fnets unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' There also exists sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted which generates the factor-driven component under the restricted factor model in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For all data generation functions, the default is to use the standard normal 9 distribution for generating ut and εt, while supplying the argument heavy = TRUE, the innovations are generated from √ 3/5 · t5, the t-distribution with 5 degrees of freedom scaled to have unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Calling fnets with default parameters The function fnets can be called with the p × n data matrix x as the only input, which sets all other arguments to their default choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' out <- fnets(x) By default, it performs the factor-adjustment under the unrestricted model in (2) with q estimated by minimising the IC in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The VAR parameter matrix is estimated via the Lasso estimator in (10) with d = 1 as the VAR order and the tuning parameters λ and η chosen via CV, and no thresholding is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' This returns an object of class fnets whose entries are described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Table 1: Entries of S3 objects of class fnets Name Description Type q Factor number integer spec Spectral density matrices for Xt, χt and ξt (when fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE) list acv Autocovariance matrices for Xt, χt and ξt list loadings Estimates of Bℓ, 0 ≤ ℓ ≤ K (when fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE) array or Λ (when fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = TRUE) factors Estimates of {ut} (when fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE) array or {Ft} (when fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = TRUE) idio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var Estimates of Aℓ, 1 ≤ ℓ ≤ d and Γ, and d and λ used list lrpc Estimates of ∆, Ω, (long-run) partial correlations and η used list mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='x Sample mean vector vector var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='method Estimation method for Aℓ string do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='lrpc Whether to estimate the long-run partial correlations (input parameter) Boolean kern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='bw Kernel bandwidth (when fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE, input parameter) double Calling fnets with optional parameters We can also specify the arguments of fnets to control how Steps 1–3 of FNETS are to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The full model call is as follows: out <- fnets(x, center = TRUE, fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE, q = c("ic", "er"), ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='op = NULL, kern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='bw = NULL, common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args = list(factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='order = NULL, max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='order = NULL, trunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='lags = 20, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='perm = 10), var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='order = 1, var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='method = c("lasso", "ds"), var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args = list(n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='iter = NULL, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='cores = min(parallel::detectCores() - 1, 3)), do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='threshold = FALSE, do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='lrpc = TRUE, lrpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='adaptive = FALSE, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args = list(tuning = c("cv", "bic"), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='folds = 1, penalty = NULL, path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='length = 10, do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='plot = FALSE) ) Here, we discuss a selection of input arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The center argument will de-mean the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted determines whether to perform the factor-adjustment under the restricted factor model in (3) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' If the number of factors is known, we can specify q with a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Otherwise, it can be set as "ic" or "er" which selects the factor number estimator to be used between (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' When q = "ic", setting the argument ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='op as an integer between 1 and 6 specifies the choice of the IC (see Appendix A) where the default is ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='op = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' kern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='bw takes a positive integer which specifies the bandwidth to be used in Step 1 of FNETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The list common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args specifies arguments for estimating Bℓ and ut under (2), and relates to the low-rank VAR representation of χt under the unrestricted factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='order specifies a vector of positive integers to be considered in VAR order selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='method determines the method for VAR parameter estimation, which can be either "lasso" (for the estimator in (10)) or "ds" (for that in (11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The list var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args takes additional parameters for Step 2 of FNETS, such as the number of gradient descent steps (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='iter, when var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='method = "lasso") or the number of cores to use for parallel computing (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='cores, when var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='method = "ds").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='threshold selects whether to threshold the estimators of Aℓ, 1 ≤ ℓ ≤ d, ∆ and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' It is possible to perform Steps 1–2 of FNETS only without estimating ∆ and Ω by setting do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='lrpc = FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' If do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='lrpc = TRUE, lrpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='adaptive specifies whether to use the non-adaptive estimator in (13) or the ACLIME estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 10 Granger causal network Long-run partial correlation heatmap 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 Figure 5: Estimated networks for data simulated as in Data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Left: Granger causal net- work N G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' A directed arrow from node i to node i′ indicates that variable i Granger causes node i′, and the edge weights proportional to the size of estimated coefficients are visualised by the edge width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Right: Long-run partial correlation network N L where the edge weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' partial correlations) are visualised by the colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The list tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args supplies arguments to the CV or eBIC procedures, including the number of folds L (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='folds), the eBIC parameter α (penalty, see (20)) and the length of the grid of values for λ and/or η (path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Finally, it is possible to set only a subset of the arguments of var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args, common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args, and tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args whereby the unspecified arguments are set to their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The factor adjustment (Step 1) and VAR parameter estimation (Step 2) functionalities can be accessed individually by calling fnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='model and fnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The latter is equiv- alent to calling fnets with q = 0 and do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='lrpc = FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The former returns an object of class fm which contains the entries of the fnets object in Table 1 that relate to the factor-driven component only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Network visualisation Using the plot method available for the objects of class fnets, we can visualise the Granger network N G induced by the estimated VAR parameter matrices, see the left panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' plot(out, type = "granger", display = "network") Setting the argument type to "pc" or "lrpc", we can visualise N C given by the partial correlations of VAR innovations or N L given by the long-run partial correlations of ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' This displays an igraph object from igraph (Csardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We can instead visualise the networks as a heat map, with the edge weights colour-coded by setting display = "heatmap".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We plot N L as a heat map in the right panel of Figure 5 using the following command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' plot(out, type = "lrpc", display = "heatmap") Forecasting The fnets objects are supported by the predict method with which we can perform h-step ahead forecasting of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For example, we can produce a one-step ahead forecast of Xn+1 as pr <- predict(out, x, h = 1, fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = TRUE) pr$forecast The argument fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted specifies whether to use the estimator �χres n+h|n in (16) generated under a restricted factor model (3), or �χunr n+h|n in (15) generated without such a restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Internally, predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='fnets calls common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='predict and idio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='predict to sequentially produce forecasts and in- sample estimators of the factor-driven component and the VAR process, and the outputs are reported together with the h-step ahead forecast of the input data, see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 11 Table 2: Entries of the output from predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='fnets Name Description Type forecast The h-step ahead forecast of Xt list common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='predict Output of common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='pred containing list $is p × n matrix containing the in-sample estimator of χt $fc p × h matrix containing the h-step ahead forecasts of χt $h Input parameter $r Factor number (only produced when fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = TRUE) idio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='predict Output of idio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='pred containing is, fc and h list mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='x Sample mean vector vector Factor number estimation It is of independent interest to estimate the number of factors (if any) in the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The function factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='number provides access to the two methods for selecting q described in Factor numbers q and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The code fn <- factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='number(x, fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE, do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='plot = TRUE) calls the information criterion-based factor number estimation method in (18), and setting do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='plot = TRUE returns Figure 3 which visualises the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Alternatively, we call the eigenvalue ratio-based method in (19) as fn <- factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='number(x, method = "er", fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE) In this case, setting do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='plot = TRUE produces a plot of ER(b) against the candidate factor number b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , ¯q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Visualisation of the tuning parameter selection procedure We provide tools for visualising the tuning parameter selection results adopted in Steps 2 and 3 of FNETS (see VAR order d, λ and η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' These tools are accessible from both fnets and fnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var by setting tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args = list(do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='plot = TRUE), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='seed(111) n <- 500 p <- 10 x <- sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='var(n, p)$data out <- fnets(x, q = 0, var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='order = 1:3, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args = list(tuning = "cv", do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='plot = TRUE)) This generates the two plots reported in Figure 6 which visualise the CV errors computed as described in Cross validation and, in particular, the left plot shows that the VAR order is correctly selected by this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' When tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='args = list(tuning = "bic"), the results from the eBIC method described in Extended Bayesian information criterion adopted in Step 2, is similarly visualised in place of the left panel of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Simulations Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022) provide comprehensive simulation results on the estimation and forecasting performance of FNETS in comparison with competing methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Therefore in this paper, we focus on assessing the performance of the methods for selecting tuning parameters such as the threshold and VAR order, which are implemented in fnets and discussed in Tuning parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Additionally in Appendix B, we compare the adaptive and the non-adaptive estimators in estimating ∆ and also investigate how their performance is carried over to estimating Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Settings We consider the following data generating processes for the factor-driven component χt: 12 5e-04 5e-03 5e-02 5e-01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='8 CV for VAR parameter estimation 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0 5 10 15 20 25 CV for (LR)PC matrix estimation Figure 6: Plots of CV(λ, b) against λ with b ∈ {1, 2, 3} (left) and CV(η) against η (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Vertical lines denote where the minimum CV measure is attained with respect to λ and η, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (C1) Taken from Forni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2017), χit is generated as a sum of AR processes χit = ∑ q j=1 aij(1 − αijL)−1ujt with q = 2, where ujt ∼iid N (0, 1), aij ∼iid U[−1, 1] and αij ∼iid U[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='8] with U[a, b] denoting a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then, χt does not admit a static representation in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (C2) χt = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' the VAR process is directly observed as Xt = ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For generating a VAR(d) process ξt, we first generate a directed Erd˝os-Rényi random graph N = (V, E) on V = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , p} with the link probability 1/p, and set entries of Ad such that Ad,ii′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='275 when (i, i′) ∈ E and Ad,ii′ = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Also, we set Aℓ = O for ℓ < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The VAR innovations are generated as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (E1) Gaussian with the covariance matrix Γ = ∆−1 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (E2) Gaussian with the covariance matrix Γ = ∆−1 such that δii = 1, δi,i+1 = δi+1,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='6, δi,i+2 = δi+2,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='3, and δii′ = 0 for |i − i′| ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' For each setting, we generate 100 realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Results: Threshold selection We assess the performance of the adaptive threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We generate χt as in (C1) and fix d = 1 for gener- ating ξt and further, treat d as known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We consider (n, p) ∈ {(200, 50), (200, 100), (500, 100), (500, 200)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then we estimate Ω using the thresholded Lasso estimator of A1 (see (10) and (12)) with two choices of thresholds, t = tada generated as described in Threshold t and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' To assess the performance of �Ω = [ �ωii′] in recovering of the support of Ω = [ωii′], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' {(i, i′) : ωii′ ̸= 0}, we plot the receiver operating characteristic (ROC) curves of true positive rate (TPR) against false positive rate (FPR), where TPR = |{(i, i′) : �ωii′ ̸= 0 and ωii′ ̸= 0}| |{(i, i′) : ωii′ ̸= 0}| and FPR = |{(i, i′) : �ωii′ ̸= 0 and ωii′ = 0}| |{(i, i′) : ωii′ = 0}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Figure 7 plots the ROC curves averaged over 100 realisations when t = tada and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' When ∆ = I under (E1), we see little improvement from adopting tada as the support recovery performance is already good even without thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' However, when ∆ ̸= I under (E2), the adaptive threshold leads to improved support recovery especially when the sample size is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Tables 3 and 4 in Appendix C additionally report the errors in estimating A1 and Ω with and without thresholding, where we see little change is brought by thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In summary, we conclude that the estimators already perform reasonably well without thresholding, and the adaptive threshold tada brings marginal improvement in support recovery which is of interest in network estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 model E1 E2 method adaptive threshold threshold=0 n = 200, p = 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 n = 200, p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 n = 500, p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 n = 500, p = 200 Figure 7: ROC curves of TPR against FPR for �β(t) (12) (with �β = �βlas) when t = tada and t = 0 in recovering the support of Ω, averaged over 100 realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Vertical lines indicate FPR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='05 Results: VAR order selection We compare the performance of the CV and eBIC methods proposed in VAR order d, λ and η for selecting the order of the VAR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Here, we consider the case when χt = 0 (setting (C2)) and when ξt is generated under (E1) with d ∈ {1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We set (n, p) ∈ {(200, 10), (200, 20), (500, 10), (500, 20)} where the range of p is in line with the simulation studies conducted in the relevant literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Zheng (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We consider {1, 2, 3, 4} as the candidate VAR orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Figure 8 and Table 5 in Appendix C show that CV works reasonably well regardless of d ∈ {1, 3}, with slightly better performance observed together with the DS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' On the other hand, eBIC tends to over-estimate the VAR order when d = 1 while under-estimating it when d = 3, and hence is less reliable compared to the CV method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 0 25 50 75 100 0 1 2 3 n = 200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p = 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' d = 1 0 25 50 75 100 2 1 0 1 method BIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' DS BIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Lasso CV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' DS CV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Lasso n = 200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p = 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' d = 3 0 25 50 75 100 0 1 2 3 n = 200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' d = 1 0 25 50 75 100 2 1 0 1 n = 200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' d = 3 0 25 50 75 100 0 1 2 3 n = 500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p = 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' d = 1 0 25 50 75 100 2 1 0 1 n = 500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p = 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' d = 3 0 25 50 75 100 0 1 2 3 n = 500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' d = 1 0 25 50 75 100 2 1 0 1 n = 500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' d = 3 Figure 8: Box plots of �d − d over 100 realisations when the VAR order is selected by the CV and eBIC methods in combination with the Lasso (10) and the DS (11) estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 14 Data example Energy price data Electricity is more difficult to store than physical commodities, which results in high volatility and seasonality in spot prices (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Global market deregulation has increased the volume of electricity trading, which promotes the development of better forecasting and risk management methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We analyse a dataset of node-specific prices in the PJM (Pennsylvania, New Jersey and Maryland) power pool area in the United States, accessed using dataminer2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='pjm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' There are four node types in the panel, which are Zone, Aggregate, Hub and Extra High Voltage (EHV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' for their definitions, see Table 8 and for the names and types of p = 50 nodes, see Table 9, all found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The series we model is the sum of the real time congestion price and marginal loss price or, equivalently, the difference between the spot price at a given location and the overall system price, where the latter can be thought of as an observed factor in the local spot price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' These are obtained as hourly prices and then averaged over each day as per Maciejowska and Weron (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We remove any short-term seasonality by subtracting a separate mean for each day of the week, then stabilise the variance by applying the inverse hyperbolic sine transformation (Uniejewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Network estimation We select the data collected from 01/01/2021 to 19/07/2021 (n = 200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The information criterion in (18) returns a single factor (�q = 1), and �d = 1 is selected by CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' See Figure 9 for the heat maps visualising the three networks N G, N C and N L described in Networks, which are produced by fnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='N G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='N C ' metadata={'source': 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maps of the three networks underlying the energy price data collected over the period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='01/01/2021–19/07/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Left: N G obtained with the Lasso estimator (10) combined with the adaptive threshold tada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Middle: N C obtained with the ACLIME estimator of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Right: N L obtained by combining the estimators of VAR parameters and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The heat maps in the left column are in the scale of [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='6] while those in the middle and right columns are in the scale of [−1, 1], with red hues denoting large positive values and blue hues large negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In the axis labels, Zone-type nodes are coloured in blue, Aggregate-types in green, Hub-types in red, and EHV-types in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The non-zero entries of the VAR parameter matrix estimates tend to take positive values, indicating that high energy prices are persistent and spill over to other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Considering the node types, Zone- type nodes (blue) tend not to have causal links from other nodes in N G, but do have causal links outwards to nodes of other types, which reflects the behaviour of the electrical transmission system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Aggregate-types (green) have strong causal links from other nodes, while EHV-types (black) have links inward from Zone-types but not from Aggregrate-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' This carries forward to N L where we observe that EHV-type nodes do not have long-run dependence with nodes belonging to Aggregate-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Summary We introduce the R package fnets which implements the FNETS methodology proposed by Barigozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2022) for network estimation and forecasting of high-dimensional time series exhibiting strong correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' It further implements data-driven methods for selecting tuning parameters, and provides tools for high-dimensional time series factor modelling under the GDFM which are of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The efficacy of our package is demonstrated on both real and simulated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 15 Bibliography S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Ahn and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Horenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Eigenvalue ratio test for the number of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Econometrica, 81(3): 1203–1227, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' [p7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Alessi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Barigozzi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Capasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Improved penalization for determining the number of factors in approximate factor models.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='org/ package=sparsevar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' R package version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' [p2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Wilms, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Basu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Bien, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Matteson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' bigtime: Sparse estimation of large time series models, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' R package version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' [p2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' An interpretable and efficient infinite-order vector autoregressive model for high- dimensional time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='01172, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' [p8, 14] 17 Dom Owens School of Mathematics, University of Bristol Supported by EPSRC Centre for Doctoral Training (EP/S023569/1) dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='owens@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='uk Haeran Cho School of Mathematics, University of Bristol Supported by the Leverhulme Trust (RPG-2019-390) haeran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='cho@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='uk Matteo Barigozzi Department of Economics, Università di Bologna Supported by MIUR (PRIN 2017, Grant 2017TA7TYC) matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='barigozzi@unibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='it 18 Appendix A: Information criteria for factor number selection Here we list information criteria for factor number estimation which are implemented in fnets and accessible by the functions fnets, fnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='model and factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='number by setting the argument ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='op at an integer belonging to {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' When fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' we have IC1: � 1 p ∑ p j=b+1 1 2m+1 ∑m k=−m �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j(ωk) � + b · c · (m−2 + √ m/n + p−1) · log(min(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' √ n/m)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC2: � 1 p ∑ p j=b+1 1 2m+1 ∑m k=−m �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j(ωk) � + b · c · (min(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' √ n/m))−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC3: � 1 p ∑ p j=b+1 1 2m+1 ∑m k=−m �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j(ωk) � + b · c · (min(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' √ n/m))−1 · log(min(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' √ n/m)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC4: log � 1 p ∑ p j=b+1 1 2m+1 ∑m k=−m �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j(ωk) � + b · c · (m−2 + √ m/n + p−1) · log(min(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' √ n/m)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC5: log � 1 p ∑ p j=b+1 1 2m+1 ∑m k=−m �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j(ωk) � + b · c · (min(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' √ n/m))−1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC6: log � 1 p ∑ p j=b+1 1 2m+1 ∑m k=−m �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j(ωk) � + b · c · (min(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' √ n/m))−1 · log(min(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' √ n/m)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' When fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = TRUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' we use one of IC1: � 1 p ∑ p j=b+1 �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j � + b · c · (n + p)/(np) · log(np/(n + p)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC2: � 1 p ∑ p j=b+1 �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j � + b · c · (n + p)/(np) · log(np/(n + p)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC3: � 1 p ∑ p j=b+1 �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j � + b · c · log(min(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p))/(min(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC4: log � 1 p ∑ p j=b+1 �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j � + b · c · (n + p)/(np) · log(np/(n + p)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC5: log � 1 p ∑ p j=b+1 �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j � + b · c · (n + p)/(np) · log(np/(n + p)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' IC6: log � 1 p ∑ p j=b+1 �µx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='j � + b · c · log(min(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p))/(min(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Whether fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='restricted = FALSE or not, the default choice is ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='op = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Appendix B: ACLIME estimator We provide a detailed description of the adaptive extension of the CLIME estimator of ∆ in (13), extending the methodology proposed in Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (2016) for precision matrix estimation in the independent setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Let �Γ∗ = �Γ + n−1I and η1 = 2 � log(p)/n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Step 1: Let ˇ∆(1) = [ ˇδ(1) ii′ ] be the solution to ˇ∆(1) i′ = arg minm∈Rp|m|1 subject to (21) ���(�Γ∗m − ei′)i ��� ≤ η1(�γii ∨ �γi′i′)mi′ ∀ 1 ≤ i ≤ p and mi′ > 0, for i′ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Then we obtain truncated estimates �δ(1) ii = ˇδ(1) ii I{|�γii|≤√ n/ log(p)} + � log(p) n I{|�γii|>√ n/ log(p)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Step 2: We obtain ˇ∆(2) i′ = arg minm∈Rp|m|1 subject to ���(�Γ∗m − ei′)i ��� ≤ η2 � �γii�δ(1) i′i′ ∀ 1 ≤ i ≤ p, where η2 > 0 is a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Since ˇ∆(2) is not guaranteed to be symmetric, the final estimator is obtained after a symmetrisation step: �∆ada = [�δii′, 1 ≤ i, i′ ≤ p] with �δ(2) ii′ = ˇδ(2) ii′ · I{| ˇδ(2) ii′ |≤| ˇδ(2) i′i |} + ˇδ(2) i′i · I{| ˇδ(2) i′i |<| ˇδ(2) ii′ |}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' (22) 19 The constraints in (21) incorporate the parameter in the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' To use linear programming software to solve this, we formulate the constraints for each 1 ≤ i′ ≤ p as ∀1 ≤ i ≤ p, ((�Γ∗ − Qi′)m − ei′)i ≤ 0, ∀1 ≤ i ≤ p, −((�Γ∗ + Qi′)m − ei′)i ≤ 0, mi′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' where Qi′ has entries qii′ = η1(�γii ∨ �γi′i′) in column i′ and 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Appendix C: Additional simulation results Threshold selection Tables 3 and 4 report the errors in estimating A1 and Ω when the threshold t = tada or t = 0 is applied to the estimator of A1 obtained by either the Lasso (10) or the DS (11) estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' With a matrix γ as an estimand we measure the estimation error of its estimator �γ using the following (scaled) matrix norms: LF = ∥�γ − γ∥F ∥γ∥F and L2 = ∥�γ − γ∥ ∥γ∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Table 3: Errors in estimating A1 with t ∈ {0, tada} in combination with the Lasso (10) and the DS (11) estimators, measured by LF and L2, averaged over 100 realisations (with standard errors reported in brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We also report the average TPR when FPR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='05 and the corresponding standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' See Results: Threshold selection in the main text for further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' t = 0 t = tada �βlas �βDS �βlas �βDS Model n p TPR LF L2 TPR LF L2 TPR LF L2 TPR LF L2 (E1) 200 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='9681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='6234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='7204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='8991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='4299 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='068) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='093) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='205) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='159) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='145) Table 4: Errors in estimating Ω with t ∈ {0, tada} applied to the estimator of A1 in combination with the Lasso (10) and the DS (11) estimators, measured by LF and L2, averaged over 100 realisations (with standard errors reported in brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We also report the average TPR when FPR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='05 and the corresponding standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' See Results: Threshold selection in the main text for further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' t = 0 t = tada �βlas �βDS �βlas �βDS Model n p TPR LF L2 TPR LF L2 TPR LF L2 TPR LF L2 (E1) 200 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='8714 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='182) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='090) 20 VAR order selection Table 5 reports the results of VAR order estimation over 100 realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Table 5: Distribution of �d − d over 100 realisations when the VAR order is selected by the CV and eBIC methods in combination with the Lasso (10) and the DS (11) estimators, see Results: VAR order selection in the main text for further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='CLIME vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' ACLIME estimators We compare the performance of the adaptive and non-adaptive estimators for the VAR innovation precision matrix ∆ and its impact on the estimation of Ω, the inverse of the long-run covariance matrix of the data (see Step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We generate χt as in (C1), fix d = 1 and treat it as known and consider (n, p) ∈ {(200, 50), (200, 100), (500, 100), (500, 200)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' In Tables 6 and 7, we report the errors of ∆ and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We consider both the Lasso (10) and DS (11) estimators of VAR parameters, and CLIME and ACLIME estimators for ∆, which lead to four different estimators for ∆ and Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Overall, we observe that with increasing n, the performance of all estimators improve according to all metrics regardless of the scenarios (E1) or (E2), while increasing p has an adverse effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' The two methods perform similarly in setting (E1) when ∆ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' There is marginal improvement for adopting the ACLIME estimator noticeable under (E2), particularly in TPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Figures 10 and 11 shows the ROC curves for the support recovery of ∆ and Ω when the Lasso estimator is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Table 6: Errors in estimating ∆ using CLIME and ACLIME estimators, measured by LF and L2, averaged over 100 realisations (with standard errors reported in brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' We also report the average TPR when FPR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='05 and the corresponding standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' CLIME ACLIME �βlas �βDS �βlas �βDS Model n p TPR LF L2 TPR LF L2 TPR LF L2 TPR LF L2 (E1) 200 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='489 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='497 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='000 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 n = 200, p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 n = 500, p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 n = 200, p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='00 n = 500, p = 200 Figure 11: ROC curves of TPR against FPR for �Ω with CLIME and ACLIME estimators in recovering the support of Ω, averaged over 100 realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Vertical lines indicate FPR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 23 Appendix D: Dataset information Table 8 defines the four node types in the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Table 9 describes the dataset analysed in Data example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Table 8: Node type definitions for energy price data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Name Definition Zone A transmission owner’s area within the PJM Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Aggregate A group of more than one individual bus into a pricing node (pnode) that is considered as a whole in the Energy Market and other various systems and Markets within PJM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Hub A group of more than one individual bus into a regional pricing node (pnode) developed to produce a stable price signal in the Energy Market and other various systems and Markets within PJM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' Extra High Voltage (EHV) Nodes at 345kV and above on the PJM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' 24 Table 9: Names, IDs and Types for the 50 power nodes in the energy price dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='Node ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='Node Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='PJM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} +page_content='25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFJT4oBgHgl3EQf6y0f/content/2301.11675v1.pdf'} diff --git a/_NE5T4oBgHgl3EQfSA5Z/content/tmp_files/2301.05525v1.pdf.txt b/_NE5T4oBgHgl3EQfSA5Z/content/tmp_files/2301.05525v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b6702e5653ebb432b7e82c796af023bcc8df5de --- /dev/null +++ b/_NE5T4oBgHgl3EQfSA5Z/content/tmp_files/2301.05525v1.pdf.txt @@ -0,0 +1,1198 @@ +Understanding Concept Identification as Consistent +Data Clustering Across Multiple Feature Spaces +Felix Lanfermann +Honda Research Institute Europe +Offenbach, Germany +felix.lanfermann@honda-ri.de +Sebastian Schmitt +Honda Research Institute Europe +Offenbach, Germany +sebastian.schmitt@honda-ri.de +Patricia Wollstadt +Honda Research Institute Europe +Offenbach, Germany +patricia.wollstadt@honda-ri.de +Abstract—Identifying meaningful concepts in large data sets +can provide valuable insights into engineering design problems. +Concept identification aims at identifying non-overlapping groups +of design instances that are similar in a joint space of all +features, but which are also similar when considering only +subsets of features. These subsets usually comprise features that +characterize a design with respect to one specific context, for +example, constructive design parameters, performance values, +or operation modes. It is desirable to evaluate the quality of +design concepts by considering several of these feature subsets +in isolation. In particular, meaningful concepts should not only +identify dense, well separated groups of data instances, but +also provide non-overlapping groups of data that persist when +considering pre-defined feature subsets separately. In this work, +we propose to view concept identification as a special form of +clustering algorithm with a broad range of potential applications +beyond engineering design. To illustrate the differences between +concept identification and classical clustering algorithms, we +apply a recently proposed concept identification algorithm to two +synthetic data sets and show the differences in identified solutions. +In addition, we introduce the mutual information measure as a +metric to evaluate whether solutions return consistent clusters +across relevant subsets. To support the novel understanding of +concept identification, we consider a simulated data set from +a decision-making problem in the energy management domain +and show that the identified clusters are more interpretable with +respect to relevant feature subsets than clusters found by common +clustering algorithms and are thus more suitable to support a +decision maker. +Index +Terms—concept +identification; +clustering; +decision- +making; interpretability; mutual information. +I. INTRODUCTION +In the engineering design domain, the identification of +design concepts is a central task that identifies groups of +engineering designs that share similar characteristics, typically +in terms of their specification, but also in terms of performance +or other types of descriptions, such as operation mode. When +identifying concepts, it is relevant that identified groups of +designs are similar with respect to all describing features, +but that this similarity is also preserved when considering +a subset of feature types in isolation. For example, designs +within a concept should be highly similar with respect to all +characteristics, but designs should also be similar in terms of +only their specifications or only their performance. This not +only creates insight into the relation between the distribution +of design groups and their corresponding quality criteria. It is +further relevant for identifying highly representative instances +of design concepts, also termed archetypes, which may then be +used as a starting point for further refinement of specifications, +which lead to consistent performance. +Concept identification may be viewed as a special form +of clustering problem that aims at identifying non-overlapping +groups of instances that have high similarity in a joint feature +space, but which persist when considering relevant subspaces +of features. A subspace is here defined as an arbitrary subset +of features that defines an instance in a specific context or do- +main, into which instances are projected from the joint feature +space. Hence, concept identification may be approached using +a clustering algorithm with specific constraints on the returned +solution. In particular, the solution should a) comprise dense +clusters in the joint space of all features, b) the clusters should +be non-overlapping, which can only be achieved for some data +sets by not assigning all instances to a cluster, and c), non- +overlapping clusters should persist in relevant subspaces of +the joint space. In other words, a suitable algorithm should +return non-overlapping clusters in the joint feature space and +instances that are assigned to a cluster in the joint space should +be assigned to the same cluster in arbitrary, a-priori defined +subspaces. We term this later property consistency of clusters +across different feature spaces. As a result of this consistency +constraint, instances being grouped within the same cluster, +show high similarity both in the joint and in the individual +subspaces. This further leads to also a high similarity between +subspaces for instances assigned to concepts. +We here evaluate a recently proposed algorithm based +on evolutionary optimization that was introduced to perform +concept identification in the engineering design domain [1], +[2], in terms of a general clustering algorithm which we +believe to be relevant also to other application domains. In +particular, we show that the concept identification algorithm +returns clustering solutions with the properties introduced +above and compare them to solutions returned by classical +clustering algorithms. We highlight the differences between +the two approaches, in particular, the ability of the concept +identification algorithm to identify consistent clusters across +arbitrary, predefined subspaces, a property that may be relevant +also in other application domains, for example, cross-domain +recommender systems [3]. We here demonstrate such an +application to a decision making problem from the energy +management domain. +arXiv:2301.05525v1 [cs.LG] 13 Jan 2023 + +We evaluate the concept identification algorithm’s ability +to return solutions according to the three constraints defined +above. To evaluate the algorithm’s ability to identify clusters in +subspaces with high similarity between clusters, as enforced by +the consistency constraint, we use the mutual information (MI) +to evaluate returned solutions, since traditional metrics for +evaluating clustering solutions do not account for this property. +The remainder of this paper is structured as follows: +In Section II, we review prior art in clustering and con- +cept identification. We then briefly introduce the algorithm +evaluated here and highlight how it differs from existing +clustering approaches. Last, we introduce the MI for evaluation +of clustering solutions. In Section III, we apply the algorithm +to three artificial data sets to demonstrate the properties of +clustering solutions returned by the algorithm. The first two +experimental data sets are designed to highlight the algorithm’s +ability to identify dense, non-overlapping clusters that are +consistent across subspaces. The third data set is generated +through high-fidelity simulation and illustrates a use case for +the algorithm in a decision-making problem from the energy +management domain. In Section IV, we discuss our findings +and potential relevant application scenarios of the algorithm. +We close in Section V with a conclusion and an outlook on +future work. +II. METHODS +A. Concept identification as an extended form of clustering +In short, there are three objectives that should be met +by a suitable solution for concept identification, namely a) +compactness of clusters, b) no overlap between clusters, and +c) a consistency of clusters that persists in arbitrary subspaces +of the joint clustering space. The first two objectives are +(partially) optimized by classical clustering algorithms [4], i.e., +algorithms that find a “grouping” or assignment of samples +to “clusters” that results in a low variance or compactness +within clusters and a high separation between clusters [5]. +Hence, existing clustering algorithms have objectives that +are related to those of concept identification, but differ in +one aspect: Clustering does not ensure the third objective of +consistent clusters in arbitrary subspaces. There are extensions +of the classical clustering objective, which add additional +constraints on clustering solutions, e.g., subspace [6] or multi- +view clustering [7]. However, none of these approaches return +solutions that are suitable to solve the problem of concept +identification. We will discuss these existing approaches and +their difference to concept identification below. +Before we discuss the relationship between concept identi- +fication and clustering in the next subsection, we will briefly +review existing concept identification approaches and describe +the algorithm, originally introduced in [1]. +Concept identification algorithms typically define a quality +metric to asses whether cluster assignments in the chosen +subspaces follow the constraints defined above. The quality +can, for example, be determined by calculating the interesting- +ness and significance (IS) [8] of assignments. This produces a +numerical, therefore comparable metric to compare different +data sets. Another approach to assess the quality of concepts +via a quality metric is used in multi-objective optimization [9]. +Here, concepts are judged by how well they cover the pareto- +front of optimal solutions. In particular, the hyper-volume of +the concepts is used to judge their quality. However, this +approach is only valid to assess concepts in data generated by +multi-objective optimizations and is not generally applicable +to data sets from other domains. +Although both measures provide a useful estimate of con- +cept quality, they cannot be used within a concept identifica- +tion algorithm that is applied to return solutions according +to the constraints described above, as these measures lack +the ability to evaluate multiple concepts simultaneously. The +measures cannot explicitly consider and penalize overlap of +concepts within the subspaces and do not evaluate a concept’s +extent. Identifying concepts by optimizing the referenced +metrics can therefore lead to trivial solutions given as either +indistinguishable concepts that span the entire data set (as this +would be an optimal assignment), or as insignificantly small +concepts containing a single point of data only. +The identification of concepts is nevertheless possible by +penalizing concept overlap in an arbitrary number of sub- +spaces, as well as very large and very small concepts. A +concept quality metric that integrates both of these constraints +is proposed by [2]. Here, a data set +D = {xi, i = 1, ..., ND} +(1) +with ND samples is considered. Each sample +x = (x1, . . . , xNS, x∆)T +(2) +is represented by features that are divided into NS subspaces. +Any features that are not associated with a subspace are +denoted by x∆. +For every concept α = 1, . . . , NC, the approach defines a +connected region Cα,k in each subspace k = 1, . . . , NS. All +samples that lie within this region are considered candidates +for the respective concept. A concept is then defined as the +term +Cα = +� +x ∈ +NS +� +k=1 +Cα,k +���x ̸∈ Cβ,k; ∀k, β ̸= α +� +, +(3) +where k, l = 1, . . . , NS enumerate the feature subspaces +and α, β = 1, . . . , NC enumerate the concepts. The concept +Cα therefore only contains those samples, that lie within all +regions {Cα,1, . . . , Cα,NS}, but not within any of the regions +{Cβ,1, . . . , Cβ,NS} associated with any other concept β. +The approach proposes a concept quality metric (CQM) +Q = +NC +� +α +Qα , +(4) + +where the individual metric Qα for each concept is defined as +Qα = +�NS +� +k +NS +� +|Cα| +|Cα,k| +� +· F +�|Cα| +ND +, s +� +· F +�|Pα| +|P| , p +� +. +(5) +For the second and third term in (5), a scaling function +F(x, y) = +� +� +� +� +� +� +� +� +� +� +� +� +1 − +� +x−y +y +�2 +, +if x < y, +1, +if y < x < 1 − y, +� +1 − +� +x−1+y +y +�2 +, +if x > 1 − y +(6) +is utilized. The second term penalizes concepts that contain +fewer or more samples of the data set than a predefined value s. +A optional set of samples P can be defined to further steer the +identification process towards user-preference. As this option +was not used in the conducted experiments in section III, we +refer to [2] for more details. +The CQM in (4) quantifies the quality of a given distribution +of concepts. It can therefore be used to identify an optimal +set of concepts by maximizing the CQM in a numerical +optimization problem. +One option is to apply the metric as the evaluation function +in an evolutionary optimization framework as suggested by +[1], where regions of potential concept candidates are adapted +to find an optimal distribution. In an n-dimensional subspace +of the dataset, such a region can be represented by one n- +dimensional hyper-ellipsoid. The hyper-ellipsoid has +NP,S = n(n + 3)/2 +(7) +parameters (for each n-dimensional subspace) which need to +be optimized by the evolutionary search algorithm. +The approach from [2] is implemented and used within the +experimental evaluations in section III. +B. Difference to existing clustering approaches +As introduced above, clustering algorithms in their most +general form target a grouping of data that yields compact +and well-separated clusters [4]. +Well-known examples of such approaches are k-means [10] +or k-means++ [11]. Both approaches divide all data samples +into a predefined number of clusters. However, solutions are +not necessarily non-overlapping, and clusters found in the +joint feature space do not lead to well-defined clusters in the +defined subspaces. Also, these clustering approaches assign all +instances to a cluster, which is not a requirement for concept +identification and may even prevent the finding of a suitable +solution [2]. +Fuzzy c-means [12] and Gaussian mixture models [13] +allow for a less strict allocation of samples to clusters and +thus relax the strict assignment of instances. However, these +algorithms do also not lead to consistent clusters across sub- +spaces. The approaches cannot identify corresponding groups +in projections of the full feature vector simultaneously, in +which each cluster in one projection corresponds to one cluster +in all other projections. These clustering approaches are thus +not suited to solve the defined problem, as they cannot be +easily extended or adapted to realize the required constraints +on the solutions [1]. +In contrast to the traditional clustering approaches discussed +above, subspace clustering [6] identifies clusters in lower- +dimensional projections when applied on high-dimensional +data. This is particularly beneficial when aiming at data-driven +dimensionality reduction and mitigating sparsity of the data. +Nonetheless, the user cannot predetermine the subspaces, into +which instances are projected in advance. However, such an +alignment of the projections within user-defined subspaces +would be necessary to solve concept identification prob- +lems, where subspaces represent specific domains or types +of features and thus have semantic meaning that should be +preserved. For this reason, and since subspace clustering might +(intentionally) lead to overlapping clusters [14], the approach +is not applicable for the defined problem. +Information maximization clustering (IMC) “simultaneously +learns discriminative representations while generating labels in +an unsupervised mode” [15]. Similar to subspace clustering, +the representations are learned within the process and no a- +priori defined subspaces are taken into consideration. The +desired consistency of groups across subspaces can hence also +not be assured. +The listed approaches may return compact clusters that are +separable with respect to the considered features. However, +consistency of the clusters with respect to multiple subspaces +of the features is not achieved. There are, however, other +approaches that integrate a notion of consistency into the clus- +tering process. Often, a data set is constructed from different +“views” on the data, where a view can be understood as a +group of connected features. The heterogeneous properties of +the data might hold a potential connection [16] and the views +provide complementary information about the data which can +be exploited [16]. Here, a view may be understood as a sub- +space, such that—in accordance with our constraints defined +above—multi-view clustering [7] maximizes a cluster quality +metric within each view, while accounting for clustering con- +sistency across different views [16] (subspaces, respectively). +High consistency can, for example, be achieved by alternating +the maximization and expectation step for each view of the +data [7], or optimizing a weight factor for each view in a k- +means-based clustering loss-function [17]. Combining multi- +view and subspace clustering can uncover clusters in concealed +subspaces in multi-view data [18]. However, multiple views +are utilized in the process to optimize clustering performance, +not to achieve highest possible consistencies across all views. +Consistency is factored in as a trade-off relation, but overlap- +ping clusters in individual views are not fully avoidable. This +is, nevertheless, a central requirement in concept identification +and renders the usage of multi-view clustering for concept +identification difficult. + +C. Evaluation of consistency with mutual information +As introduced in the last subsection, the concept identifica- +tion algorithm evaluated here differs from existing clustering +algorithms by returning strictly non-overlapping clusters and +by ensuring consistency of clusters in subspaces. To evaluate +whether these objectives are met, we compare the algorithm to +existing clustering algorithms and evaluate solutions by, first, +calculating the Silhouette Coefficient [19] to demonstrate that +the concept identification algorithm returns compact clusters, +with a similar separation than found in clusters returned by +existing clustering algorithms. Second, we apply the mutual +information (MI), a measure from information theory [20], to +evaluate the consistency criterion. Note that existing metrics +for validating clustering solutions quantify compactness and +separation, the objectives of classical clustering algorithms [5]. +Examples of these metrics are the Silhouette Coefficient as +used here or the Davies-Bouldin index [21]. However, these +metrics are not suitable to judge the third criterion of concept +identification, the consistency of clusters across subspaces. +The central difference between concept identification and +regular clustering algorithms is the enforced consistency of +clusters between the joint and all a-priori defined subspaces. +As a result of this consistency, instances that are assigned +to a cluster show a high similarity in the joint space, but +also in all subspaces. This similarity within subspaces should +lead to a high similarity also between subspaces. In other +words, when considering only one subspace, knowing that +an instance is assigned to a specific cluster, is informative +about the assignment of that instance in all other subspaces. +Accordingly, the properties of an instance in one subspace are +informative about the properties of that instance in another +subspace. +Hence, we here evaluate the similarity between subspaces +that is enforced by the consistency of clusters, using the MI. +MI quantifies how much information one variable provides +about a second. Thus, it can be used to measure how infor- +mative an instance’s value in one subspace is about the same +instance’s value in another subspace. Hence, we calculate the +MI between targeted subspaces for all instances assigned to a +cluster, to quantify if the algorithm satisfies our requirement +for producing clusters where instances behave similarly in +subspaces of interest. +The MI quantifies the amount of information one random +variable, X, provides about a second random variable, Y , [22] +as +I(X; Y ) = +� +x,y +p(x, y) log p(x|y) +p(x) += +� +x,y +p(x, y) log p(x, y) +p(x)p(y), +(8) +where we write p(x) as a shorthand for the probability +p(X = x) of variable X taking on the value x. The MI may +be interpreted as the Kullback–Leibler divergence between the +product of the marginal distributions, p(x)p(y), and the joint +distribution, p(x, y), and thus measures the deviation of the +two variables’ joint distribution from statistical independence. +We estimate the MI using a nearest-neighbor-based estima- +tor for continuous data proposed by [23] and implemented in +the IDTxl Python toolbox [24], which uses an estimator from +the JIDT toolbox [25]. Note that it is a well-known problem +that MI estimates from finite data suffer from estimation +bias (e.g., [26]). One approach to handle estimation bias +in continuous estimators is the use of permutation testing. +Here, we generate a Null-distribution representing the Null- +hypothesis of X and Y being statistically independent by +repeatedly shuffling one of the two variables and estimating +the MI from this shuffled data. The original estimate is then +compared against this distribution and a p-value is calculated +as the fraction of estimates from shuffled data that are higher +than the original estimate (see, e.g., [27]). +III. EXPERIMENTS +A set of experiments was conducted to illustrate and +quantify the differences between concept identification and +clustering approaches. In each experiment, various clustering +techniques, as well as the introduced concept identification +algorithm were applied to a data set. The first two experi- +ments were conducted on artificial data sets to illustrate the +specific properties of solutions returned by the applied concept +identification algorithm. The third experiment compares the +results of one clustering technique and the referenced concept +identification approach on an application data set from the field +of energy management in a decision-making problem. Table I +provides an overview of the utilized data sets and clustering +preliminaries. +TABLE I +DETAILS OF THE EXPERIMENTAL DATA +Experiment +2D +4D +3D +data source +artificial +artificial +building simulations +# data samples +34000 +30000 +20699 +# subspaces NS +2 +2 +2 +# total features +2 +4 +3 +# features per space +1, 1 +2, 2 +1, 2 +# clusters / concepts NC +3 +3 +3 +# opt. parameters NP +12 +30 +21 +A. Two-dimensional artificial data +To illustrate the main differences between the solutions +given by traditional clustering techniques and the evaluated +concept identification algorithm, a simple synthetic data set, +composed of 34000 samples, was created. Each point of data +is represented by two features, f1 and f2, (Fig. 1). The data +set was uniformly sampled from an area given by +A = +� +� +� +� +� +0 ≤ f1 < 4, +if 0 ≤ f2 < 4, +4 ≤ f1 < 10, +if 0 ≤ f2 < 6, +6 ≤ f1 < 10, +if 6 ≤ f2 < 10. +(9) +To demonstrate the robustness of the approaches, the exper- +iment was repeated 20 times. +According to the three constraints defined for the concept +identification problem, a solution should comprise clusters in + +the joint space, (f1, f2), which are also clearly separable based +on subspaces f1 and f2, individually. It is therefore necessary +to find three consistent groups of samples with respect to both +feature f1 and f2, individually, and to the joint feature space. +The groups should not overlap when considering either sub- +or the joint space. +Four clustering methods were applied to the two- +dimensional data set: k-means, Gaussian mixture models +(GMM), as well as a modification of each algorithm. Since the +concept identification approach does not require all samples to +be associated with a concept, this might create an unfair ad- +vantage for this method compared to clustering algorithms that +always assign all instances to a cluster. Hence, in the modified +clustering methods, only those samples were considered to be +part of the cluster, that were closest to the respective cluster +center1. +Next, we applied the concept identification algorithm to +find groups of samples in the joint spaces, that are consistent +within subspaces f1 and f2. +Since we aimed at identifying three groups in two one- +dimensional subspaces, the total number of parameters that +needed to be modified by the optimization amounts to NP = +3 · (1(1 + 3)/2 + 1(1 + 3)/2) = 12, according to (7). As +the optimization algorithm, a covariance matrix adaptation +evolutionary strategy (CMA-ES) [28], [29] was used with a +population size of 10 for 1000 generation. +The different methods lead to very different partitions +of the data set (Fig. 1). All methods found valid clusters, +but only the concept identification method identified groups +that meet the required conditions. If projected onto one of +the features, all traditional clustering solutions demonstrated +significant overlap between the clusters, showing that none +of the clustering algorithms returned solutions that led to +consistent clusters in considered subspaces. +The results were further evaluated by calculating the MI +between both features, f1 and f2, while considering only those +samples that were assigned to a cluster. Between the clustering +approaches, the modified versions of k-means and GMMs led +to higher levels of MI, compared to all other traditional clus- +tering algorithms (Fig. 2). However, the solution found by the +concept identification algorithm showed a significantly higher +MI than all other solutions. Since MI provides a quantitative +measure of consistency across subspaces, we conclude that the +concept identification algorithm is suited best for assigning +instances to clusters such that instances behave highly similar +across subspaces. +To investigate whether the identified concepts also satisfy +the need for compactness and separation imposed on common +clustering methods, we calculated the silhouette coefficients +for all results (Fig. 3). Since the clustering method and the +1From the regular k-means method, the maximum inner-cluster distance +was calculated. For the modified k-means approach, only those samples, that +lay within a radius of 20% of this maximum distance to the center, were +assigned to the cluster. For the modified GMM approach, only those samples, +that are allocated to the original groups with a probability rate of larger than +90%, are assigned to the cluster +concept identification approach lead to comparable scores, we +follow that the requirement to produce compact and separable +groups is met. We further emphasize that the concept identifi- +cation approach leads to non-overlapping groups by definition +(3). +B. Four-dimensional artificial data +The identified concepts resulting from the first experiment +might look trivial at first glance. In a one-dimensional sub- +space, the elliptic representation describes each concept by a +mean value and a radius. The distribution of concepts is hence +given as a direct segmentation of the one-dimensional data. +Illustrating the clusters in the (full) two-dimensional data set +then naturally creates a simple grid structure (Fig. 1). +In data sets where the subspaces of interest contain more +than one dimension, the results are less trivial. To demonstrate +this effect, we conducted a second experiment involving a +second artificial data set. +The data set contained 30000 samples, each described +by four features f1, f2, f3, and f4, and was created by +sampling from the normal distribution N(µi, σ2) around +three centers µ1 = (0, 0, 0, 0)T , µ2 = (10, 10, 10, 10)T , and +µ3 = (10, 10, 0, 0)T . +Again, the same four clustering approaches and the concept +identification method were applied to identify three groups of +samples. The modified k-means and GMM again only consider +those samples to be part of the cluster, that are closest2 to +the cluster center. On the concept identification algorithm, we +again imposed the constraint that groups should be consistent +within two subspaces. The first subspace was defined as a +combination of features f1 and f2, while the second subspace +comprised f3 and f4. An ideal partition would lead to groups +that do not overlap when projected onto the subspaces. +For the concept identification approach, each concept is +again represented by the combination of one ellipse for each +of the two subspaces. According to (7), the total number of +parameters to be optimized amounts to NP = 3·(2(2+3)/2+ +2(2 + 3)/2) = 30, since both subspaces are two-dimensional. +The algorithm—again a CMA-ES with a population size +of 10 and 1000 generations—hence optimizes the concept +distribution by adapting in total 30 parameters. +The solutions given by all clustering methods show sub- +stantial overlap in the individual subspaces (Fig. 4). All +clustering approaches identify sample groups that are dis- +tinguishable within the full four-dimensional data set. The +cluster centers are close to the means µ1 = (0, 0, 0, 0)T , +µ2 = (10, 10, 10, 10)T , and µ3 = (10, 10, 0, 0)T that were +used to create the data set. However, when projected into the +subspaces, two of the three clusters are indistinguishable. In +contrast, the groups discovered by the concept identification +approach (i.e. the concepts) do not overlap within the sub- +spaces. +2From the regular k-means and GMM approaches, the maximum inner- +cluster distance is calculated. For the modified approaches, only those samples, +that lie within a radius of 10% of this maximum distance to the center, are +assigned to the cluster + +Fig. 1. Partitions of the data set into clusters and concepts based on different methods: Four clustering and one concept identification method are applied to +an artificial two-dimensional data set to divide the data into consistent and compact groups. The goal to derive a separation into groups that do not overlap +when the data set is projected onto one a single feature is achieved only by the concept identification approach. The clusters and concepts are represented by +the purple, green and yellow samples. The projections of the clusters and concepts onto the features are depicted next to the axes. +Fig. 2. +Mutual information (mean and standard deviation for all repeated +experiments) between the identified concepts and clusters: The concept iden- +tification approach leads to significantly larger MI than the other approaches. +The top shows the results for the two-dimensional data set, the bottom shows +the results for the four-dimensional data set. +Besides the visual assessment, the results were again +evaluated by calculating the MI. In detail, it is calculated how +much information is shared between the concepts and clusters +in both subspaces. Variables X and Y are given as the joint +features f1 and f2, as well as f3 and f4, respectively. As in the +previous experiment, the concept identification method found +groups with the largest MI (Fig. 2), thereby showing that the +chosen approach lead to consistent concepts within the two +subspaces. +C. Three-dimensional energy management configuration data +Last, algorithms were compared on a data set from a +realistic application scenario. +The data set was created by +simulating several thousand building energy management con- +figurations [30]. A common challenge when optimizing such +configurations is to support a decision maker in selecting +an optimal solution given the user’s specific preferences or +requirements. Here, concept identification may be used to +Fig. 3. Silhouette Coefficient (mean and standard deviation for all repeated +experiments) for the identified concepts and clusters: The concept identifi- +cation approach and the clustering methods lead to comparable silhouette +coefficients. The top shows the results for the two-dimensional data set, the +bottom shows the results for the four-dimensional data set. +identify different types or categories of solutions that make +the selection of a preferable solution easier. +The data set was generated using a many-objective evo- +lutionary algorithm that adapted nine different parameters +to achieve optimal configurations, where the quality of an +individual configuration was judged by ten partially conflicting +objectives. One configuration was defined by parameters for +the size, orientation, and inclination of a photovoltaic (PV) +system, the size and operating conditions of a battery energy +storage system, and the size of an integrated combined heat +and power plant (CHP). Each configuration was evaluated +based on various objectives: the first objective was the nec- +essary investment costs for the PV system, the battery, and +the CHP. The second objective was the yearly total costs from +buying electricity from the grid, buying gas to power the CHP, +as well as additional maintenance and operating costs. The +third objective was the resilience of the configuration, referring +in this case to the amount of time, that a building is able to + +K-means +K-means core +GM +GM core +Concepts +10 +10 +10 +10 +10 +f2 +f2 +f2 +f2 +f2 +0 +0 +0 +0 +0 +0 +10 +0 +10 +0 +10 +0 +10 +0 +10 +f1 +f1 +f1 +fi1 +f12D Artificial Data +All samples +K-means +K-means core +GM +GM core +Concepts +0.00 +0.25 +0.50 +0.75 +1.00 +Mutual Information +4D Artificial Data +All samples +K-means +K-means core +GM +GM core +Concepts +0.00 +0.25 +0.50 +0.75 +1.00 +Mutual Information2D Artificial Data +K-means +K-means core +GM +GM core +Concepts +0.0 +0.2 +0.4 +0.6 +Silhouette Coeficient +4D Artificial Data +K-means +K-means core +GM +GM core +Concepts +0.0 +0.2 +0.4 +0.6 +0.8 +Silhouette CoefficientFig. 4. Partitions of the data set into clusters and concepts based on different methods: Four clustering and one concept identification method are applied to +an artificial four-dimensional data set to divide the data into consistent and compact groups. The goal to derive a separation into groups that do not overlap +when the data set is projected onto the two separate subspaces is achieved only by the concept identification approach. The first subspace is given as the +combination of f1 and f2, the second subspace is given as the combination of f3 and f4. The clusters and concepts are represented by the purple, green and +yellow samples. +operate without energy supply from the grid. Note that in the +present data set, the resilience is given as a negative value were +more negative numbers indicate a higher resilience (lower is +better). +Within the resulting data set of the pareto-optimal solutions, +i.e., solutions that each are an optimal trade-off between +different constraints, we applied the concept identification +algorithm and k-means clustering to identify three differ- +ent configuration concepts or clusters. We considered three +features, the initial investment costs, yearly total costs, and +resilience. Additionally, we defined two relevant subspaces, a) +the investment costs, and b) a combination of the expected +yearly total costs for operation and the solutions’ resilience. +The total number of parameters that are optimized for the +concept identification approach hence amounts to NP = 3 · +(1(1+3)/2+2(2+3)/2) = 21. The CMA-ES is again applied +with a population size of 10 and 1000 generations. +As in the previous experiments, both, the clustering +algorithm and the concept identification approach lead to +distinguishable groups with respect to the full data set (Fig. +5). However, when projected into the subspaces of interest, +a significant overlap of the clusters found by k-means was +evident (Fig. 6). With respect to investment costs, all three +clusters overlapped, where the green and yellow clusters +almost covered the same feature range. In the second subspace +also a significant overlap was visible and a unique allocation +of samples to clusters was not possible on the basis of yearly +total costs and resilience. +On the other hand, the concept identification algorithm +identified concepts that were clearly separable within both +spaces. With respect to the first subspace, investment cost, +the purple, green, and yellow concepts represent high, low, +and medium investment solutions, respectively. Clusters were +also unique, i.e., non-overlapping, with respect to the second +subspace, presenting trade-off solutions for yearly total costs +and resilience. Within this second subspace, inferred clusters +comprised solutions that showed an anticorrelated trend, where +higher costs were associated with lower resilience values, +indicating more robust solutions. +One purpose for the identification of concepts in engi- +neering design tasks is the selection of highly representative +instances. Those instances can be understood as concept +archetypes, which, for example, may be used as starting +points or prototypes for further refinement of multiple design +variations. From the configuration concepts, we derived rep- +resentatives by identifying the instance that is closest to the +corresponding concept’s mean with respect to the full data set +(Fig. 6). +In conclusion, the concept identification process provided +the decision maker with different configuration options that +meet the initially defined requirements and thus led to an +interpretable clustering solution. In particular, clusters were +non-overlapping with respect to their investment cost, allowing +for a clear separation into three cost levels. Furthermore, +within each level, solutions followed a trend where higher +operation cost led to higher resilience. For example, choosing +the purple concept allows for realizing configurations with +either very low yearly costs, very good resilience values, +or a good trade-off between the two (at the cost of a high +investment). Choosing the green concept cannot achieve the +same performance with respect to yearly costs and resilience, +is however realizable with a smaller investment budget. The + +K-means +K-means core +GM +GM core +Concepts +10 +10 +10 +10 +10 +f2 +f2 +f2 +f2 +f2 +0 +0 +0 +0 +0 +10 +10 +10 +10 +0 +0 +0 +0 +0 +10 +f1 +f1 +f1 +f1 +f1 +10 +10 +10 +10 +10 +f4 +f4 +4 +f4 +0 +0 +0 +0 +0 +0 +10 +0 +10 +0 +10 +0 +10 +0 +10 +f3 +f3 +f3 +f3 +f3yellow concept is essentially a trade-off concept in between +the other two. We conclude that enforcing non-overlapping +solutions that are also clearly separable within relevant feature +subspaces may aid decision-making processes as solutions +allow for a clearer interpretation in terms of relevant feature +types. +IV. DISCUSSION +The identification of designs sharing similar characteristics +in different meaningful description spaces is a central compo- +nent of the engineering design process. In the present paper, +we propose that this concept identification problem is a special +form of clustering problem, where additional constraints are +imposed on the clustering solution. These constraints are— +besides identifying dense and well-separated clusters—a non- +overlap of clusters and that non-overlapping clusters persist +when only subsets of features are considered. We here demon- +strate that a recently proposed concept identification algorithm +returns such clustering solutions and may be applicable to +problems outside of the engineering design domain. +In three experiments, we show that this recently proposed +concept identification algorithm fulfills the three constraints, +which is not achieved by classical clustering algorithms. We +further introduce the MI as an additional evaluation metric, +next to classical evaluation metrics for clustering solutions, +which specifically tests whether identified solutions comprise +consistent clusters across subspaces. Last, we demonstrate +the application of the concept identification algorithm in a +decision making problem on optimization results from the +energy management domain, where the concept identification +algorithm identified more interpretable clusters and thus eased +the decision-making process. +From the fact that the distribution of identified concepts +demonstrates higher MI values for the two artificial data +sets and the energy management application scenario, we +conclude that the proposed method is well suited to find +concepts with high consistency across defined subspaces. The +method can be set up to identify groups that share similar +properties based on an arbitrary combination of subspaces +(i.e. subsets of features). The experiments further show, that +traditional clustering methods do not lead to solutions with +similar properties. The proposed clusters showed significant +overlap in the subspaces, hence providing no consistent solu- +tion across subspaces. Consistency may be a desirable property +in additional applications, such as cross-domain recommender +systems. +Aside from a visual inspection, the quality of concepts can +be evaluated by estimating the MI between features defining +the subspaces. The MI provides an assessment of consistency, +essentially evaluating how similar the features of sample +groups are in predefined subspaces of the full feature set. In all +experiments, the MI was persistently higher for the solution +returned by the concept identification algorithm, compared to +the clusters found by traditional clustering algorithms. +MI proved to be a good estimate for concept quality +by evaluating consistency across multiple subspaces. This, +however, leads to the question if MI can directly be used to +optimize the distribution of concepts. Instead of evaluating the +performance of a separate concept identification method, one +option would be to implement an estimation of the MI as +the evaluation function of a concept optimization algorithm. +Such a process leads to groups of data samples that are op- +timized based on their consistency across multiple subspaces. +However, the found groups are likely to be indistinguishable +as concepts—while a process that optimizes MI across mul- +tiple feature sets can assure consistency, it neglects the two +other requirements for reasonable clusters: compactness and +uniqueness (i.e. non-overlapping groups). An assignment of +samples into highly overlapping groups is not suitable for the +identification of concepts, as the groups are likely to be too +similar to function as distinct alternatives. The benefit for the +user would hence be very limited. +While the identification process provides the user with +consistent clusters, the choice of subspaces in the first place +remains a challenging task. A sensible split of features into +separate subspaces requires in-depth domain knowledge, aside +from a thorough understanding of the application domain. The +choice of subspaces has a high influence on the potential +identification outcome and defines the constitution of a concept +in the particular task. +Another topic that allows for further research is the choice +of concept representatives. In many application scenarios, +the user wants to select a manageable amount of individual +samples from each concept (which can consist of several +thousands of samples or more). Depending on the design task, +various choices are reasonable. If, for example, the concept +identification is part of an iterative optimization process, it is +sensible to pick archetypal configurations from each concept +as prototypes for further optimization steps. Samples that are +closest to the mean of the full feature vector can be understood +as archetypes for each concept. But a selection of representa- +tives based on fewer features, or an entire different criterion +can be equally reasonable and depends on the engineering task. +V. CONCLUSION +Concept identification, originally proposed in the engineer- +ing design domain, may be understood as a special form +of clustering problem. Thus, concept identification algorithms +may be applied as clustering algorithms with properties that +are relevant to application domains other than engineering +design. In particular, solutions comprise a consistent clustering +of instances across the full feature space and a-priori defined +subspaces that are relevant to the specific problem. We demon- +strate how concept identification solutions differ from classical +clustering solutions on two artificial data sets, designed to +illustrate these differences. Further, we introduce the Mutual +Information (MI) as an evaluation criterion of the concept +identification algorithm. The MI quantifies this consistency +across subspaces for a given clustering solution. We conclude +that the MI is a suitable metric to compare various clustering +approaches, when consistency across subspaces is required. +Last, we show how concept identification algorithms may be + +Fig. 5. Comparison of clusters found by the k-means algorithm and the identified concepts: The data set of energy management configuration is shown with +respect to three objectives. The three clusters and concepts are marked in purple, green, and yellow. In both cases, the groups are distinguishable when all +three features are considered as the basis for distinction. +used in application domains other than engineering design, +and successfully apply the algorithm in a decision-making task +from energy management optimization. +In three experiments we demonstrated that a) concept +identification leads to consistent clusters in selectable sub- +spaces and b) common clustering algorithms do not have this +property. From this, we further conclude that concept identi- +fication can generally be understood as a clustering technique +that not only generates compact and clearly distinguishable +groups, but also allows to integrate a third clustering objective, +namely, ensuring the consistency of the groups with respect +to predefined subspaces of the data. +In conclusion, we propose concept identification algorithms +as a suitable tool to generate clustering solutions with desirable +properties, such as increased interpretability and explainability. +We further introduce Mutual Information as a suitable metric +to assess these properties. +REFERENCES +[1] F. Lanfermann, S. Schmitt, and S. Menzel, “An Effective Measure to +Identify Meaningful Concepts in Engineering Design optimization,” in +2020 IEEE Symposium Series on Computational Intelligence (SSCI). +IEEE, dec 2020, pp. 934–941. +[2] F. Lanfermann and S. Schmitt, “Concept identification for complex +engineering datasets,” Advanced Engineering Informatics, vol. 53, p. +101704, 2022. +[3] M. M. Khan, R. Ibrahim, and I. Ghani, “Cross domain recommender sys- +tems: a systematic literature review,” ACM Computing Surveys (CSUR), +vol. 50, no. 3, pp. 1–34, 2017. +[4] T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical +Learning, 2nd ed. +New York: Springer, 2009. +[5] Y. Liu, Z. Li, H. Xiong, X. Gao, and J. Wu, “Understanding of internal +clustering validation measures,” in 2010 IEEE International Conference +on Data Mining. +IEEE, 2010, pp. 911–916. +[6] L. Parsons, E. Haque, and H. Liu, “Subspace clustering for high +dimensional data,” ACM SIGKDD Explorations Newsletter, vol. 6, no. 1, +pp. 90–105, jun 2004. +[7] S. Bickel and T. Scheffer, “Multi-View Clustering,” in Fourth IEEE +International Conference on Data Mining (ICDM’04). +IEEE, 2004, +pp. 19–26. +[8] P. Tan and V. Kumar, “Interestingness Measures for Association Pat- +terns: A Perspective,” in KDD Workshop on Postprocessing in Machine +Learning and Data Mining, 2000. +[9] L. Graening and B. Sendhoff, “Shape Mining: A Holistic Data Mining +Approach for Engineering Design,” Advanced Engineering Informatics, +vol. 28, no. 2, pp. 166–185, 2014. +[10] J. MacQueen, “Some methods for classification and analysis of multi- +variate observations,” Proceedings of the Fifth Berkeley Symposium on +Mathematical Statistics and Probability, vol. 1, pp. 281—-297, 1967. +[11] D. Arthur and S. Vassilvitskii, “k-means++: The Advantages of Careful +Seeding,” Proceedings of the eighteenth annual ACM-SIAM symposium +on Discrete algorithms, pp. 1027–1035, 2007. +[12] J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algo- +rithms. +Boston, MA: Springer US, 1981. +[13] C. E. Rasmussen, “The infinite Gaussian mixture model,” Advances in +Neural Information Processing Systems, no. 1, pp. 554–559, 2000. +[14] K. Sim, V. Gopalkrishnan, A. Zimek, and G. Cong, “A survey on +enhanced subspace clustering,” Data Mining and Knowledge Discovery, +vol. 26, no. 2, pp. 332–397, 2013. +[15] F. Ntelemis, Y. Jin, and S. A. Thomas, “Information maximization +clustering via multi-view self-labelling,” Knowledge-Based Systems, p. +109042, 2022. +[16] Y. Yang and H. Wang, “Multi-view clustering: A survey,” Big Data +Mining and Analytics, vol. 1, no. 2, pp. 83–107, jun 2018. +[17] X. Cai, F. Nie, and H. Huang, “Multi-View K-Means Clustering on +Big Data Xiao,” in Proceedings of the Twenty-Third International Joint +Conference on Artificial Intelligence, sep 2013. +[18] H. Gao, F. Nie, X. Li, and H. Huang, “Multi-view Subspace Clustering,” +in 2015 IEEE International Conference on Computer Vision (ICCV). +IEEE, dec 2015, pp. 4238–4246. +[19] P. J. Rousseeuw, “Silhouettes: A graphical aid to the interpretation and +validation of cluster analysis,” Journal of Computational and Applied +Mathematics, vol. 20, no. C, pp. 53–65, 1987. +[20] C. E. Shannon, “A mathematical theory of communication,” The Bell +System Technical Journal, vol. 27, pp. 379–423, 1948. + +Clusters +Concepts +0 +0 +Resilience +Resilience +2000 +2000 +4000 +4000 +6000 +6000 +0 +0 +-8000 +8000 +200000 +200000 +vestment +400000 +340000 +400000 +340000 +335000 +335000 +330000 +330000 +600000 +600000 +325000 +325000 +4SO +Cost +320000 +320000 +ly total costs +cly total costs +S' +LS +Veay +YearlFig. 6. Comparison of clusters found by the k-means algorithm and the identified concepts in the two subspaces of interest: The distribution of the three +clusters and concepts (marked purple, green, and yellow) are shown in the one-dimensional space of investment costs. The box indicates the 25-75%-percentile +and the mean, the whiskers show the minimum and maximum values. In the two-dimensional space of resilience over yearly total costs, the groups are shown +as the corresponding kernel-density estimation in addition to the convex hull represented by a dashed line. While there is a significant overlap visible between +the found clusters, the concept identification leads to a non-overlapping distribution of concepts in both spaces. The stars mark selected concept representatives. +[21] D. L. Davies and D. W. Bouldin, “A cluster separation measure,” IEEE +Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI- +1, no. 2, pp. 224–227, 1979. +[22] D. J. C. MacKay, Information Theory, Inference, and Learning Algo- +rithms. +Cambridge, UK: Cambridge University Press, 2005. +[23] A. Kraskov, H. St¨ogbauer, and P. Grassberger, “Estimating mutual +information,” Physical Review E, vol. 69, no. 6, p. 16, 2004. +[24] P. Wollstadt, J. T. Lizier, R. Vicente, C. Finn, M. Mart´ınez-Zarzuela, +P. A. M. Mediano, L. Novelli, and M. Wibral, “IDTxl: The Information +Dynamics Toolkit xl: a Python package for the efficient analysis +of multivariate information dynamics in networks,” Journal of Open +Source Software, vol. 4, no. 34, p. 1081, 2019. [Online]. Available: +https://github.com/pwollstadt/IDTxl +[25] J. T. Lizier, “JIDT: An information-theoretic toolkit for studying the +dynamics of complex systems,” Frontiers in Robotics and AI, vol. 1, +p. 11, 2014. +[26] L. Paninski, “Estimation of entropy and mutual information,” Neural +Computation, vol. 15, no. 6, pp. 1191–1253, 2003. +[27] R. Vicente, M. Wibral, M. Lindner, and G. Pipa, “Transfer entropy- +a model-free measure of effective connectivity for the neurosciences,” +Journal of Computational Neuroscience, vol. 30, no. 1, pp. 45–67, 2011. +[28] N. Hansen, The CMA Evolution Strategy: A Comparing Review. Berlin, +Heidelberg: Springer Berlin Heidelberg, 2006, pp. 75–102. +[29] N. Hansen and A. Ostermeier, “Completely Derandomized Self- +Adaptation in Evolution Strategies,” Evolutionary Computation, vol. 9, +no. 2, pp. 159–195, 2001. +[30] Q. Liu, F. Lanfermann, T. Rodemann, and Y. Jin, “Surrogate-assisted +many-objective optimization of building energy management,” in revi- +sion. + +0 +1 +2 +5000 +3 +0 +500000 +320000 +325000 +330000 +335000 +340000 +Yearly total costs +Investment costs +0 +Resilience +2 +5000 +3 +0 +500000 +320000 +325000 +330000 +335000 +340000 +Yearly total costs +Investment costs \ No newline at end of file diff --git a/_NE5T4oBgHgl3EQfSA5Z/content/tmp_files/load_file.txt b/_NE5T4oBgHgl3EQfSA5Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..269a3467d4a2cdb19c11f921207502132790ec72 --- /dev/null +++ b/_NE5T4oBgHgl3EQfSA5Z/content/tmp_files/load_file.txt @@ -0,0 +1,567 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf,len=566 +page_content='Understanding Concept Identification as Consistent Data Clustering Across Multiple Feature Spaces Felix Lanfermann Honda Research Institute Europe Offenbach, Germany felix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='lanfermann@honda-ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='de Sebastian Schmitt Honda Research Institute Europe Offenbach, Germany sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='schmitt@honda-ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='de Patricia Wollstadt Honda Research Institute Europe Offenbach, Germany patricia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='wollstadt@honda-ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='de Abstract—Identifying meaningful concepts in large data sets can provide valuable insights into engineering design problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Concept identification aims at identifying non-overlapping groups of design instances that are similar in a joint space of all features, but which are also similar when considering only subsets of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' These subsets usually comprise features that characterize a design with respect to one specific context, for example, constructive design parameters, performance values, or operation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' It is desirable to evaluate the quality of design concepts by considering several of these feature subsets in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In particular, meaningful concepts should not only identify dense, well separated groups of data instances, but also provide non-overlapping groups of data that persist when considering pre-defined feature subsets separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In this work, we propose to view concept identification as a special form of clustering algorithm with a broad range of potential applications beyond engineering design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' To illustrate the differences between concept identification and classical clustering algorithms, we apply a recently proposed concept identification algorithm to two synthetic data sets and show the differences in identified solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In addition, we introduce the mutual information measure as a metric to evaluate whether solutions return consistent clusters across relevant subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' To support the novel understanding of concept identification, we consider a simulated data set from a decision-making problem in the energy management domain and show that the identified clusters are more interpretable with respect to relevant feature subsets than clusters found by common clustering algorithms and are thus more suitable to support a decision maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Index Terms—concept identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' clustering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' decision- making;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' interpretability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' INTRODUCTION In the engineering design domain, the identification of design concepts is a central task that identifies groups of engineering designs that share similar characteristics, typically in terms of their specification, but also in terms of performance or other types of descriptions, such as operation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' When identifying concepts, it is relevant that identified groups of designs are similar with respect to all describing features, but that this similarity is also preserved when considering a subset of feature types in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' For example, designs within a concept should be highly similar with respect to all characteristics, but designs should also be similar in terms of only their specifications or only their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' This not only creates insight into the relation between the distribution of design groups and their corresponding quality criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' It is further relevant for identifying highly representative instances of design concepts, also termed archetypes, which may then be used as a starting point for further refinement of specifications, which lead to consistent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Concept identification may be viewed as a special form of clustering problem that aims at identifying non-overlapping groups of instances that have high similarity in a joint feature space, but which persist when considering relevant subspaces of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' A subspace is here defined as an arbitrary subset of features that defines an instance in a specific context or do- main, into which instances are projected from the joint feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Hence, concept identification may be approached using a clustering algorithm with specific constraints on the returned solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In particular, the solution should a) comprise dense clusters in the joint space of all features, b) the clusters should be non-overlapping, which can only be achieved for some data sets by not assigning all instances to a cluster, and c), non- overlapping clusters should persist in relevant subspaces of the joint space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In other words, a suitable algorithm should return non-overlapping clusters in the joint feature space and instances that are assigned to a cluster in the joint space should be assigned to the same cluster in arbitrary, a-priori defined subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We term this later property consistency of clusters across different feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' As a result of this consistency constraint, instances being grouped within the same cluster, show high similarity both in the joint and in the individual subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' This further leads to also a high similarity between subspaces for instances assigned to concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We here evaluate a recently proposed algorithm based on evolutionary optimization that was introduced to perform concept identification in the engineering design domain [1], [2], in terms of a general clustering algorithm which we believe to be relevant also to other application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In particular, we show that the concept identification algorithm returns clustering solutions with the properties introduced above and compare them to solutions returned by classical clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We highlight the differences between the two approaches, in particular, the ability of the concept identification algorithm to identify consistent clusters across arbitrary, predefined subspaces, a property that may be relevant also in other application domains, for example, cross-domain recommender systems [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We here demonstrate such an application to a decision making problem from the energy management domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='05525v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='LG] 13 Jan 2023 We evaluate the concept identification algorithm’s ability to return solutions according to the three constraints defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' To evaluate the algorithm’s ability to identify clusters in subspaces with high similarity between clusters, as enforced by the consistency constraint, we use the mutual information (MI) to evaluate returned solutions, since traditional metrics for evaluating clustering solutions do not account for this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The remainder of this paper is structured as follows: In Section II, we review prior art in clustering and con- cept identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We then briefly introduce the algorithm evaluated here and highlight how it differs from existing clustering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Last, we introduce the MI for evaluation of clustering solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In Section III, we apply the algorithm to three artificial data sets to demonstrate the properties of clustering solutions returned by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The first two experimental data sets are designed to highlight the algorithm’s ability to identify dense, non-overlapping clusters that are consistent across subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The third data set is generated through high-fidelity simulation and illustrates a use case for the algorithm in a decision-making problem from the energy management domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In Section IV, we discuss our findings and potential relevant application scenarios of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We close in Section V with a conclusion and an outlook on future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Concept identification as an extended form of clustering In short, there are three objectives that should be met by a suitable solution for concept identification, namely a) compactness of clusters, b) no overlap between clusters, and c) a consistency of clusters that persists in arbitrary subspaces of the joint clustering space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The first two objectives are (partially) optimized by classical clustering algorithms [4], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=', algorithms that find a “grouping” or assignment of samples to “clusters” that results in a low variance or compactness within clusters and a high separation between clusters [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Hence, existing clustering algorithms have objectives that are related to those of concept identification, but differ in one aspect: Clustering does not ensure the third objective of consistent clusters in arbitrary subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' There are extensions of the classical clustering objective, which add additional constraints on clustering solutions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=', subspace [6] or multi- view clustering [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, none of these approaches return solutions that are suitable to solve the problem of concept identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We will discuss these existing approaches and their difference to concept identification below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Before we discuss the relationship between concept identi- fication and clustering in the next subsection, we will briefly review existing concept identification approaches and describe the algorithm, originally introduced in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Concept identification algorithms typically define a quality metric to asses whether cluster assignments in the chosen subspaces follow the constraints defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The quality can, for example, be determined by calculating the interesting- ness and significance (IS) [8] of assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' This produces a numerical, therefore comparable metric to compare different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Another approach to assess the quality of concepts via a quality metric is used in multi-objective optimization [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Here, concepts are judged by how well they cover the pareto- front of optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In particular, the hyper-volume of the concepts is used to judge their quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, this approach is only valid to assess concepts in data generated by multi-objective optimizations and is not generally applicable to data sets from other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Although both measures provide a useful estimate of con- cept quality, they cannot be used within a concept identifica- tion algorithm that is applied to return solutions according to the constraints described above, as these measures lack the ability to evaluate multiple concepts simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The measures cannot explicitly consider and penalize overlap of concepts within the subspaces and do not evaluate a concept’s extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Identifying concepts by optimizing the referenced metrics can therefore lead to trivial solutions given as either indistinguishable concepts that span the entire data set (as this would be an optimal assignment), or as insignificantly small concepts containing a single point of data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The identification of concepts is nevertheless possible by penalizing concept overlap in an arbitrary number of sub- spaces, as well as very large and very small concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' A concept quality metric that integrates both of these constraints is proposed by [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Here, a data set D = {xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=', ND} (1) with ND samples is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Each sample x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' , xNS, x∆)T (2) is represented by features that are divided into NS subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Any features that are not associated with a subspace are denoted by x∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' For every concept α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' , NC, the approach defines a connected region Cα,k in each subspace k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' , NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' All samples that lie within this region are considered candidates for the respective concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' A concept is then defined as the term Cα = � x ∈ NS � k=1 Cα,k ���x ̸∈ Cβ,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' ∀k, β ̸= α � , (3) where k, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' , NS enumerate the feature subspaces and α, β = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' , NC enumerate the concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The concept Cα therefore only contains those samples, that lie within all regions {Cα,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' , Cα,NS}, but not within any of the regions {Cβ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' , Cβ,NS} associated with any other concept β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The approach proposes a concept quality metric (CQM) Q = NC � α Qα , (4) where the individual metric Qα for each concept is defined as Qα = �NS � k NS � |Cα| |Cα,k| � F �|Cα| ND , s � F �|Pα| |P| , p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' (5) For the second and third term in (5), a scaling function F(x, y) = � � � � � � � � � � � � 1 − � x−y y �2 , if x < y, 1, if y < x < 1 − y, � 1 − � x−1+y y �2 , if x > 1 − y (6) is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The second term penalizes concepts that contain fewer or more samples of the data set than a predefined value s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' A optional set of samples P can be defined to further steer the identification process towards user-preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' As this option was not used in the conducted experiments in section III, we refer to [2] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The CQM in (4) quantifies the quality of a given distribution of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' It can therefore be used to identify an optimal set of concepts by maximizing the CQM in a numerical optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' One option is to apply the metric as the evaluation function in an evolutionary optimization framework as suggested by [1], where regions of potential concept candidates are adapted to find an optimal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In an n-dimensional subspace of the dataset, such a region can be represented by one n- dimensional hyper-ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The hyper-ellipsoid has NP,S = n(n + 3)/2 (7) parameters (for each n-dimensional subspace) which need to be optimized by the evolutionary search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The approach from [2] is implemented and used within the experimental evaluations in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Difference to existing clustering approaches As introduced above, clustering algorithms in their most general form target a grouping of data that yields compact and well-separated clusters [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Well-known examples of such approaches are k-means [10] or k-means++ [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Both approaches divide all data samples into a predefined number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, solutions are not necessarily non-overlapping, and clusters found in the joint feature space do not lead to well-defined clusters in the defined subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Also, these clustering approaches assign all instances to a cluster, which is not a requirement for concept identification and may even prevent the finding of a suitable solution [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Fuzzy c-means [12] and Gaussian mixture models [13] allow for a less strict allocation of samples to clusters and thus relax the strict assignment of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, these algorithms do also not lead to consistent clusters across sub- spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The approaches cannot identify corresponding groups in projections of the full feature vector simultaneously, in which each cluster in one projection corresponds to one cluster in all other projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' These clustering approaches are thus not suited to solve the defined problem, as they cannot be easily extended or adapted to realize the required constraints on the solutions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In contrast to the traditional clustering approaches discussed above, subspace clustering [6] identifies clusters in lower- dimensional projections when applied on high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' This is particularly beneficial when aiming at data-driven dimensionality reduction and mitigating sparsity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Nonetheless, the user cannot predetermine the subspaces, into which instances are projected in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, such an alignment of the projections within user-defined subspaces would be necessary to solve concept identification prob- lems, where subspaces represent specific domains or types of features and thus have semantic meaning that should be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' For this reason, and since subspace clustering might (intentionally) lead to overlapping clusters [14], the approach is not applicable for the defined problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Information maximization clustering (IMC) “simultaneously learns discriminative representations while generating labels in an unsupervised mode” [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Similar to subspace clustering, the representations are learned within the process and no a- priori defined subspaces are taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The desired consistency of groups across subspaces can hence also not be assured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The listed approaches may return compact clusters that are separable with respect to the considered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, consistency of the clusters with respect to multiple subspaces of the features is not achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' There are, however, other approaches that integrate a notion of consistency into the clus- tering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Often, a data set is constructed from different “views” on the data, where a view can be understood as a group of connected features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The heterogeneous properties of the data might hold a potential connection [16] and the views provide complementary information about the data which can be exploited [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Here, a view may be understood as a sub- space, such that—in accordance with our constraints defined above—multi-view clustering [7] maximizes a cluster quality metric within each view, while accounting for clustering con- sistency across different views [16] (subspaces, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' High consistency can, for example, be achieved by alternating the maximization and expectation step for each view of the data [7], or optimizing a weight factor for each view in a k- means-based clustering loss-function [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Combining multi- view and subspace clustering can uncover clusters in concealed subspaces in multi-view data [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, multiple views are utilized in the process to optimize clustering performance, not to achieve highest possible consistencies across all views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Consistency is factored in as a trade-off relation, but overlap- ping clusters in individual views are not fully avoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' This is, nevertheless, a central requirement in concept identification and renders the usage of multi-view clustering for concept identification difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Evaluation of consistency with mutual information As introduced in the last subsection, the concept identifica- tion algorithm evaluated here differs from existing clustering algorithms by returning strictly non-overlapping clusters and by ensuring consistency of clusters in subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' To evaluate whether these objectives are met, we compare the algorithm to existing clustering algorithms and evaluate solutions by, first, calculating the Silhouette Coefficient [19] to demonstrate that the concept identification algorithm returns compact clusters, with a similar separation than found in clusters returned by existing clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Second, we apply the mutual information (MI), a measure from information theory [20], to evaluate the consistency criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Note that existing metrics for validating clustering solutions quantify compactness and separation, the objectives of classical clustering algorithms [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Examples of these metrics are the Silhouette Coefficient as used here or the Davies-Bouldin index [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, these metrics are not suitable to judge the third criterion of concept identification, the consistency of clusters across subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The central difference between concept identification and regular clustering algorithms is the enforced consistency of clusters between the joint and all a-priori defined subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' As a result of this consistency, instances that are assigned to a cluster show a high similarity in the joint space, but also in all subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' This similarity within subspaces should lead to a high similarity also between subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In other words, when considering only one subspace, knowing that an instance is assigned to a specific cluster, is informative about the assignment of that instance in all other subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Accordingly, the properties of an instance in one subspace are informative about the properties of that instance in another subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Hence, we here evaluate the similarity between subspaces that is enforced by the consistency of clusters, using the MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' MI quantifies how much information one variable provides about a second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Thus, it can be used to measure how infor- mative an instance’s value in one subspace is about the same instance’s value in another subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Hence, we calculate the MI between targeted subspaces for all instances assigned to a cluster, to quantify if the algorithm satisfies our requirement for producing clusters where instances behave similarly in subspaces of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The MI quantifies the amount of information one random variable, X, provides about a second random variable, Y , [22] as I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Y ) = � x,y p(x, y) log p(x|y) p(x) = � x,y p(x, y) log p(x, y) p(x)p(y), (8) where we write p(x) as a shorthand for the probability p(X = x) of variable X taking on the value x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The MI may be interpreted as the Kullback–Leibler divergence between the product of the marginal distributions, p(x)p(y), and the joint distribution, p(x, y), and thus measures the deviation of the two variables’ joint distribution from statistical independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We estimate the MI using a nearest-neighbor-based estima- tor for continuous data proposed by [23] and implemented in the IDTxl Python toolbox [24], which uses an estimator from the JIDT toolbox [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Note that it is a well-known problem that MI estimates from finite data suffer from estimation bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=', [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' One approach to handle estimation bias in continuous estimators is the use of permutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Here, we generate a Null-distribution representing the Null- hypothesis of X and Y being statistically independent by repeatedly shuffling one of the two variables and estimating the MI from this shuffled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The original estimate is then compared against this distribution and a p-value is calculated as the fraction of estimates from shuffled data that are higher than the original estimate (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=', [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' EXPERIMENTS A set of experiments was conducted to illustrate and quantify the differences between concept identification and clustering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In each experiment, various clustering techniques, as well as the introduced concept identification algorithm were applied to a data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The first two experi- ments were conducted on artificial data sets to illustrate the specific properties of solutions returned by the applied concept identification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The third experiment compares the results of one clustering technique and the referenced concept identification approach on an application data set from the field of energy management in a decision-making problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Table I provides an overview of the utilized data sets and clustering preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' TABLE I DETAILS OF THE EXPERIMENTAL DATA Experiment 2D 4D 3D data source artificial artificial building simulations # data samples 34000 30000 20699 # subspaces NS 2 2 2 # total features 2 4 3 # features per space 1, 1 2, 2 1, 2 # clusters / concepts NC 3 3 3 # opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' parameters NP 12 30 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Two-dimensional artificial data To illustrate the main differences between the solutions given by traditional clustering techniques and the evaluated concept identification algorithm, a simple synthetic data set, composed of 34000 samples, was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Each point of data is represented by two features, f1 and f2, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The data set was uniformly sampled from an area given by A = � � � � � 0 ≤ f1 < 4, if 0 ≤ f2 < 4, 4 ≤ f1 < 10, if 0 ≤ f2 < 6, 6 ≤ f1 < 10, if 6 ≤ f2 < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' (9) To demonstrate the robustness of the approaches, the exper- iment was repeated 20 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' According to the three constraints defined for the concept identification problem, a solution should comprise clusters in the joint space, (f1, f2), which are also clearly separable based on subspaces f1 and f2, individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' It is therefore necessary to find three consistent groups of samples with respect to both feature f1 and f2, individually, and to the joint feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The groups should not overlap when considering either sub- or the joint space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Four clustering methods were applied to the two- dimensional data set: k-means, Gaussian mixture models (GMM), as well as a modification of each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Since the concept identification approach does not require all samples to be associated with a concept, this might create an unfair ad- vantage for this method compared to clustering algorithms that always assign all instances to a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Hence, in the modified clustering methods, only those samples were considered to be part of the cluster, that were closest to the respective cluster center1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Next, we applied the concept identification algorithm to find groups of samples in the joint spaces, that are consistent within subspaces f1 and f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Since we aimed at identifying three groups in two one- dimensional subspaces, the total number of parameters that needed to be modified by the optimization amounts to NP = 3 · (1(1 + 3)/2 + 1(1 + 3)/2) = 12, according to (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' As the optimization algorithm, a covariance matrix adaptation evolutionary strategy (CMA-ES) [28], [29] was used with a population size of 10 for 1000 generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The different methods lead to very different partitions of the data set (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' All methods found valid clusters, but only the concept identification method identified groups that meet the required conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' If projected onto one of the features, all traditional clustering solutions demonstrated significant overlap between the clusters, showing that none of the clustering algorithms returned solutions that led to consistent clusters in considered subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The results were further evaluated by calculating the MI between both features, f1 and f2, while considering only those samples that were assigned to a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Between the clustering approaches, the modified versions of k-means and GMMs led to higher levels of MI, compared to all other traditional clus- tering algorithms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, the solution found by the concept identification algorithm showed a significantly higher MI than all other solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Since MI provides a quantitative measure of consistency across subspaces, we conclude that the concept identification algorithm is suited best for assigning instances to clusters such that instances behave highly similar across subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' To investigate whether the identified concepts also satisfy the need for compactness and separation imposed on common clustering methods, we calculated the silhouette coefficients for all results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Since the clustering method and the 1From the regular k-means method, the maximum inner-cluster distance was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' For the modified k-means approach, only those samples, that lay within a radius of 20% of this maximum distance to the center, were assigned to the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' For the modified GMM approach, only those samples, that are allocated to the original groups with a probability rate of larger than 90%, are assigned to the cluster concept identification approach lead to comparable scores, we follow that the requirement to produce compact and separable groups is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We further emphasize that the concept identifi- cation approach leads to non-overlapping groups by definition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Four-dimensional artificial data The identified concepts resulting from the first experiment might look trivial at first glance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In a one-dimensional sub- space, the elliptic representation describes each concept by a mean value and a radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The distribution of concepts is hence given as a direct segmentation of the one-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Illustrating the clusters in the (full) two-dimensional data set then naturally creates a simple grid structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In data sets where the subspaces of interest contain more than one dimension, the results are less trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' To demonstrate this effect, we conducted a second experiment involving a second artificial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The data set contained 30000 samples, each described by four features f1, f2, f3, and f4, and was created by sampling from the normal distribution N(µi, σ2) around three centers µ1 = (0, 0, 0, 0)T , µ2 = (10, 10, 10, 10)T , and µ3 = (10, 10, 0, 0)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Again, the same four clustering approaches and the concept identification method were applied to identify three groups of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The modified k-means and GMM again only consider those samples to be part of the cluster, that are closest2 to the cluster center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' On the concept identification algorithm, we again imposed the constraint that groups should be consistent within two subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The first subspace was defined as a combination of features f1 and f2, while the second subspace comprised f3 and f4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' An ideal partition would lead to groups that do not overlap when projected onto the subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' For the concept identification approach, each concept is again represented by the combination of one ellipse for each of the two subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' According to (7), the total number of parameters to be optimized amounts to NP = 3·(2(2+3)/2+ 2(2 + 3)/2) = 30, since both subspaces are two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The algorithm—again a CMA-ES with a population size of 10 and 1000 generations—hence optimizes the concept distribution by adapting in total 30 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The solutions given by all clustering methods show sub- stantial overlap in the individual subspaces (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' All clustering approaches identify sample groups that are dis- tinguishable within the full four-dimensional data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The cluster centers are close to the means µ1 = (0, 0, 0, 0)T , µ2 = (10, 10, 10, 10)T , and µ3 = (10, 10, 0, 0)T that were used to create the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, when projected into the subspaces, two of the three clusters are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In contrast, the groups discovered by the concept identification approach (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' the concepts) do not overlap within the sub- spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 2From the regular k-means and GMM approaches, the maximum inner- cluster distance is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' For the modified approaches, only those samples, that lie within a radius of 10% of this maximum distance to the center, are assigned to the cluster Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Partitions of the data set into clusters and concepts based on different methods: Four clustering and one concept identification method are applied to an artificial two-dimensional data set to divide the data into consistent and compact groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The goal to derive a separation into groups that do not overlap when the data set is projected onto one a single feature is achieved only by the concept identification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The clusters and concepts are represented by the purple, green and yellow samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The projections of the clusters and concepts onto the features are depicted next to the axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Mutual information (mean and standard deviation for all repeated experiments) between the identified concepts and clusters: The concept iden- tification approach leads to significantly larger MI than the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The top shows the results for the two-dimensional data set, the bottom shows the results for the four-dimensional data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Besides the visual assessment, the results were again evaluated by calculating the MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In detail, it is calculated how much information is shared between the concepts and clusters in both subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Variables X and Y are given as the joint features f1 and f2, as well as f3 and f4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' As in the previous experiment, the concept identification method found groups with the largest MI (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 2), thereby showing that the chosen approach lead to consistent concepts within the two subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Three-dimensional energy management configuration data Last, algorithms were compared on a data set from a realistic application scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The data set was created by simulating several thousand building energy management con- figurations [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' A common challenge when optimizing such configurations is to support a decision maker in selecting an optimal solution given the user’s specific preferences or requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Here, concept identification may be used to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Silhouette Coefficient (mean and standard deviation for all repeated experiments) for the identified concepts and clusters: The concept identifi- cation approach and the clustering methods lead to comparable silhouette coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The top shows the results for the two-dimensional data set, the bottom shows the results for the four-dimensional data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' identify different types or categories of solutions that make the selection of a preferable solution easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The data set was generated using a many-objective evo- lutionary algorithm that adapted nine different parameters to achieve optimal configurations, where the quality of an individual configuration was judged by ten partially conflicting objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' One configuration was defined by parameters for the size, orientation, and inclination of a photovoltaic (PV) system, the size and operating conditions of a battery energy storage system, and the size of an integrated combined heat and power plant (CHP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Each configuration was evaluated based on various objectives: the first objective was the nec- essary investment costs for the PV system, the battery, and the CHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The second objective was the yearly total costs from buying electricity from the grid, buying gas to power the CHP, as well as additional maintenance and operating costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The third objective was the resilience of the configuration, referring in this case to the amount of time, that a building is able to K-means K-means core GM GM core Concepts 10 10 10 10 10 f2 f2 f2 f2 f2 0 0 0 0 0 0 10 0 10 0 10 0 10 0 10 f1 f1 f1 fi1 f12D Artificial Data All samples K-means K-means core GM GM core Concepts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='00 Mutual Information 4D Artificial Data All samples K-means K-means core GM GM core Concepts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='00 Mutual Information2D Artificial Data K-means K-means core GM GM core Concepts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='6 Silhouette Coeficient 4D Artificial Data K-means K-means core GM GM core Concepts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='8 Silhouette CoefficientFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Partitions of the data set into clusters and concepts based on different methods: Four clustering and one concept identification method are applied to an artificial four-dimensional data set to divide the data into consistent and compact groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The goal to derive a separation into groups that do not overlap when the data set is projected onto the two separate subspaces is achieved only by the concept identification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The first subspace is given as the combination of f1 and f2, the second subspace is given as the combination of f3 and f4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The clusters and concepts are represented by the purple, green and yellow samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' operate without energy supply from the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Note that in the present data set, the resilience is given as a negative value were more negative numbers indicate a higher resilience (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Within the resulting data set of the pareto-optimal solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=', solutions that each are an optimal trade-off between different constraints, we applied the concept identification algorithm and k-means clustering to identify three differ- ent configuration concepts or clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We considered three features, the initial investment costs, yearly total costs, and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Additionally, we defined two relevant subspaces, a) the investment costs, and b) a combination of the expected yearly total costs for operation and the solutions’ resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The total number of parameters that are optimized for the concept identification approach hence amounts to NP = 3 · (1(1+3)/2+2(2+3)/2) = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The CMA-ES is again applied with a population size of 10 and 1000 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' As in the previous experiments, both, the clustering algorithm and the concept identification approach lead to distinguishable groups with respect to the full data set (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, when projected into the subspaces of interest, a significant overlap of the clusters found by k-means was evident (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' With respect to investment costs, all three clusters overlapped, where the green and yellow clusters almost covered the same feature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In the second subspace also a significant overlap was visible and a unique allocation of samples to clusters was not possible on the basis of yearly total costs and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' On the other hand, the concept identification algorithm identified concepts that were clearly separable within both spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' With respect to the first subspace, investment cost, the purple, green, and yellow concepts represent high, low, and medium investment solutions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Clusters were also unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=', non-overlapping, with respect to the second subspace, presenting trade-off solutions for yearly total costs and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Within this second subspace, inferred clusters comprised solutions that showed an anticorrelated trend, where higher costs were associated with lower resilience values, indicating more robust solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' One purpose for the identification of concepts in engi- neering design tasks is the selection of highly representative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Those instances can be understood as concept archetypes, which, for example, may be used as starting points or prototypes for further refinement of multiple design variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' From the configuration concepts, we derived rep- resentatives by identifying the instance that is closest to the corresponding concept’s mean with respect to the full data set (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In conclusion, the concept identification process provided the decision maker with different configuration options that meet the initially defined requirements and thus led to an interpretable clustering solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In particular, clusters were non-overlapping with respect to their investment cost, allowing for a clear separation into three cost levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Furthermore, within each level, solutions followed a trend where higher operation cost led to higher resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' For example, choosing the purple concept allows for realizing configurations with either very low yearly costs, very good resilience values, or a good trade-off between the two (at the cost of a high investment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Choosing the green concept cannot achieve the same performance with respect to yearly costs and resilience, is however realizable with a smaller investment budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The K-means K-means core GM GM core Concepts 10 10 10 10 10 f2 f2 f2 f2 f2 0 0 0 0 0 10 10 10 10 0 0 0 0 0 10 f1 f1 f1 f1 f1 10 10 10 10 10 f4 f4 4 f4 0 0 0 0 0 0 10 0 10 0 10 0 10 0 10 f3 f3 f3 f3 f3yellow concept is essentially a trade-off concept in between the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We conclude that enforcing non-overlapping solutions that are also clearly separable within relevant feature subspaces may aid decision-making processes as solutions allow for a clearer interpretation in terms of relevant feature types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' DISCUSSION The identification of designs sharing similar characteristics in different meaningful description spaces is a central compo- nent of the engineering design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In the present paper, we propose that this concept identification problem is a special form of clustering problem, where additional constraints are imposed on the clustering solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' These constraints are— besides identifying dense and well-separated clusters—a non- overlap of clusters and that non-overlapping clusters persist when only subsets of features are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We here demon- strate that a recently proposed concept identification algorithm returns such clustering solutions and may be applicable to problems outside of the engineering design domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In three experiments, we show that this recently proposed concept identification algorithm fulfills the three constraints, which is not achieved by classical clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We further introduce the MI as an additional evaluation metric, next to classical evaluation metrics for clustering solutions, which specifically tests whether identified solutions comprise consistent clusters across subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Last, we demonstrate the application of the concept identification algorithm in a decision making problem on optimization results from the energy management domain, where the concept identification algorithm identified more interpretable clusters and thus eased the decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' From the fact that the distribution of identified concepts demonstrates higher MI values for the two artificial data sets and the energy management application scenario, we conclude that the proposed method is well suited to find concepts with high consistency across defined subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The method can be set up to identify groups that share similar properties based on an arbitrary combination of subspaces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' subsets of features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The experiments further show, that traditional clustering methods do not lead to solutions with similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The proposed clusters showed significant overlap in the subspaces, hence providing no consistent solu- tion across subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Consistency may be a desirable property in additional applications, such as cross-domain recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Aside from a visual inspection, the quality of concepts can be evaluated by estimating the MI between features defining the subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The MI provides an assessment of consistency, essentially evaluating how similar the features of sample groups are in predefined subspaces of the full feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In all experiments, the MI was persistently higher for the solution returned by the concept identification algorithm, compared to the clusters found by traditional clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' MI proved to be a good estimate for concept quality by evaluating consistency across multiple subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' This, however, leads to the question if MI can directly be used to optimize the distribution of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Instead of evaluating the performance of a separate concept identification method, one option would be to implement an estimation of the MI as the evaluation function of a concept optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Such a process leads to groups of data samples that are op- timized based on their consistency across multiple subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' However, the found groups are likely to be indistinguishable as concepts—while a process that optimizes MI across mul- tiple feature sets can assure consistency, it neglects the two other requirements for reasonable clusters: compactness and uniqueness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' non-overlapping groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' An assignment of samples into highly overlapping groups is not suitable for the identification of concepts, as the groups are likely to be too similar to function as distinct alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The benefit for the user would hence be very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' While the identification process provides the user with consistent clusters, the choice of subspaces in the first place remains a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' A sensible split of features into separate subspaces requires in-depth domain knowledge, aside from a thorough understanding of the application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The choice of subspaces has a high influence on the potential identification outcome and defines the constitution of a concept in the particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Another topic that allows for further research is the choice of concept representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In many application scenarios, the user wants to select a manageable amount of individual samples from each concept (which can consist of several thousands of samples or more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Depending on the design task, various choices are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' If, for example, the concept identification is part of an iterative optimization process, it is sensible to pick archetypal configurations from each concept as prototypes for further optimization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Samples that are closest to the mean of the full feature vector can be understood as archetypes for each concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' But a selection of representa- tives based on fewer features, or an entire different criterion can be equally reasonable and depends on the engineering task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' CONCLUSION Concept identification, originally proposed in the engineer- ing design domain, may be understood as a special form of clustering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Thus, concept identification algorithms may be applied as clustering algorithms with properties that are relevant to application domains other than engineering design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In particular, solutions comprise a consistent clustering of instances across the full feature space and a-priori defined subspaces that are relevant to the specific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We demon- strate how concept identification solutions differ from classical clustering solutions on two artificial data sets, designed to illustrate these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Further, we introduce the Mutual Information (MI) as an evaluation criterion of the concept identification algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The MI quantifies this consistency across subspaces for a given clustering solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' We conclude that the MI is a suitable metric to compare various clustering approaches, when consistency across subspaces is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Last, we show how concept identification algorithms may be Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' Comparison of clusters found by the k-means algorithm and the identified concepts: The data set of energy management configuration is shown with respect to three objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' The three clusters and concepts are marked in purple, green, and yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In both cases, the groups are distinguishable when all three features are considered as the basis for distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' used in application domains other than engineering design, and successfully apply the algorithm in a decision-making task from energy management optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' In three experiments we demonstrated that a) concept identification leads to consistent clusters in selectable sub- spaces and b) common clustering algorithms do not have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE5T4oBgHgl3EQfSA5Z/content/2301.05525v1.pdf'} +page_content=' From this, we further conclude that concept identi- fication can generally be understood as a clustering technique that not only generates compact and clearly distinguishable groups, but also allows to integrate a third clustering objective, namely, ensuring the consistency of the groups with respect to predefined subspaces of the data.' metadata={'source': 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a/edE_T4oBgHgl3EQf1hxM/content/tmp_files/2301.08335v1.pdf.txt b/edE_T4oBgHgl3EQf1hxM/content/tmp_files/2301.08335v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f240d79670223f75fa6e53c9cef2fdabe568592e --- /dev/null +++ b/edE_T4oBgHgl3EQf1hxM/content/tmp_files/2301.08335v1.pdf.txt @@ -0,0 +1,11948 @@ +Université de Lorraine +UNIVERSAL HIGHER LIE ALGEBRAS OF SINGULAR SPACES +AND THEIR SYMMETRIES +Auteur +Ruben Louis +Directeur +Prof. Camille Laurent-Gengoux +Thèse +présentée et soutenue publiquement le 12 novembre 2022 pour l’obtention du +Doctorat de l’Université de Lorraine +(mention Mathématiques) +Membres du jury : +Directeur de thèse : +M. Camille Laurent-Gengoux +Professeur, Université de Lorraine, Metz +Président de jury : +M. Robert Yuncken +Professeur, Université de Lorraine, Metz +Rapporteurs : +M. Marco Zambon +Professeur, KU Leuven, Leuven +Mme. Chenchang Zhu +Professeure, Université de Göttingen, Göttingen +Examinateurs : +Mme. Claire Debord +Professeure, Université de Paris, Paris +M. Pol Vanhaecke +Professeur, Université de Poitiers, Poitiers +Membre invité : +M. Rajan Mehta +Professeur, Smith College, Massachusetts +arXiv:2301.08335v1 [math.DG] 19 Jan 2023 + +UNIVERSITE +IAEM +DELORRAINEInstitut +ELIECARTAN1 +ABSTRACT +This thesis breaks into two main parts. Let me describe them. +• We show that there is an equivalence of categories between Lie-Rinehart algebras over a commu- +tative algebra O and homotopy equivalence classes of negatively graded acyclic Lie 8-algebroids. +Therefore, this result makes sense of the universal Lie 8-algebroid of every singular foliation, +without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. This +extends to a purely algebraic setting the construction of the universal Q-manifold of a locally +real analytic singular foliation of [LLS20, Lav17]. Also, to any ideal I Ă O preserved by the +anchor map of a Lie-Rinehart algebra A, we associate a homotopy equivalence class of negatively +graded Lie 8-algebroids over complexes computing TorOpA, O{Iq. Several explicit examples are +given. +• The second part is dedicated to some applications of the results on Lie-Rinehart algebras. +1. We associate to any affine variety a universal Lie 8-algebroid of the Lie-Rinehart algebra +of its vector fields. We study the effect of some common operations on affine varieties such +as blow-ups, germs at a point, etc. +2. We give an interpretation of the blow-up of a singular foliation F in the sense of Mohsen +[Moh21] in terms of the universal Lie 8-algebroid of F, in fact an almost Lie algebroid over +a geometric resolution of F. +3. We introduce the notion of longitudinal vector fields on a graded manifold over a singular +foliation, and study their cohomology. We prove that the cohomology groups of the latter +vanish. +4. We study symmetries of singular foliations through universal Lie 8-algebroids. More pre- +cisely, we prove that a weak symmetry action of a Lie algebra g on a singular foliation F +(which is morally an action of g on the leaf space M{F) induces a unique up to homotopy +Lie 8-morphism from g to the Differential Graded Lie Algebra (DGLA) of vector fields +on a universal Lie 8-algebroid of F (such morphim is known under the name "L8-algebra +action" in [MZ12]). We deduce from this general result several geometrical consequences. +For instance, We give an example of a Lie algebra action on an affine sub-variety which +cannot be extended on the ambient space. Last, we present the notion of bi-submersion +towers over a singular foliation and lift symmetries to those. +Keywords: Homotopy algebras, Lie 8-algebroids, dg-manifolds, singular foliations, algebraic geom- +etry, singularities. + +2 +ACKNOWLEDGEMENTS +I wish to express my gratitude to God and to several people for their help on the accomplishment of +this thesis. +First, I would like to express my gratitude to my supervisor Camille Laurent-Gengoux for ac- +cepting me as his Ph.D. student. He was patient, attentive to me. He was a great support for +me and the one on which I could lean when I feel lost in my ideas. He encouraged me in my +research. He gave me confidence. He knew how to motivate me. He gave me valuable advice and +suggested directions to take, articles to read in order to lead better my research. He transmitted +to me knowledge and the ethics of a researcher, he is therefore my mathematic father. I also +thank him for the time he devoted to me and for his comments and his suggestions throughout +of this Ph.D. I feel very lucky for this opportunity I had to work with him. +My gratitude also goes to Chenchang Zhu and Marco Zambon for accepting to be reporters for +my thesis. I also thank Claire Debord, Pol Vanhaecke, Robert Yuncken, and Rajan Amit Mehta +for taking part in the jury. +I would like to acknowledge the full financial support for this Ph.D from Région Grand Est. +Likewise, I thank the managers and staff of Université de Lorraine for their collaboration and +their help in the administrative procedures. I particularly would like to warmly thank Prof. +Philippe Bonneau for helping me find accommodation and settling me comfortably in Metz to +begin my PhD. +Thanks to the CNRS MITI 80Prime project GRANUM also to Franco-German PHC project +Procope. I would like to thank the Institut Henri Poincaré for hosting me in november 2021. +I want to thank S. Lavau for his advice and comments on my paper "Symmetries of singular +foliations". I would like to thank C. Ospel, P. Vanhaecke and V. Salnikov for giving the possibility +to present the results on Lie-Rinehart algebras at the “Rencontre Poisson à La Rochelle, 21-22 +October 2021”. I was glad to meet my mathematic grand father Claude Roger at this conference +and I would like to thank him for his articles on the Lie-Rinehart algebras that he gave me. Also, +I thank the organizers of the Geometry Seminar at the Aristotle University of Thessaloniki, in +particular Panagiotis Batakidis for inviting me in January 28th, 2022 to give a talk on my work. +I would like to thank Iakovos Androulidakis with whom I had the honor of discussing my work +at the "Foliations, pseudodifferential operators and groupoids" school, Göttingen, Germany in +February 28th-March 4th, 2022. I also would like to thank Leonid Ryvkin for always being ready +to discuss with me when I needed it. +I am also grateful to the organizers of Poisson 2022 Advanced School at CRM (Centre de Recerca +Matemàtica) Barcelona, for giving this precious opportunity to be in charge of the problems +sessions for the Lecture "Singular Foliations". Likewise, I would like to thank the organizers of +Poisson 2022 Conference - ICMAT, Madrid for giving the possibility to present my results at the +Poster session. Last, I would like to express sincere gratitude to Université d’État d’Haïti and +more precisely the department of mathematics of École Normale Supérieure (ENS), for giving a +golden opportunity to meet mathematics. I would like to mention a few names of ENS professors +who have contributed to this achievement: Prof. Bérard Cenatus, Prof. Lesly Dejean, Prof. Dr. + +3 +Antonine Phigareau, Prof. Dr. Pierre Timothe, Prof. Dr. J.B Antenord, Prof. Dr. OLguine +Yacinthe, Prof. Dr. Oscar Walguen, Prof. Steeve Germain, Prof. Dr. Aril Milce, Prof. Dr. +Yvesner Marcelin and Prof. Dr. J. K Innocent and Prof. Dr. Dieuseul Predelus. I specially +would like to thank Prof. Dr. Achis Chery and Prof. Dr. Pol Vanhaecke for supporting me by +writing letters in my favor to obtain this grant. + +4 +Je tiens à remercier Anderson Augusma ainsi que Dor Dieunel pour m’avoir aidé à vérifier les +erreurs typographiques dans le manuscrit. +Je remercie chaleureusement Wanglaise Fateon qui m’a également aidé à relire mes articles et à +identifier les coquilles. La rencontrer a été l’une des meilleures choses qui me soient arrivées. +Enfin, je dédie cette thèse à mes parents, Madame et Monsieur Wisner Louis qui ont toujours +été présents pour m’encourager au tout début et tout au long de ces années de thèse. Je remercie +mon seul et unique frère et petit frère Benjamin Louis pour ses blagues drôles qui m’ont aidé à +ne pas devenir fou dans la foulée. Je t’aime mon frère. +À tous mes proches ! + +Contents +I +Lie-Rinehart algebras ” Acyclic Lie 8-algebroids +13 +1 +Preliminaries +14 +1.1 +Graded symmetric algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +1.2 +Graded symmetric co-algebras and their morphisms +. . . . . . . . . . . . . . . . . . . +16 +1.2.1 +Co-morphisms and co-derivations . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +1.2.2 +Richardson-Nijenhuis bracket and co-derivations +. . . . . . . . . . . . . . . . . +21 +2 +Lie 8-algebroids and their morphisms +23 +2.1 +Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +2.1.1 +dg-almost Lie algebroids over O . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +2.1.2 +Lie 8-algebroids over O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +2.2 +Lie 8-algebroids as homological co-derivations +. . . . . . . . . . . . . . . . . . . . . . +28 +2.3 +Morphisms of Lie 8-algebroids and homotopies . . . . . . . . . . . . . . . . . . . . . . +30 +2.3.1 +Morphisms of Lie 8-algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +2.3.2 +Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +2.3.3 +More technical lemmas and propositions . . . . . . . . . . . . . . . . . . . . . . +39 +3 +Lie-Rinehart algebras and their morphisms +43 +3.1 +Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +3.2 +Algebraic and geometric examples +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +3.2.1 +Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +46 +3.2.2 +On free resolutions of length ď 2 and Lie-Rinehart algebras . . . . . . . . . . . +48 +4 +Main results of Part I +52 +4.1 +Presentation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +4.2 +Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +4.3 +Proof of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +4.3.1 +A crucial bi-complex: PagepnqpE1, Eq . . . . . . . . . . . . . . . . . . . . . . . . +56 +4.3.2 +Proof on the existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +4.3.3 +Proof of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +5 + +CONTENTS +6 +4.4 +Examples of universal Lie 8-algebroids of Lie-Rinehart algebras +. . . . . . . . . . . . +68 +4.4.1 +New constructions from old ones . . . . . . . . . . . . . . . . . . . . . . . . . . +68 +4.4.2 +Sections vanishing on a codimension 1 subvariety . . . . . . . . . . . . . . . . . +70 +4.4.3 +Algebra extension +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +71 +4.4.4 +Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +71 +4.4.5 +Universal Lie 8-algebroids of some singular foliations +. . . . . . . . . . . . . . +72 +4.4.6 +Vector fields annihilating a Koszul function ϕ +. . . . . . . . . . . . . . . . . . +74 +4.4.7 +Restriction to ϕ “ 0 of vector fields annihilating ϕ . . . . . . . . . . . . . . . . +76 +4.4.8 +Vector fields vanishing on subsets of a vector space . . . . . . . . . . . . . . . . +76 +4.4.9 +Vector fields vanishing on a complete intersection . . . . . . . . . . . . . . . . . +78 +II +Geometric Applications +80 +5 +Universal Lie 8-algebroids of affine varieties +81 +5.1 +Background on affine varieties and some constructions . . . . . . . . . . . . . . . . . . +81 +5.1.1 +Three main constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +86 +5.2 +Universal Lie 8-algebroid of an affine variety . . . . . . . . . . . . . . . . . . . . . . . +91 +5.3 +Some examples of universal Lie-algebroids over an affine variety . . . . . . . . . . . . . +93 +5.3.1 +Vector fields tangent to W: a codimension one example . . . . . . . . . . . . . +93 +6 +Universal Q-manifolds of a singular foliation +96 +6.1 +Q-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +96 +6.1.1 +Graded manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +96 +6.1.2 +NQ-manifolds +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +100 +6.1.3 +NQ-manifolds - Lie 8-algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . +102 +6.2 +Universal Q-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +103 +6.2.1 +The complex defined by adQp0q +. . . . . . . . . . . . . . . . . . . . . . . . . . . +104 +6.2.2 +A result on longitudinal vector fields and examples . . . . . . . . . . . . . . . . +104 +7 +Isotropy Lie algebras of a singular foliation +110 +7.1 +Specialization of a Lie 8-algebroid at a point . . . . . . . . . . . . . . . . . . . . . . . +111 +7.2 +A blow-up procedure for a singular foliation . . . . . . . . . . . . . . . . . . . . . . . . +115 +7.2.1 +Grassmann bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +115 +7.2.2 +A blow-up procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +116 +8 +Symmetries of singular foliations through Lie 8-algebroids +123 +8.1 +Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +123 +8.2 +A Lie 8-morphism lifting a weak symmetry of a foliation +. . . . . . . . . . . . . . . . +126 +8.2.1 +Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +128 +8.3 +Main statements +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +130 +8.3.1 +Proof of 8.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +132 +8.3.2 +Particular examples +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +136 +8.4 +Lifts of weak symmetry actions and Lie 8-algebroids . . . . . . . . . . . . . . . . . . . +136 +8.4.1 +A more general statement of Proposition 8.4.1 +. . . . . . . . . . . . . . . . . . +138 + +CONTENTS +7 +9 +On weak and strict symmetries: an obstruction theory +142 +9.1 +Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +142 +9.2 +An obstruction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +145 +10 Bi-submersion towers +151 +10.1 Symmetries of bi-submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +151 +10.1.1 Lift of a symmetry to the bi-submersion tower +. . . . . . . . . . . . . . . . . . +155 +10.1.2 Lifts of actions of a Lie algebra on a bi-submersion tower +. . . . . . . . . . . . +158 +Appendices +160 +A Tensor algebra +161 +A.1 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +161 +A.2 The tensor algebra of a linear space +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +164 +A.2.1 +T ‚ +OV as a co-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +164 +B Homological algebra +166 +B.1 +Complexes of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +166 +B.1.1 +Operations on complexes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +170 +B.2 +Resolutions of a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +173 +B.3 +Geometric resolutions of a singular foliation . . . . . . . . . . . . . . . . . . . . . . . . +177 + +Introduction +The recent studies about Lie 8-algebras or Lie 8-groups, their morphisms and their -oids equivalent +(i.e. Lie 8-algebroids [Cam19, Vor05, Vor10] and “higher groupoids” [SS20]) is usually justified by +their use in various fields of theoretical physics and mathematics. Lie 8-algebras or -oids often appear +where, at first look, they do not seem to be part of the story, but end up to be needed to answer +natural questions, in particular questions where no higher-structure concept seems a priori involved. +Among examples of such a situation, let us cite deformation quantization of Poisson manifolds [Kon03] +and many recent developments of BV operator theory, e.g. +[Cam17], deformations of coisotropic +submanifolds [CF04], integration problems of Lie algebroids by stacky-groupoids [TZ06], complex +submanifolds and Atiyah classes [CSX16, Kap99, LGSX21]. The list could be longer. +For instance, it appears in theory of singular foliations. Singular foliations arise frequently in differ- +ential or algebraic geometry. Here following [AS09, AZ14, Cer79, Deb01, LLS20] we define a singular +foliation on a smooth, complex, algebraic, real analytic manifold M with sheaf of functions O to be +a subsheaf F: U ÝÑ FpUq of the sheaf of vector fields X, which is closed under the Lie bracket and +locally finitely generated as an O-module. By Hermann’s theorem [Her62], this is enough to induce +a partition of the manifold M into embedded submanifolds of possibly different dimensions, called +leaves of the singular foliation. Singular foliations appear for instance as orbits of Lie group actions +with possibly different dimensions or as symplectic leaves of a Poisson structure. When all the leaves +have the same dimension, we recover the usual “regular foliations”[DHH86, LGLR22]. +In [LLS20]-[Lav17], it is proven that “behind” several singular foliations F there is a natural ho- +motopy class of Lie 8-algebroids, called universal Lie 8-algebroid of F, and that the latter answers +natural basic questions about the existence of “good” generators and relations for a singular folia- +tion.The first part of this thesis is mainly an algebraization of [LLS20]-[Lav17], that allows to enlarge +widely the classes of examples. More precisely, Theorems 4.2.1 and 4.2.4 are similar to the main +theorems Theorem 2.8. and Theorem 2.9 in [LLS20]: +1. Theorem 4.2.1 equips any free O-resolution of a Lie-Rinehart algebra A with a Lie 8-algebroid +structure (Theorem 2.8. in [LLS20] was a statement for geometric resolutions of locally real +analytic singular foliation on an open subset with compact closure). This is a sort of homotopy +transfer theorem, except that no existing homotopy transfer theorem applies in the context of +generic O-modules (for instance, [Cam19] deals only with projective O-modules). +The difficulty +8 + +CONTENTS +9 +is that we cannot apply the explicit transfer formulas that appear in the homological perturbation +lemma because there is in general no O-linear section of A to its projective resolutions. +2. Theorem 4.2.4 states that any Lie 8-algebroid structure that terminates in A comes equipped +with a unique up to homotopy Lie 8-algebroid morphism to any structure as in the first item +(Theorem 2.8. in [LLS20] was a similar statement for Lie 8-algebroids whose anchor takes values +in a given singular foliation). +As in [LLS20], an immediate corollary of the result is that any two Lie 8-algebroids as in the first item +are homotopy equivalent in a unique up to homotopy manner, defining therefore a class canonically +associated to the Lie-Rinehart algebra, that deserve in view of the second item to be called “universal”. +However: +1. While [LLS20] dealt with Lie 8-algebroids over projective resolutions of finite length and finite +dimension, we work here with Lie 8-algebroids over any free resolution -even those of infinite +length and of infinite dimension in every degree. +2. In particular, since we are in a context where taking twice the dual does not bring back the +initial space, we can not work with Q-manifolds (those being the “dual” of Lie 8-algebroids): it +is much complicated to deal with morphisms and homotopies. +By doing so, several limitations of [LLS20] are overcome. While [LLS20] only applied to singular +foliations which were algebraic or locally real analytic on a relatively compact open subset, the present +thesis associates a natural homotopy class of Lie 8-algebroids to any Lie-Rinehart algebra, and in +particular +a) to any singular foliation on a smooth manifold, (finitely generated or not). This construction still +works with singular foliations in the sense of Stefan-Sussmann for instance, or to any involutive +C8pMq-module in ΓpAq. +b) to any affine variety, to which we associate its Lie-Rinehart algebra of vector fields), and more +generally to derivations of any commutative algebra, +c) to singular Lie algebroids in the sense of Androulidakis and Zambon [AZ18], +d) to unexpected various contexts, e.g. +Poisson vector fields of a Poisson manifold, seen as a +Lie-Rinehart algebra over Casimir functions, or symmetries of a singular foliation, seen as a +Lie-Rinehart algebra over functions constant on the leaves. +These Lie 8-algebroids are constructed on O-free resolutions of the initial Lie-Rinehart algebra over +O. They are universal in some sense (see Section 4.2), and they also are in particular unique up to +homotopy equivalence. +A similar algebraization of the main results of [LLS20], using semi-models category, appeared re- +cently in Yaël Frégier and Rigel A. Juarez-Ojeda [FJO18]. There are strong similarities between our +results and theirs, but morphisms and homotopies in [FJO18] do not match ours. It is highly possi- +ble, however, that Theorem 4.2.1 could be recovered using their results. Luca Vitagliano [Vit15] also +constructed Lie 8-algebra structures out of regular foliations, which are of course a particular case + +CONTENTS +10 +of Lie-Rinehart algebra. These constructions do not have the same purposes. For regular foliations, +our Lie 8-algebroid structure is trivial in the sense that it is a Lie algebroid, while his structures +become trivial when a good transverse submanifold exists. Lars Kjeseth [Kje01b, Kje01a] also has a +notion of resolutions of Lie-Rinehart algebras. But his construction is more in the spirit of Koszul-Tate +resolution: Definition 1. in [Kje01b] defines Lie-Rinehart algebras resolutions as resolutions of their +Chevalley-Eilenberg coalgebra, not of the Lie-Rinehart algebra itself as a module. It answers a differ- +ent category of questions, related to BRST and the search of cohomological model for Lie-Rinehart +algebra cohomology. +These results can be used to understand the geometry of singular foliations such as their symme- +tries. More precisely, Let pM, Fq be a foliated manifold. A global symmetry of a singular foliation +F on M is a diffeomorphism φ: M ÝÑ M which preserves F, that is, φ˚pFq “ F. The image of a +leaf through a global symmetry is again a leaf (not necessarily the same leaf). For G a Lie group, a +strict symmetry action of G on a foliated manifold pM, Fq is a smooth action G ˆ M ÝÑ M that acts +by global symmetries [GZ21]. Infinitesimally, it corresponds to a Lie algebra morphism g ÝÑ XpMq +between the Lie algebra pg, r¨ , ¨sgq of G and the Lie algebra of symmetries of F. +A strict symmetry action of G on M goes down to the leaf space M{F, even though the latter space +is not a manifold. The opposite direction is more sophisticated, since a strict symmetry action of G +on M{F does not induce a strict action over M in general. However, it makes sense to consider linear +maps ϱ: g ÝÑ XpMq that satisfy rϱpxq, Fs Ă F for all x P g, and which are Lie algebra morphisms +up to F, namely, ϱprx, ysgq ´ rϱpxq, ϱpyqs P F for all x, y P g. The latter linear maps are called “weak +symmetry actions”. These actions induce a “strict action”on the leaf space i.e. a Lie algebra morphism +g ÝÑ XpM{Fq, whenever M{F is a manifold. +Let us emphasize on the following observation: An infinitesimal action of a Lie algebra g on a +manifold M is a Lie algebra morphism g ÝÑ XpMq. Replacing M by a Lie 8-algebroid pE, Qq, one +expects to define them as Lie 8-algebra morphisms g ÝÑ XpE, Qq, the latter space being a DGLA. +Various results about those are given in Mehta-Zambon [MZ12]. In particular, these authors give +several equivalent definitions and interpretations of those. +In view of [LLS20, Lav17] it is shown that behind every singular foliation or more generally any Lie- +Rinehart algebras [LGL22b] there exists a Lie 8-algebroid structure which is unique up to homotopy +called the universal Lie 8-algebroid. Here is a natural question: what does a symmetry of a singular +foliation F induce on an universal Lie 8-algebroid of F? Theorem 8.3.1 of this thesis gives an answer +to that question. It states that any weak symmetry action of a Lie algebra on a singular foliation F +can be lifted to a Lie 8-morphism valued in the DGLA of vector fields on an universal Lie 8-algebroid +of F. Such Lie 8-morphisms were studied by Mehta and Zambon [MZ12] as "L8-algebra actions". +This goes in the same direction as [GZ21] who already underlined Lie-2-group structures associated +to strict symmetry action of Lie groups. Furthermore, Theorem 8.3.1 says this lift is unique modulo +homotopy equivalence. +This result gives several geometric consequences. Here is an elementary question: can a Lie algebra +action g Ñ XpWq on an affine variety W Ă Cd be extended to a Lie algebra action g Ñ XpCdq on Cd? +Said differently: it is trivial that any vector field on W extends to Cd, but can this extension be done +in such a manner that it preserves the Lie bracket? Although no “8-oids” appears in the question, +which seems to be a pure algebraic geometry question, we claim that the answer goes through Lie + +CONTENTS +11 +8-algebroids and singular foliations. More precisely, the idea is then to say that any g-action on W +induces a weak symmetry action on the singular foliation IW XpCdq of all vector fields vanishing on W +(here IW is the ideal that defines W). By Theorem 8.3.1, we know that it is possible to lift any weak +symmetry action of singular foliation into a Lie 8-morphism. But is it possible to build such a Lie +8-morphism where the polynomial-degree ´1 of the second order Taylor coefficient is zero? There are +cohomological obstructions. In some specific cases, obstruction classes appear on some cohomology, +although in general the obstruction is rather a Maurer-Cartan-like equations that may or may not +have solutions. We show that both questions are in fact related. +Here is another interesting question where we would like to apply our results: Can we desingularize +a singular affine variety W Ď Cd by making use of the universal Lie 8-algebroid of the singular foliation +F “ XpWq of vector fields tangent W? In section 7.2.2, we use the geometric resolution of the singular +foliation (i.e. the resolution on which the universal Lie 8-algebroid is built) to recover several notions +of resolution of singularities: on being due to Nash [LU81] and a second one to Mohsen [Moh21]. But +the meaning of these spaces are unclear. We would like to relate them with the higher brackets of the +universal Lie 8-algebroid, e.g. to understand the role of the 3-ary bracket in this procedure. +In the last chapter of the thesis, we introduce the notion of "bi-submersion towers" over singular +foliations that we denote by TB. The latter notion as the name suggests is a family of "bi-submersions" +which are built one over the other. The concept of bi-submersion over singular foliations has been +introduced in [AS09] and it is used in K-theory [AS19] or differential geometry [AS11, AZ14, GZ19]. +We show that such a bi-submersion tower over a singular foliation F exists if F admits a geometric +resolution. Provided that it exists, we show in Theorem 10.1.25 that any infinitesimal action of a +Lie algebra g on the singular foliation F lifts to the bi-submersion tower TB. This lift looks like the +beginning of a kind of Lie 8-morphism. We wonder if we can continue the construction in Theorem +10.1.25 to a Lie 8-morphism. +The thesis is organized as follows. +In chapter 1, we recall some basics on graded co-algebra +structures on the graded symmetry algebra and their morphisms. +We also review the notion of +co-derivations and their properties. +In Chapter 2, we introduce the notion of Lie 8-algebroid on +an arbitrary commutative unital algebra O over a field of characteristic zero, and also define their +morphisms in terms of co-derivations. We define the notion of homotopy between them. Some technical +Lemmas and Propositions are given. In Chapter 3, we fix notations and review definitions, examples, +and give main properties of Lie-Rinehart algebras. Besides, we construct a Lie 2-algebroid structure +over any Lie-Rinehart algebra that admits a free resolution of length less or equal to 2. In Chapter +4, we state and prove the main results of the first part of the thesis, i.e. the equivalence of categories +between Lie-Rinehart algebras and homotopy classes of free acyclic Lie 8-algebroids, which justifies +the name universal Lie 8-algebroid of a Lie-Rinehart algebra. Also, we describe the universal Lie +8-algebroids of several Lie-Rinehart algebras. The complexity reached by the higher brackets in these +examples should convince us that it is not a trivial structure, even for relatively simple Lie-Rinehart +algebras. The Chapter 5 is devoted to the applications of the results of the previous chapters to +affine varieties. We recall some basics definitions and theorems on affine varieties W. We present +three main constructions of Lie-Rinehart associated to W and relate their universal Lie 8-algebroids +together. Also, some examples of the universal Lie 8-algebroids are given, such as blow-ups, vector + +CONTENTS +12 +fields vanishing on a complete intersection, etc. In Chapter 6, we recall the notion of Q-manifolds +and apply the results on Lie-Rinehart algebras to recover the universal Lie 8-algebroids of a singular +foliation [LLS20]. This chapter ends with a result on the cohomology of longitudinal vector fields. In +Chapter 7, we recall the definition of Androulidakis and Skandalis isotropy Lie algebra of a singular +foliation at a point and recall from [LLS20] its relation with the Universal Lie 8-algebroids of a +singular foliation. We end with a blow-up procedure for a singular foliation inspired by O. Mohsen. +In Chapter 8, we study symmetries of singular foliations. We present some definitions and facts on +weak symmetry actions of Lie algebras on singular foliations and give some examples. We state the +main results of the second part of the thesis and present their proofs. In Chapter 9, we define an +obstruction class for extending a Lie algebra action on an affine variety to ambient space and also give +some examples. In Chapter 10, we look at symmetries of bi-submersions. Afterwards, we introduce +the notion of bi-submersion towers over a singular foliation and point out some observations related +to their symmetries. +Finally, in Appendix A, we recall the definition of tensor algebra and fix notations. In Appendix +B, we recall some general facts on homological algebra and give some geometric constructions. + +Part I +Lie-Rinehart algebras ” Acyclic Lie +8-algebroids +13 + +CHAPTER 1 +Preliminaries +This chapter sets the ground for the whole thesis, especially chapters 2, 3 and 4. It fixes terminologies, +conventions and notations. For more details, we also refer the reader to Appendix A and B. +Throughout this thesis, K is a field of characteristic zero, and O is an associative commutative +unital K-algebra unless otherwise mentioned. Also, an O-module E is seen as K-vector space in the +natural way, λ¨e :“ pλ¨1Oq¨e, where 1O ” 1 is the unit of O. In the sequel, we will drop the notation "¨". +Geometrically, O can be understood as the algebra of smooth functions on a manifold M, or on +an open subset U Ă M of a complex manifold, or the coordinate ring of an affine variety W. +1.1 +Graded symmetric algebras +For E “ ‘iPZEi be a Z-graded module, we denote by |x| P Z the degree of a homogeneous element +x P E. +• We denote by Ä‚ E and call (reduced) graded symmetric algebra of E over O the quotient +T ‚ +OE{xx bO y ´ p´1q|x||y|y bO xy +of the tensor algebra (see Appendix A) over O of E, namely +T ‚ +OE :“ +8 +à +k“1 +E bO ¨ ¨ ¨ bO E +looooooomooooooon +k times +by the ideal generated by x bO y ´ p´1q|x||y|y bO x, with x, y arbitrary homogeneous elements +of E. This quotient is a graded commutative algebra. We denote its product by d. +• Similarly, we denote by S‚ +KpEq and call (reduced) graded symmetric algebra of E over the field K +the quotient of the tensor algebra (over K) of E, i.e., +T ‚ +KE :“ +8 +à +k“1 +E bK ¨ ¨ ¨ bK E +looooooomooooooon +k times +14 + +CHAPTER 1. PRELIMINARIES +15 +by the ideal generated by x bK y ´ p´1q|x||y|y bK x, with x, y arbitrary homogeneous elements +of E. We denote by ¨ the product in S‚ +KpEq. +The algebras Ä‚ E and S‚ +KpEq come equipped with two different “grading” . Let us make them +explicit. +1. We define the degree of x “ x1 d ¨ ¨ ¨ d xn P Än E or x “ x1 ¨ ¨ ¨ ¨ ¨ xn P Sn +KpEq by +|x1 ¨ ¨ ¨ ¨ ¨ xn| “ |x1 d ¨ ¨ ¨ d xn| “ |x1| ` ¨ ¨ ¨ ` |xn| +for any homogeneous x1, . . . , xn P E. With respect to this degree, Ä‚ E and S‚ +KpEq are graded +commutative algebras. +2. The second grading is called "polynomial-degree". The polynomial-degree of x1 d¨ ¨ ¨dxn P Än E +or x1¨¨ ¨ ¨¨xn P Sn +KpEq is defined to be n. We have the following polynomial-degree decomposition +Ä‚ E “ ‘kě1 +Äk E and S‚ +KpEq “ ‘kě1Sk +KpEq, where Äk E and Sk +KpEq stand for the O-module +of elements of polynomial-degree k and the K-vector space of elements of polynomial-degree k, +respectively. +Convention 1.1.1. For E a graded O-module, the set of elements of polynomial-degree k and degree +d in Ä‚ E (resp. Sk +KpEq) shall be denoted by Äk E|d (resp. Sk +KpEq|d). For example, +k +ä +E|d “ +à +i1`¨¨¨`ik“d +Ei1 bO ¨ ¨ ¨ bO Eik. +• For any homogeneous elements x1, . . . , xk P E and σ P Sk a permutation of t1, . . . , ku, the Koszul +sign ϵpσ; x1, . . . , xkq is defined by: +xσp1q d ¨ ¨ ¨ d xσpkq “ ϵpσ; x1, . . . , xkq x1 d ¨ ¨ ¨ d xk. +We often write ϵpσq for ϵpσ; x1, . . . , xkq. +• For i, j P N, a pi, jq-shuffle is a permutation σ P Si`j such that σp1q ă . . . ă σpiq and σpi ` 1q ă +. . . ă σpi`jq, and the set of all pi, jq-shuffles is denoted by Spi, jq. More generally, a pi1, . . . , ikq- +shuffle is a permutation σ P Si1`¨¨¨`ik such that +σp1q ă ¨ ¨ ¨ ă σpi1q +σpi1 ` 1q ă ¨ ¨ ¨ ă σpi1 ` i2q, +... +σpi1 ` i2 ` ¨ ¨ ¨ ` ij´1 ` 1q ă ¨ ¨ ¨ ă σpi1 ` i2 ` ¨ ¨ ¨ ` ijq, +... +σpi1 ` i2 ` ¨ ¨ ¨ ` ik´1 ` 1q ă ¨ ¨ ¨ ă σpi1 ` i2 ` ¨ ¨ ¨ ` ikq +The set of all pi1, . . . ikq-shuffles is denoted by Spi1, . . . , ikq. + +CHAPTER 1. PRELIMINARIES +16 +1.2 +Graded symmetric co-algebras and their morphisms +Graded co-algebra structures are the dual version of graded algebra structures, i.e., they are obtained +by reversing all arrows and permutes the order of composition. Let us recall the definition of a co- +algebra structure on an O-module. We refer the reader to, e.g [JLL, Kas12, Man] or Chapter 8 of +[Man05b] for more details on this topic. +Definition 1.2.1. A coassociative graded cocommutative co-algebra structure on an graded O-module +C “ ‘iPZCi is +• a linear map of degree zero called (graded) coproduct ∆: C ÝÑ C b C with +∆pCkq Ă +à +i`j“k +Ci b Cj +is such that ∆pCkq has non-zero1 intersection with only finitely many spaces Ci b Cj +• p∆ b idq ˝ ∆ “ pid b ∆q ˝ ∆: C Ñ C b C b C, called graded coassociativity. +• τ ˝ ∆ “ ∆, called graded cocommutativity, +where τ : C b C Ñ C b C is defined by τpa b bq “ p´1q|a||b|b b a. +Equivalently, the following diagrams commute +C +∆ +� +∆ +� C b C +∆bid +� +C b C +idb∆ +� C b C b C +C +∆ +� +∆ +� +C b C +τ +� C b C. +The pair pC, ∆q is called graded O-co-algebra. We say that pC, ∆q is counital if there is a morphism +of graded vector spaces u: C Ñ K such that pu b idq ˝ ∆ “ pid b uq ˝ ∆ “ id i.e. the diagram below +commutes +C +id +� +∆ +� +∆ +� C b C +idbu +� +C b C +ubid� K b C » C » C b K. +Proposition 1.2.2. Let pC, ∆q and pC1, ∆1q be two co-algebras over O. +Then, the tensor product +pC b C1, pidC b τ b idC1q ˝ ∆ b ∆1q is a co-algebra over O. +Example 1.2.3. The polynomial ring Krts in one indeterminate t is a co-algebra with the coproduct +∆ given on monomials tn, n ě 0 by: +∆ptnq “ +nÿ +k“0 +ˆn +k +˙ +tk b tn´k +and extends by linearity. +The graded symmetric algebra is also an example of co-algebra. +1Note that this condition is automatically satisfied when the module is concentrated in positive degree. + +CHAPTER 1. PRELIMINARIES +17 +Lemma 1.2.4. Let E be a O-module. Both Ä‚ E and S‚ +KpEq admit natural co-algebra structures with +respect to the deconcatenation ∆ defined by: +∆px1 d ¨ ¨ ¨ d xnq “ +n´1 +ÿ +i“1 +ÿ +σPSpi,n´iq +ϵpσqxσp1q d ¨ ¨ ¨ d xσpiq b xσpi`1q d ¨ ¨ ¨ d xσpnq +(1.1) +for every x1, . . . , xn P E. +Example 1.2.5. For small n. The formula of Equation (1.1) on homogeneous elements x, y, z P E +means that +• ∆pxq “ 0 +• ∆px d yq “ x b y ` p´1q|x||y|y b x +• ∆px d y d zq “ x d y b z ` p´1q|x||z|x d z b y ` p´1q|x|p|z|`|y|qy d z b x ` +x b y d z ` p´1q|x||y|y b x d z ` p´1q|z|p|x|`|y|qz b x d y +Remark 1.2.6. According to Definition 1.2.1, the coproduct ∆ is graded with respect to the positive +grading given by the "polynomial-degree" of monomials in the symmetric algebras Ä‚ E and S‚ +KpEq. +Example 1.2.7. Consider M “ Rd with global coordinates px1, . . . , xdq. The algebra of differential +forms pΩpMq, ^q over M has a co-algebra structure ∆Ω : ΩpMq Ñ ΩpMq b ΩpMq given as in Lemma +1.2.4. Namely, +∆pdx1 ^ ¨ ¨ ¨ ^ dxnq “ +n´1 +ÿ +i“1 +ÿ +σPSpi,n´iq +ϵpσqdxσp1q ^ ¨ ¨ ¨ ^ dxσpiq b dxσpi`1q ^ ¨ ¨ ¨ ^ dxσpnq. +This matches (1.1) if we consider E “ Ω1pMq is concentrated in degree `1. +1.2.1 +Co-morphisms and co-derivations +We recall the definitions of co-morphisms, co-derivations of graded co-algebra structures. +Definition 1.2.8. Let pC1, ∆1q and pC, ∆q be two graded co-algebras. +1. A morphism of co-algebras or co-morphism from pC1, ∆1q to pC, ∆q is a linear map Φ: C1 ÝÑ C +of degree 0 such that the following diagram commutes +C1 +∆1 +� +Φ +� C +∆ +� +C1 b C1 ΦbΦ � C b C +(1.2) +In formula: ∆ ˝ Φ “ pΦ b Φq ˝ ∆1. +2. Let Φ: C1 ÝÑ C be a co-morphism. A Φ-co-derivation of degree l is a linear map H: C1 ÝÑ C of +degree l such that the following diagram commutes, +C1 +∆1 +� +H +� C +∆ +� +C1 b C1 HbΦ`ΦbH � C b C +(1.3) + +CHAPTER 1. PRELIMINARIES +18 +that is, the so-called co-Leibniz identity is satisfied: +∆ ˝ H “ pH b Φ ` Φ b Hq ˝ ∆1. +(1.4) +When C1 “ C and Φ “ id we speak of a co-derivation (of degree l). +Remark 1.2.9. It is easily checked that the composition of two co-morphisms C2 Ψ +Ñ C1 ΦÑ C is again +a co-morphism. +Proposition 1.2.10. We denote by CoDerpCq the linear space of co-derivations of pC, ∆q. The set +CoDerpCq is a graded Lie sub-algebra of HompC, Cq when equipped with the graded commutator. +Proof. For two co-derivations H1, H2 P CoDerpCq one has, +∆ ˝ H1 ˝ H2 “ pH1 b id ` id b H1q ˝ pH2 b id ` id b H2q ˝ ∆ +“ pH1 ˝ H2 b id ` H1 b H2 ` p´1q|H1||H2|H2 b H1 ` id b H1 ˝ H2q ˝ ∆. +Also, we obtain a similar equation for p´1q|H1||H2|∆ ˝ H2 ˝ H1 by changing the roles of H1 and H2. +Subtracting both terms, one get the co-Leibniz identity for rH1, H2s “ H1˝H2´p´1q|H1||H2|H2˝H1. +Let us describe Definition 1.2.8 in the context of the thesis. Let E1 and E be graded O-modules. +Definition 1.2.11. A linear map Φ: S‚ +KpE1q Ñ S‚ +KpEq is said to be of polynomial-degree/degree r P Z, +if it sends polynomials of Sk +KpEq to those of Sk´r +K +pEq/elements of S‚ +KpEq|d to S‚ +KpEq|d`r. In formulas, +for very k ě 0, (resp. d P Z) one has +Φ +´ +Sk +KpEq +¯ +Ď Sk´r +K +pEq, +resp. +Φ +` +S‚ +KpEq|d +˘ +Ď S‚ +KpEq|d`r. +Remark 1.2.12. Any linear map Φ: S‚ +KpE1q Ñ S‚ +KpEq of degree l can be decomposed with respect to +the polynomial-degree as formal sum: +Φ “ +ÿ +rPZ +Φprq +(1.5) +where, for all k P N0, Φprq : S‚ +KpE1q Ñ S‚´r +K +pEq is a linear map of polynomial-degree r and of degree l. +• It should be understood that Φprqpvq “ 0 for all v P Sk +KpE1q with k ď r. Also, the decomposition +(1.5) makes sense in particular when the polynomial-degrees of Φ are lower bounded that is, +there is an integer N P Z, such that Φprq “ 0 for all r ă N. In that case, for every v P Sk +KpEq, +ÿ +rPZ +Φprqpvq “ +ÿ +Nďrăk +Φprqpvq, +is a finite sum. Hence, Equation (1.5) becomes +Φ “ +ÿ +rěN +Φprq, +for some N P Z, since the l.h.s of (1.5) is finite. Notice that a linear map Φ: S‚ +KpE1q Ñ S‚ +KpEq +is of polynomial-degree N if and only if ΦpNq is the unique non-zero term, namely Φprq “ 0, for +r ‰ N. +• Also, for the composition of two linear maps S‚ +KpE2q Ψ +Ñ S‚ +KpE1q ΦÑ S‚ +KpEq, we have +pΦ ˝ Ψqprq “ +ÿ +i`j“r +Φpiq ˝ Ψpjq. +(1.6) +for every r P Z. Here, although the sum is infinite, it becomes finite when applied to a given +element in S‚ +KpE1q. + +CHAPTER 1. PRELIMINARIES +19 +Co-morphisms on symmetric algebras +Let us have a discussion on co-morphisms from the symmetric graded co-algebras pS‚ +KpE1q, ∆1q and +pS‚ +KpEq, ∆q, where ∆1 and ∆ are the respective coproducts like in Lemma 1.2.4. Given any linear map +F : S‚ +KpE1q Ñ E. Denoting by Fr : Sr`1 +K +pE1q Ñ E for r P N0, the restriction of F to Sr`1 +K +pE1q. The +linear map F can be extended to a unique co-morphism ¯F : pS‚ +KpE1q, ∆1q Ñ pS‚ +KpEq, ∆q by taking for +k P N the component on Sk +KpEq to be, for any homogeneous x1, . . . , xn P E1 +ÿ +i1`¨¨¨`ik“n +ÿ +σPSpi1,...,irq +ϵpσq 1 +k! +k +ź +j“1 +Fijpxσpi1`¨¨¨`ij´1`1q, . . . , xσpi1`¨¨¨`ijqq. +(1.7) +where Spi1, . . . , ikq is the set of pi1, . . . , ikq-shuffles, with i1, . . . , ik P N. +Example 1.2.13. Let us compute ¯F : +` +Sk +KpE1q, ∆1˘ +ÝÑ pS‚ +KpEq, ∆q for k “ 1, 2, 3. For x, y, z P E1, +• ¯Fpxq “ F0pxq. +• ¯Fpx ¨ yq “ F1px ¨ yq ` F0pxq ¨ F0pyq. +• ¯Fpx ¨ y ¨ zq “ F2px ¨ y ¨ zq ` F0pxq ¨ F1py ¨ zq ` p´1q|x||y|F0pyq ¨ F1px ¨ zq ` +p´1qp|x|`|y|q|z|F0pzq ¨ F1px ¨ yq ` F0pxq ¨ F0pyq ¨ F0pzq. +We can easily check that ∆ ˝ ¯F “ p ¯F b ¯Fq ˝ ∆1, e.g +• ∆ ˝ ¯Fpxq “ ∆ ˝ F0pxq “ 0 “ p ¯F b ¯Fq ˝ ∆1pxq +• On one side, +∆ ˝ ¯Fpx ¨ yq “ ∆ ˝ F1px ¨ yq ` ∆pF0pxq ¨ F0pyqq +“ F0pxq b F0pyq ` p´1q|x||y|F0pyq b F0pxq. +On the other side, +p ¯F b ¯Fq ˝ ∆1px, yq “ p ¯F b ¯Fqpx b y ` p´1q|x||y|y b xq +“ p ¯F b ¯Fqpx b yq ` p´1q|x||y|p ¯F b ¯Fqpy b xq +“ F0pxq b F0pyq ` p´1q|x||y|F0pyq ¨ F0pxq. +The following proposition claims that every co-morphism from S‚ +KpE1q to S‚ +KpEq is of the form +described in (1.7). For a proof, see [JLL, Kas12, Man05b]. +Proposition 1.2.14. Let E1, E be two graded O-modules. For any morphism of graded vector space +F : SKpE1q ÝÑ E there exists an (unique) morphism of graded co-algebras ¯F : pSKpE1q, ∆1q ÝÑ pSKpEq, ∆q +that satisfies pr ˝ ¯F “ F. +Here pr is the canonical projection onto the term of polynomial-degree 1, i.e., pr: S‚ +KpEq Ñ S1 +KpEq » E. +Remark 1.2.15. The following facts are crucial, and will be used in the sequel. + +CHAPTER 1. PRELIMINARIES +20 +1. By Proposition 1.2.14, a co-morphism Φ: S‚ +KpE1q Ñ S‚ +KpEq is entirely determined by the collection +indexed by r P N0 of maps called its r-th Taylor coefficients: +Φr : Sr`1 +K +pE1q +Φ +ÝÑ S‚ +KpEq +pr +ÝÑ E, +(1.8) +Notice that the component Φprq of polynomial-degree r ě 0 of Φ coincides with r-th Taylor +coefficient Φr on Sr`1 +K +pE1q. In fact we have, Φr “ ppr ˝ Φprqq|Sr`1 +K +pE1q. +2. By item 1, a co-morphism Φ: S‚ +KpE1q Ñ S‚ +KpEq is completely determined by its components of +polynomial-degree r ě 0, hence it admits a polynomial decomposition of the form: +Φ “ +ÿ +rě0 +Φprq. +(1.9) +3. Similar results as in Proposition 1.2.14 hold for Φ-co-derivations or for co-derivations, e.g. see +Corollary VIII.34 in [Man05b] or [JLL]. +Given a co-morphism Φ: S‚ +KpE1q Ñ S‚ +KpEq a Φ-co- +derivation H of degree l is entirely determined by the Taylor coefficients defined as in (1.8) +Hr : Sr`1 +K +pE1q Ñ E, r P N0. +For k P N0 and x1, . . . , xn P E1, the component on Sk +KpEq of Hpx1 ¨ ¨ ¨ ¨ ¨ xnq takes the form +ÿ +i1`¨¨¨`ik“n +ÿ +σPSpi1,...,ikq +ϵpσq 1 +k!Hi1pxσp1q, ¨ ¨ ¨ , xσpi1qq ¨ +k´1 +ź +j“1 +Φij´1pxσpi1`¨¨¨`ij´1`1q, . . . , xσpi1`¨¨¨`ijqq. +(1.10) +Also, H has a decomposition into polynomial-degree of the form : +H “ +ÿ +rě0 +Hprq. +(1.11) +Example 1.2.16. Every linear map H : S‚ +KpEq Ñ E of degree l admits an unique extension to a +co-derivation ¯H : S‚ +KpEq Ñ S‚ +KpEq of degree l. The latter is given explicitly by the formula +¯Hpx1 ¨ ¨ ¨ ¨ ¨ xnq “ +nÿ +i“1 +ÿ +σPSpi,n´iq +ϵpσqHpxσp1q ¨ ¨ ¨ ¨ ¨ xσpiqq ¨ xσpi`1q ¨ ¨ ¨ ¨ ¨ xσpnq. +(1.12) +The following lemma-definition is helpful in the sequel. +Lemma 1.2.17 (Definition). We say that a co-algebra morphism Φ: S‚ +KpE1q Ñ S‚ +KpEq is O-multilinear +when the equivalent conditions below are satisfied: +(i) For every n ě 0, the n-th Taylor coefficient Φpnq : Sn`1 +K +pE1q ÝÑ E of Φ is O-multilinear +(ii) There exists an induced co-algebra morphism ΦO : Ä‚ E1 ÝÑ Ä‚ E making the following diagram +commutative : +S‚ +KpE1q +� +Φ +� S‚ +KpEq +� +Ä‚ E1 +ΦO +� Ä‚ E +When it is the case we still write Φ for ΦO. +Proof. The equivalence is straightforward. +Remark 1.2.18. Similarly, the formula (1.10) shows that the Taylor coefficients of H are O-multilinear +if and only if H induces a ΦO-co-derivation HO : Ä‚ E1 ÝÑ Ä‚ E. + +CHAPTER 1. PRELIMINARIES +21 +1.2.2 +Richardson-Nijenhuis bracket and co-derivations +We need to use the Richardson-Nijenhuis bracket to express our results and explain some proofs in the +coming chapters. This bracket was defined mainly to understand Lie algebra structures on a vector +space and their deformations [NR66] as well as Poisson structures and their generalizations. Let us +recall the definition in our context. +Let E be an O-module. For k P N0, homomorphisms of degree j from Sk`1 +K +pEq to E shall be, by +definition, the space +Homj +K +´ +Sk`1 +K +pEq , E +¯ +:“ ‘mPZHomK +´ +Sk`1 +K +pEq |m´j, Em +¯ +. +We refer the reader for example to [KMS93, KS04] for the following definition. +Definition 1.2.19. The Richardson-Nihenhuis bracket is a K-bilinear map +r ¨ , ¨ sRN : Homi +K +´ +Sp`1 +K +pEq , E +¯ +b Homj +K +´ +Sq`1 +K +pEq , E +¯ +Ñ Homi`j +K +´ +Sp`q`1 +K +pEq , E +¯ +which is defined on homogeneous elements P P Hom‚ +K +´ +Sp`1 +K +pEq , E +¯ +and R P Hom‚ +K +´ +Sq`1 +K +pEq , E +¯ +by +rP, RsRN “ P ˝ R ´ p´1q|P||R|R ˝ P, +(1.13) +with the understanding that P ˝ R is defined on x0 ¨ ¨ ¨ ¨ xp`q P Sp`q`1 +K +pEq by +pP ˝ Rqpx0 ¨ ¨ ¨ ¨ xp`qq :“ +ÿ +σPSpq`1,pq +ϵpσqPpRpxσp0q ¨ ¨ ¨ xσpqqq ¨ xσpq`1q ¨ ¨ ¨ ¨ xσpp`qqq. +(1.14) +The bracket is extended by bilinearity. +Remark 1.2.20. The bracket r¨ , ¨sRN should not be confused with the graded commutator r¨ , ¨s +of the graded vector space HomK pS‚ +KpEq, S‚ +KpEqq. In fact, the graded commutator of two elements +P, R P Hom‚ +K pS‚ +KpEq, S‚ +KpEqq is rP, Rs “ P ˝R´p´1q|P||R|R˝P, but here P ˝R is the usual composition +of homomorphism. +It is easy to check that +Lemma 1.2.21. The graded vector space Hom‚ +K pS‚ +KpEq , Eq together with the Richardson-Nihenhuis +bracket r ¨ , ¨ sRN is a graded Lie algebra structure. +For a given i P Z, and a given P “ ř +kě0 Pk with Pk P Homi +K +´ +Sk`1 +K +pEq, E +¯ +, we denote by +¯P P CoDerpS‚ +KpEqq the unique co-derivation of degree i with Taylor coefficients pPkqkPN0. The relation +between the Richardson-Nijenhuis bracket an co-derivations is described in the following lemma: +Lemma 1.2.22. The map P ÞÑ ¯P is a graded Lie algebra morphism that is, for every P, R of degrees +l, r as above, we have +Ğ +rP, RsRN “ r ¯P, ¯Rs “ ¯P ˝ ¯R ´ p´1qlr ¯R ˝ ¯P. +Proof. Linearity follows from uniqueness of the extension ¯P described in Proposition 1.2.14. Indeed, +for P, R as above, one should have pr ˝ p ¯P ` ¯Qq “ P ` R “ pr ˝ p Ğ +P ` Rq. By unicity, we must have +Ğ +P ` R “ ¯P ` ¯R. Likewise, one has for any λ P K, Ď +λP “ λ ¯P. + +CHAPTER 1. PRELIMINARIES +22 +To show that it is a graded Lie algebra morphism, it suffices to check that pr ˝ r ¯P, ¯Rs “ rP, RsRN. For +x “ x0 ¨ ¨ ¨ ¨ xr P Sr`1 +K +pEq +` +pr ˝ r ¯P, ¯Rs +˘ +r pxq “ P ˝ ¯Rpxq ´ p´1q|P||R|R ˝ ¯Ppxq +(1.15) +By using Formula (1.12), we obtain for example +P ˝ ¯Rpxq “ +kÿ +j“0 +ÿ +σPSpj`1,k´jq +ϵpσqPpRjpxσp0q ¨ ¨ ¨ ¨ ¨ xσpjqq ¨ xσpj`1q ¨ ¨ ¨ ¨ ¨ xσpkqq +“ +ÿ +i ` j “ k +i, j ě 0 +ÿ +σPSpj`1,iq +ϵpσqPipRjpxσp0q ¨ ¨ ¨ ¨ ¨ xσpjqq ¨ xσpj`1q ¨ ¨ ¨ ¨ ¨ xσprqq +“ +ÿ +i ` j “ r +i, j ě 0 +pPi ˝ Rjqpxq +with the understanding that Pipx0 ¨ ¨ ¨ ¨ xmq “ 0 for m ‰ i. As a consequence, the right-hand side of +Equation (1.15) becomes +ÿ +i ` j “ r +i, j ě 0 +´ +pPi ˝ Rjqpxq ´ p´1q|P||R|pRj ˝ Piqpxq +¯ +“ +ÿ +i ` j “ r +i, j ě 0 +rPi, RjsRNpxq +“ prP, RsRNqr pxq. +Conclusion: +In this chapter, we recalled the co-algebra language, in particular comorphisms, Richardson- +Nijenhuis bracket. An important point is not to confuse the tensor products bK and bO, hence +the graded symmetric algebra S‚ +KpEq and S‚ +OpEq, which is denoted by S‚pEq. This introduction +is completed in Appendix A. + +CHAPTER 2 +Lie 8-algebroids and their morphisms +2.1 +Definitions +In this section, we present the notion Lie 8-algebroid over the algebra O. We also give geometrical +examples. We refer the reader to Appendix B for some generalities on graded modules. +By definition, Lie 8-algebras on a graded vector space V are co-derivations of degree ´1 squaring +to 0 of the graded symmetric algebra SpV q (e.g see [LS93, Ryv16]). Lie 8-algebroids generalize Lie +8-algebras and Lie algebroids, but the situation is more sophisticated, because the 2-ary bracket is +not O-linear. Let us make some preparations before introducing the definition. +Definition 2.1.1. A derivation of O is a K-linear map X : O Ñ O that satisfies the so-called Leibniz +identity +Xpfgq “ Xpfqg ` fXpgq +for all f, g P O. The set of all derivations of O inherits a Lie K-algebra structure from the Lie algebra +EndKpOq, whose Lie bracket is the commutator, that is, +rX, Y s “ X ˝ Y ´ Y ˝ X, +for all X, Y derivations of O. +We denote by DerpOq the Lie algebra of derivations of O. Also, for X P DerpOq, Xrfs stands for +the derivation X applied to f P O. +Example 2.1.2. Geometrically, derivations of O can be understood as follows: +1. when O “ C8pMq is the algebra of functions of a smooth manifold M, as vector fields on a +smooth manifold M; +23 + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +24 +2. when O is the algebra of holomorphic functions on an open subset U of Cd, as vector fields on +U; +3. when O “ OW is the coordinate ring of an affine variety W, as vector fields on the affine +variety W. +2.1.1 +dg-almost Lie algebroids over O +Definition 2.1.3. [Lav17, LLS20] A (negatively graded) almost differential graded Lie algebroid +pE‚, ℓ1, ℓ2, ρq over O is a complex +pE, ℓ1, ρq : +¨ ¨ ¨ +ℓ1 +ÝÑ E´3 +ℓ1 +ÝÑ E´2 +ℓ1 +ÝÑ E´1 +ρ +ÝÑ DerpOq. +of projective O-modules equipped with a graded symmetric degree `1 K-bilinear bracket +ℓ2 “ r¨ , ¨s: d2 E Ñ E +such that: +1. ℓ2 satisfies the Leibniz identity with respect to ρ: E´1 ÝÑ DerpOq, i.e. +ℓ2px, fyq “ fℓ2px, yq ` ρpxqrfsy +(2.1) +for all x P E´1, y P E and f P O. +2. ℓ1 is degree `1-derivation of ℓ2, i.e. for all x P E´i, y P E: +ℓ1pℓ2px, yqq ` ℓ2pℓ1pxq, yq ` p´1qiℓ2px, ℓ1pyqq “ 0, +3. ρ is a morphism, i.e. for all x, y P E´1 +ρpℓ2px, yqq “ rρpxq, ρpyqs. +The O-linear map ρ is called the anchor map, and ℓ1 the differential. +Remark 2.1.4. For any almost graded Lie-algebroid pE‚, ℓ1, ℓ2, ρq define the Jacobiator +Jacpx, y, zq “ ℓ2pℓ2px, yq, zq ` p´1q|y||z|ℓ2pℓ2px, zq, yq ` p´1q|x|p|y|`|z|qℓ2pℓ2py, zq, xq, +x, y, z P E +(2.2) +For any triple of elements x, y, z P E´1 of degree ´1. Equivalently: +Jacpx, y, zq “ ℓ2pℓ2px, yq, zq` ö px, y, zq, +where ö px, y, zq means that we sum on the circular permutations of x, y, z with Koszul signs. It is +graded symmetric of degree ´2. By axiom 3, the Jacobiator takes values in the kernel of the anchor +map, since +ρpJacpx, y, zqq “ ρpℓ2pℓ2px, yq, zqq` ö px, y, zq +“ rρpℓ2px, yqq, ρpzqs` ö px, y, zq +“ rrρpxq, ρpyqs, ρpzqs` ö px, y, zq “ 0, +since r¨ , ¨s satisfies Jacobi identity. +It is easy to see that the bracket ℓ2 goes to the quotient and endows, +E´1 +ker ρ and +E´1 +ℓ1pE´2q with a Lie +K-algebra bracket ¯ℓ2. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +25 +Remark 2.1.5. The map px, y, zq ÞÑ Jacpx, y, zq is O-trilinear: indeed, it is clear if x, y, z P Eď´2. +By graded symmetry, we need to verify this point when the at least one of the variables is of degree +´1. Let f P O, +Case 1: x P E´1 and y, z P Eď´2 +Jacpx, y, fzq “ ℓ2pℓ2px, yq, fzq ` p´1q|z||y|ℓ2pℓ2px, fzq, yq ` p´1q|z|`|y|ℓ2pℓ2py, fzq, xq +“ fℓ2pℓ2px, yq, zq ` p´1q|z||y|pℓ2pfℓ2px, zq, yq ` ℓ2pρpxqrfsz, yqq ` p´1q|z|`|y|fℓ2pℓ2py, zq, xq` +p´1q|z|`|y|p´1q|z|`|y|`1ρpxqrfsℓ2py, zq +“ fJacpx, y, zq ` ((((((((((( +( +p´1q|z||y|ρpxqrfsℓ2pz, yq ´ ((((((( +( +ρpxqrfsℓ2py, zq +Case 2: x, y P E´1 and z P Eď´2 +Jacpx, y, fzq “ ℓ2pℓ2px, yq, fzq ` p´1q|z|ℓ2pℓ2px, fzq, yq ` p´1q|z|`1ℓ2pℓ2py, fzq, xq +“ fℓ2pℓ2px, yq, zq ` ρpℓ2px, yqqrfsz ` p´1q|z|ℓ2pfℓ2px, zq ` ρpxqrfsz, yqq` +p´1q|z|`1ℓ2pfℓ2py, zq ` ρpyqrfsz, xq +“ fJacpx, y, zq ` ρpℓ2px, yqqrfsz ` p´1q|z|pp´1q|z|(((((((( +ρpyqrfs ℓ2px, zq ` ((((((( +( +ρpxqrfsℓ2pz, yq +` p´1q|z|ρpyqrρpxqrfsszq ` +(((((((((((((((( +p´1q|z|`1pp´1q|z|ρpxqrfs ℓ2py, zqq ` ((((((( +( +ρpyqrfsℓ2pz, xq +` p´1q|z|ρpxqrρpyqrfsszq +“ fJacpx, y, zq ` ((((((((((((( +pρpℓ2px, yqq ´ rρpxq, ρpyqsqrfsz +“ fJacpx, y, zq. +Case 3: for x, y, z P E´1, the proof is identical to case 2. +Example 2.1.6. A differential graded Lie algebra (DGLA) (see e.g [Man05a]) is a graded vector space +g “ +à +jPZ +gj over K together with a bilinear map called graded Lie bracket r¨ , ¨s : gi ˆ gj Ñ gi`j and a +differential map d: gi Ñ gi´1 such that for all homogeneous elements v1, v2, v3 P g has the properties +• graded commutativity: rv1, v2s “ ´p´1q|v1||v2|rv2, v1s, +• Jacobi’s identity: p´1q|v1||v3|rv1, rv2, v3ss ` p´1q|v2||v1|rv2, rv3, v1ss ` p´1q|v3||v2|rv3, rv1, v2ss “ 0, +• Leibniz’s identity: drv1, v2s “ rdv1, v2s ` p´1q|v1|rv1, dv2s. +DGLAs are examples of almost differential graded Lie algebroid (ADGLA) with O “ K (and +gj “ 0 for j ě 0). We take E´i :“ gir1s for i ě 1, ℓ1pxq :“ ´dpxq, and ℓ2px, yq :“ p´1q|x|rx, ys. Of +course, here ρ “ 0. Note that ADGLAs are more general than DGLAs, since it does not impose Jacobi +identity. +Example 2.1.7. [Hue03] An almost-Lie algebroid over a manifold M is a triple pA Ñ M, r¨ , ¨sA , ρAq +made of a vector bundle A Ñ M, a skew-symmetric bracket r¨ , ¨sA : ΓpAq ˆ ΓpAq Ñ ΓpAq, fulfilling +the Leibniz identity, i.e. for all a, b P ΓpAq, f P C8pMq +ra, fbsA “ fra, bsA ` ρApaqrfsb, + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +26 +and a vector bundle morphism ρ: A Ñ TM, that satisfies, +ρpra, bsAq “ rρApaq, ρApbqs. +We say that ρA is the anchor of pA Ñ M, r¨ , ¨sA , ρAq. When r¨ , ¨sA satisfies Jacobi’s identity, we speak +of a Lie algebroid over M [Mac05]. +Almost Lie algebroids over a manifold M give examples of almost differential graded Lie algebroid +over O “ C8pMq with E´1 “ ΓpAq, E´i “ 0 for i ‰ 1, ℓ2 “ r¨ , ¨sA, and the anchor is ρA. +2.1.2 +Lie 8-algebroids over O +Lie 8-algebroids over manifolds were introduced (explicitly or implicitly) by various authors, e.g. +[SSS12], [Vor10], and [Še01]. We refer to Giuseppe Bonavolontà and Norbert Poncin for a complete +overview of the matter [BP13]. It extends the notion of almost differential graded Lie algebroids. +Definition 2.1.8. [LS93] A negatively graded Lie 8-algebroid over O is a collection E “ pE´iqiě1 of +projective O-modules, equipped with: +1. a collection of linear maps ℓi : Si +KpEq ÝÑ E of degree `1 called i-ary brackets +2. a O-linear map E´1 ÝÑ DerpOq called anchor map +satisfying the following axioms : +piq the higher Jacobi identity: +nÿ +i“1 +ÿ +σPSi,n´i +ϵpσq ℓn´i`1pℓipxσp1q, . . . , xσpiqq, xσpi`1q, . . . , xσpnqq “ 0, +(2.3) +for all n ě 1 and homogeneous elements x1, . . . , xn P E, +piiq for i ‰ 2, the bracket ℓi is O-linear, while for i “ 2, +ℓ2px, fyq “ ρpxqrfs y ` fℓ2px, yq for all x, y P E, f P O , +where, by convention, ρE is extended by zero on E´i for all i ě 2, +piiiq ρ ˝ ℓ1 “ 0 on E´2, +pivq ρ is a morphism of brackets, i.e., ρpℓ2px, yqq “ rρpxq, ρpyqs for all x, y P E´1. +We denote it by pE‚, ℓ‚, ρq. +Convention 2.1.9. From now on, we simply say “Lie 8-algebroid” for “negatively graded Lie 8- +algebroid”. +Let us explain these axioms. +Remark 2.1.10. It follows from Definition 2.1.8 that: + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +27 +1. The higher Jacobi identity is equivalent to +nÿ +i“1 +rℓi, ℓn`1´isRN “ 0, +for all positive integer n. See Definition 1.2.19 a definition of r¨ , ¨sRN. +2. For n “ 1, higher Jacobi identity yields ℓ1 ˝ ℓ1 “ 0. Thus, +¨ ¨ ¨ +ℓ1 +ÝÑ E´3 +ℓ1 +ÝÑ E´2 +ℓ1 +ÝÑ E´1 +(2.4) +is a complex of projective O-modules. +3. Higher Jacobi identity for n “ 2 reads ℓ1pℓ2px, yqq ` ℓ2pℓ1pxq, yq ` p´1q|x|ℓ2px, ℓ1pyqq “ 0. +4. The third and the fourth axiom are consequences of item piq, and piiq if O has no zero divisors1 +on E´1. Indeed, for all x P E´2 and y P E´1 and f P O. Higher Jacobi identity specialized at +n “ 2 and Leibniz identity read +ℓ1pℓ2px, fyqq “ ℓ2pℓ1pxq, fyq +“ fℓ2pℓ1pxq, yq ` ρpℓ1pxqqrfsy +Using Leibniz identity again, the left-hand side of the equation above also reads ℓ1pℓ2px, fyqq “ +fℓ1pℓ2px, yqq. Hence: +ρpℓ1pxqqrfsy “ 0. +If O has no zero divisors on E´1, then ρpℓ1pxqqrfs “ 0. Since x and f are arbitrary, we have +ρ ˝ ℓ1|E´2 “ 0. +Also, by writing higher Jacobi for n “ 3 on elements x, y, z P E´1 and f P O while using Leibniz +identity we get +0 “ ℓ1pℓ3px, y, fzqq ` ℓ2pℓ2px, yq, fzq ´ ℓ2pℓ2px, fzq, yq ` ℓ2pℓ2py, fzq, xq +“ fℓ1pℓ3px, y, zqq ` fℓ2pℓ2px, yq, zq ` ρpℓ2px, yqqrfsz ´ ℓ2pfℓ2px, zq, yq ´ ℓ2pρpxqrfsz, yq +` ℓ2pfℓ2py, zq, xq ` ℓ2pρpyqrfsz, xq +“ f pℓ1pℓ3px, y, fzqq ` ℓ2pℓ2px, yq, zq ´ ℓ2pℓ2px, zq, yq ` ℓ2pℓ2py, zq, xqq +loooooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooooon +“0 +`ρpℓ2px, yqqrfsz +` ρpyqrρpxqrfssz ´ ρpxqrρpyqrfssz ` ((((((( +( +ρpyqrfsℓ2px, zq ` ((((((( +( +ρpyqrfsℓ2pz, xq ´ ((((((( +( +ρpxqrfsℓ2pz, yq +´ ((((((( +( +ρpxqrfsℓ2py, zq. +This implies, pρpℓ2px, yqqrfs ´ rρpxq, ρpyqsrfsqz “ 0. Therefore, when O has no zero divisors on +E´1, we obtain that +ρpℓ2px, yqq “ rρpxq, ρpyqs. +Definition 2.1.11. A Lie 8-algebroid is said to be acyclic if the complex (2.4) has no cohomology +in degree ď ´2. +1i.e. for every f P O the linear map E´1 +f +� E´1 given by multiplication by f, is injective. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +28 +Example 2.1.12. When O “ K we have only the axiom piq of 2.1.8, since the other axioms are trivial. +Therefore, we recover the axioms that define Lie 8-algebras [Sta92, LS93]. +Remark 2.1.13. Geometrically, we are in the case where the projective modules pE´iqiě1 of Definition +2.1.8 are finitely generated. Serre-Swan theorem [Swa62] assures for every i ě 1, E´i “ ΓpE´iq for +some vector bundle E´i Ñ M over a manifold M. Hence, Definition 2.1.8 is rewritten word by word +as follows: A (finitely generated) negatively graded Lie 8-algebroid pE, pℓkqkě1, ρq over a manifold M +is +1. a collection of vector bundles E “ pE´iqiě1 over M +2. together with a sheaf of Lie 8-algebra structures pℓkqkě1 over the sheaf of sections of E +3. that comes with a vector bundle morphism ρ: E´1 ÝÑ TM, called the anchor, +4. such that the k-ary-brackets are all O-multilinear except when k “ 2 and at least one of the +arguments is of degree ´1. The 2-ary bracket satisfies the Leibniz identity +ℓ2px, fyq “ ρpxqrfsy ` fℓ2px, yq, x P ΓpE´1q, y P ΓpEq. +(2.5) +2.2 +Lie 8-algebroids as homological co-derivations +When it comes to manipulating morphisms of Lie 8-alegbroids, it quickly becomes quite tedious. +Therefore, it is very useful to dualize in order to make the notion of morphisms clearer. +In the finite dimensional case [Vor10], it is usual to see it as a derivation of the symmetric algebra +of the dual, i.e. as a Q-manifold (see Chapter 6.1). The duality finite rank Lie 8-algebroids and +Q-manifolds is especially efficient to deal with morphisms. +In infinite dimension case, we cannot dualize Lie 8-algebroids in the sense of T. Voronov [Vor10, +BP13] anymore, since the identification of SpV q˚ with SpV ˚q does not hold. What I do to deal with this +problem in the infinite case, is to stay in the world of squared to zero co-derivations and impose some +particular additionnal properties on their Taylor coeficients, related to O-linearity and the Leibniz rule. +Now we give an alternative description of Lie 8-algebroids in terms of co-derivation (extending the +usual [Vor05, Vor10, BP13] correspondence between Lie 8-algebroids and Q-manifolds in the finite +rank case). +Definition 2.2.1. A co-derivation Q P CoDerpSKpEqq of degree `1 is said to be an homological co- +derivation or co-differential when Q2 “ 0. +Given such a co-derivation Q, the triplet pSKpEq, ∆, Qq is then called a differential graded co-algebra. +The following lemma is important. +Lemma 2.2.2. A co-derivation Q P CoDerpSKpEqq of degree `1 is an homological co-derivation if +and only if pr ˝ Q2 “ 0. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +29 +Proof. By Proposition 1.2.10, the bracket +rQ, Qs “ Q ˝ Q ´ p´1q|Q||Q|Q ˝ Q “ 2Q2 +is a co-derivation, since co-derivations of a graded co-algebra is closed under the graded commutator +bracket r¨ , ¨s. Therefore, Q2 “ 1 +2rQ, Qs is a co-derivation, and it is completely determined by pr ˝ +Q2. +Remark 2.2.3. The composition of two co-derivations is not a co-derivation in general. In particular, +the composition of a co-derivation with itself is not a co-derivation unless it is of odd degree. +For a good understanding of the next proposition, see notations of Taylor coefficients in Equation +(1.8). +Proposition 2.2.4. Given a collection E “ pE´iqiě1 of projective O-modules, there is a one-to- +one correspondence between Lie 8-algebroid structures pE‚, ℓ‚, ρq on E and pairs pQE, ρq made of an +homological co-derivation QE : S‚ +KpEq Ñ S‚ +KpEq and a O-linear morphism, ρ: E´1 Ñ DerpOq called the +anchor, such that +1. for k ‰ 1 the k-th Taylor coefficient Qpkq +E : Sk`1 +K +pE1q ÝÑ E of QE is O-multilinear, +2. for all x, y P E and f P O, we have, Qp1q +E px ¨ fyq “ fQp1q +E px ¨ yq ` ρpxqrfs y, +3. ρ ˝ Qp0q +E +“ 0 on E´2, +4. ρ ˝ Qp1q +E px ¨ yq “ rρpxq, ρpyqs, for all x, y P E´1. +The correspondence consists in assigning to a Lie 8-algebroid pE‚, ℓ‚, ρq “ pℓ1, ℓ2, ℓ3, ¨ ¨ ¨ q the co- +derivation QE whose k-th Taylor coefficient is the k-ary bracket ℓk`1 for all k P N0. +Proof. Take the i-th Taylor coefficient of co-derivation QE that satisfies the requirements 1. 2. 3. and +4. of Proposition 2.2.4 as, pr ˝ Qpiq +E “ ℓi`1 : Si`1 +K pEq Ñ E. By Lemma 1.2.22 we have +2Q2 +E “ rQE, QEs “ +ÿ +ně1 +ÿ +i`j“n`1 +Ğ +rℓi, ℓjsRN. +Thus, by uniqueness of the extension as co-derivation, Q2 +E “ 0 is equivalent to +0 “ +nÿ +i“1 +rℓi, ℓn`1´isRN P Hom2 +K pSn +KpEq , Eq . +For all integer n ě 1. +Remark 2.2.5. Note that, e.g. if O “ K, we recover the equivalence between Lie 8-algebras and +co-differentials. +Convention 2.2.6. From now, when relevant, we sometime denote an underlying structure of Lie +8-algebroid pE‚, ℓ‚, ρq on E by pE, QE, ρq instead. +Remark 2.2.7. Notice that QE : S‚ +KpEq ÝÑ S‚ +KpEq does not induce a co-derivation on Ä‚ E unless +ρ “ 0. +Let us make the correspondence given by Proposition 2.2.4 explicit on the following examples. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +30 +Example 2.2.8. Differential Graded Lie Algebras. We come back to Example 2.1.6. Any differential +graded Lie algebra pg “ ‘PZgi, r¨ , ¨s , dq is Lie 8-algebroid with trivial anchor, where the unary bracket +ℓ1 and the binary bracket ℓ2 are obtained by adding a sign to the differential map d and the graded +skew-symmetric Lie bracket r¨ , ¨s respectively, and the other brackets ℓk for k ě 3 vanish. For k ě 3, +the k-th Taylor coefficient of the corresponding co-derivation Qg is zero, hence it can be written as +Qg “ Qp0q +g +` Qp1q +g . For every homogeneous monomial x1 ¨ ¨ ¨ xk P Skg +Qp0q +g px1 ¨ ¨ ¨ xkq “ +kÿ +i“1 +p´1qp|x1|`¨¨¨`|xi´1|q|xi|ℓ1pxiq ¨ x1 ¨ ¨ ¨ pxi ¨ ¨ ¨ xk, +Qp1q +g px1 ¨ ¨ ¨ xkq “ +ÿ +1ďiăjďk +p´1qp|x1|`¨¨¨`|xj´1|q|xj|`p|x1|`¨¨¨`|xi´1|q|xi|ℓ2pxi, xjq ¨ x1 ¨ ¨ ¨ rxi ¨ ¨ ¨ pxj ¨ ¨ ¨ xk. +Proposition 2.2.4 and Remark 2.2.5 say that pg “ ‘PZgi, r¨ , ¨s , dq is a DGLA if and only if +• Q2 +g “ 0; +or +• pQp0q +g q2 “ pQp1q +g q2 “ 0 and Qp0q +g +˝ Qp1q +g +` Qp1q +g +˝ Qp0q +g +“ 0. +2.3 +Morphisms of Lie 8-algebroids and homotopies +This section extends Section 3.4 of [LLS20] to the infinite dimensional setting. +2.3.1 +Morphisms of Lie 8-algebroids +Definition 2.3.1. A Lie 8-algebroid morphism (or Lie 8-morphism) from a Lie 8-algebroid pE1, QE1, ρ1q +to a Lie 8-algebroid pE, QE, ρq, is a co-morphism Φ: S‚ +KpE1q ÝÑ S‚ +KpEq such that +Φ ˝ QE1 “ QE ˝ Φ +(2.6) +and +1. all Taylor coefficients are O-multilinear, +2. which satisfies ρ ˝ Φ0 “ ρ1 on E1 +´1. +Above, Φ0 “ Φp0q +|E1 : E1 Ñ E is the 0-th Taylor coefficient of Φ. Notice that item 1. is equivalent to +saying Φ is O-multilinear in the sense of Lemma 1.2.17. +Remark 2.3.2. Recall from [LS93] that Lie 8-algebra morphisms from pS‚ +KpEq, QEq to pS‚ +KpE1q, QE1q +are defined to be co-algebra morphisms Φ: S‚ +KpE1q ÝÑ S‚ +KpEq that satisfy (2.6). Definition 2.3.1 adds +two additional assumptions to turn a Lie 8-algebra morphism into a Lie 8-algebroid morphism. +Remark 2.3.3. For a Lie 8-algebroid morphism Φ: S‚ +KpE1q ÝÑ S‚ +KpEq, the 0-th Taylor coefficient +Φ0 : pE1, ℓ1 +1q ÝÑ pE, ℓ1q of Φ is the chain map, that is, the following diagram commutes +¨ ¨ ¨ +� E1 +´3 +Φ0 +� +ℓ1 +1 +� E1 +´2 +Φ0 +� +ℓ1 +1 +� E1 +´1 +Φ0 +� +ρ1 � DerpOq +id +� +¨ ¨ ¨ +� E´3 +ℓ1 +� E´2 +ℓ1 +� E´1 +ρ � DerpOq +(2.7) + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +31 +namely, Φ0 ˝ ℓ1 +1 “ ℓ1 ˝ Φ0. +Also, by going a bit further, it satisfies for all x, y P E1 +Φ0 ˝ ℓ1 +2px, yq ` Φ1 ˝ ℓ1 +1px d yq “ ℓ1 ˝ Φ1px d yq ` ℓ2pΦ0pxq, Φ0pyqq. +(2.8) +Example 2.3.4. Let pA, r¨ , ¨sA , ρAq and pB, r¨ , ¨sB , ρBq be two Lie algebroids over a manifold M. A +Lie algebroid morphism φ: A ÝÑ B over the identity [Mac05] induces a Lie 8-algebroid morphism +S‚pφq: S‚pΓpAqq Ñ S‚pΓpBqq in the section level: where corresponding co-algebra morphism S‚pφq is +defined as +a1 ¨ ¨ ¨ ¨ an ÞÑ φpa1q ¨ ¨ ¨ ¨ φpanq, +for all a1, . . . , an P A. +Example 2.3.5. Lie 8-morphisms of Differential Graded Lie Algebras. Let us consider pg, r¨ , ¨sg , dgq +and ph, r¨ , ¨sh , dhq two differential graded Lie algebras (see Example 2.1.6), where Qg and Qh de- +note their respective associated co-derivations. +Let Φ: pS‚pgr1sq, Qgq ÝÑ pS‚phr1sq, Qhq be a Lie +8-morphism, i.e. one has +Φ ˝ Qg “ Qh ˝ Φ. +(2.9) +In particular, Φ0 ˝dg “ dh ˝Φ0. Let us write down what the restriction of Equation (2.9) to low Taylor +coefficients: Let x, y P g be two homogeneous elements. One has, +Φpx ¨ yq “ Φ1px ¨ yq ` Φ0pxq ¨ Φ0pyq, +Φpxq “ Φ0pxq. +A direct computation of the LHS of Equation (2.9) applied to x ¨ y gives, +Φ ˝ Qgpx ¨ yq “ Φ +´ +´dgpxq ¨ yq ´ p´1qp|x|´1qp|y|´1qdgpyq ¨ x ` p´1q|x|rx, ysg +¯ +“ ´Φ1pdgpxq ¨ yq ´ Φ0pdgpxqq ¨ Φ0pyq ´ p´1qp|x|´1qp|y|´1q pΦ1pdgpyq ¨ xq ´ Φ0pdgpyqq ¨ Φ0pxqq +` p´1q|x|Φ0prx, ysgq +Also, the RHS of Equation (2.9) applied to x ¨ y gives, +Qh ˝ Φpx ¨ yq “ QhpΦ1px ¨ yq ` Φ0pxq ¨ Φ0pyqq +“ ´dhpΦ1pxqq ¨ y ´ dhpΦ0pxqq ¨ Φ0pyq ´ p´1qp|x|´1qp|y|´1qdhpΦ0pyqq ¨ Φ0pxq +` p´1q|x|rΦ0pxq, Φ0pyqsh +Since both sides are equal, and Φ0 ˝ dg “ dh ˝ Φ0 we obtain, +Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqsh “ p´1qp|x|´1q|y|qΦ1pdgpyq ¨ xq ´ p´1q|x| pΦ1pdgpxq ¨ yq ´ dhpΦ1px ¨ yqqq . +2.3.2 +Homotopies +In this section, we define homotopy between Lie 8-morphisms, and between Lie 8-algebroids. This +extends [LLS20] from finite dimensional Q-manifolds to arbitrary Lie 8-algebroids. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +32 +A practical definition +Let us consider pΩ‚pra, bsq, ^, ddRq the differential graded algebra made of the forms on ra, bs together +with the wedge product and the Rham differential. If we denote by t the coordinate on ra, bs, we have +Ω‚pra, bsq “ C8pra, bsq ‘ C8pra, bsq dt. We equip Ω‚pra, bsq with a co-associative co-algebra structure +∆: Ω‚pra, bsq Ñ Ω‚pra, bsq bΩ‚pra,bsq Ω‚pra, bsq, given by ∆Ωp1q “ 1 b 1, where we extend by Ω‚pra, bsq- +linearity. +Let pE1, QE1, ρ1q and pE, QE, ρq be Lie 8-algebroids over O. +Lemma 2.3.6. The triplet +` +S‚ +KpEq bK Ω‚pra, bsq, ¯∆ “ pid b τ b idq ˝ ∆ b ∆Ω, B “ QE1 b id ` id b ddR +˘ +is a differential graded co-algebra. +Proof. One should understand that for α P C8pra, bsq and for an homogeneous element v P S‚ +KpEq, +pQE b id ` id b ddRqpv b αq “ QEpvq b α ` p´1q|v|v b α1dt +also, +pQE b id ` id b ddRqpv b αdtq “ QEpvq b αdt ` v b ddRpαdtq +loooomoooon +“0 +“ QEpvq b αdt. +Clearly, B2 “ 0, see the definition of tensor product of complexes, Appendix B. Let us check that B +is indeed a co-derivation w.r.t the co-product ¯∆. Take α P C8pra, bsq and a homogeneous element +v P S‚ +KpEq, we will use the Sweedler notation, ∆pvq “ vp1q b vp2q, to avoid a long useless text. On one +hand, +¯∆ ˝ Bpv b αq “ pid b τ b idq ˝ ∆ b ∆ΩpQEpvq b α ` p´1q|v|v b α1dtq +“ id b τ b id +´ +∆ ˝ QEpvq b α b 1 ` p´1q|v|∆pvq b α1dt b 1 +¯ +“ pid b τ b idq ˝ +´ +pQE b id ` id b QEq ˝ ∆pvq b α b 1 ` p´1q|v|∆pvq b α1dt b 1 +¯ +“ QEpvp1qq b α b vp2q b 1 ` p´1q|vp1q|vp1q b α b QEpvp2qq b 1` +p´1q|v|`|vp2q|vp1q b pα1dtq b vp2q b 1. +On the other hand, +pB b id ` id b Bq ˝ ¯∆pv b αq “ pB b id ` id b Bq ˝ pid b τ b idqp∆pvq b α b 1q +“ pB b id ` id b Bqpvp1q b α b vp2q b 1q +“ Bpvp1q b αq b vp1q b 1 ` p´1q|vp1q|vp1q b α b Bpvp2q b 1q +“ +´ +QEpvp1q b α ` p´1q|vp1q|vp1q b α1dtq +¯ +b vp2q b 1` +p´1q|vp1q|vp1q b α b QEpvp2qq b 1. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +33 +Both sides are obviously equal. Likewise, a straightforward computation shows that ¯∆ ˝ Bpv b αdtq “ +pB b id ` id b Bq ˝ ¯∆pv b αdtq. Hence +¯∆ ˝ B “ pB b id ` id b Bq ˝ ¯∆ +on S‚ +KpEq bK Ω‚pra, bsq. +Remark 2.3.7. Elements of degree k of S‚ +KpEq bK Ω‚pra, bsq are K-linear combinations of elements +of the form v b α and w b βdt, with α, β P C8pra, bsq and v P S‚ +KpEq|k and w P S‚ +KpEq|k´1. The +latter can be seen as maps, t P ra, bs ÞÑ αptqv b 1 and t P ra, bs ÞÑ βptqw b dt. Hence, elements of +degree k of this complex can be considered as element of SKpEq bK Ω‚pra, bsq that depend on t, i.e. as +t P ra, bs ÞÑ vt b 1 ` wt b dt, with vt P S‚ +KpE1q|k and wt P S‚ +KpE1q|k´1. +The following is a temporary definition. It will be generalized later to another more practical for +gluing homotopies. +Definition 2.3.8. A homotopy that joins two Lie 8-algebroid morphisms Φ, Ψ: S‚ +KpE1q Ñ S‚ +KpEq is +the datum made of an interval ra, bs Ă R and a chain map +pS‚ +KpE1q, QE1q +H +ÝÑ pS‚ +KpEq bK Ω‚pra, bsq, QE b id ` id b ddRq +v ÞÝÑ +´ +t P ra, bs ÞÑ Jtpvq b 1 ´ p´1q|v|Htpvq b dt +¯ +1. which is a co-algebra morphism, +2. and that coincides with Φ and Ψ at t “ a and b respectively, i.e. for all v P S‚ +KpE1q, one has +Hpvqpaq “ Φpvq b 1 and Hpvqpbq “ Ψpvq b 1. +Remark 2.3.9. In the definition above, the map H induces for every t P ra, bs two different O- +multilinear maps +$ +& +% +Jt : S‚ +KpE1q ÝÑ S‚ +KpEq +Ht : S‚ +KpE1q ÝÑ S‚ +KpEq. +. +Since H is of degree 0, one of the maps is of degree zero and the other one of degree ´1 respectively. +By using respectively the property of co-algebra morphisms and chain map property, we obtain the +following for every t P ra, bs: +1. Let ¯∆ be the co-product on S‚ +KpEq bK Ω‚pra, bsq like in Lemma 2.3.6. We have, +¯∆ ˝ Hpvq “ H b H ˝ ∆1pvq. +Let v “ v1 ¨ ¨ ¨ vn P S‚ +KpE1q, a direct computation of gives us +∆ b ∆Ω ˝ Hpvqptq “ ∆ ˝ Jtpvq b 1 b 1 ´ p´1q|v|∆ ˝ Htpvq b dt b 1. +Let us use the Sweedler notation just like in Lemma 2.3.6 to compute H b H ˝ ∆1pvqptq. We +have, +H b H ˝ ∆1pvqptq “ H b Hpvp1q b vp2qqptq +“ Hptqpvp1qq b Hpvp1qqptq +“ +´ +Jtpvp1qq b 1 ´ p´1q|vp1q|Htpvp1qq b dt +¯ +b +´ +Jtpvp2qq b 1 ´ p´1q|vp2q|Htpvp2qq b dt +¯ + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +34 +Hence, +pid b τ b idq´1 ˝ pHbHq ˝ ∆1pvq “ +pJt b Jtq ˝ ∆1pvq b 1 b 1 ` ((((((((((((( +Ht b Ht ˝ ∆1pvq b dt b dt +` p´1q|v| ` +Ht b Jt ˝ ∆1pvq ` Jt b Ht ˝ ∆1pvq +˘ +b dt ˆ 1 +By equating both sides, we obtain equations that say that Jt is a co-morphism and Ht a Jt-co- +derivation. +2. and for any homogeneous element v P S‚ +KpE1q we have: in one side +H ˝ QE1pvqptq “ Jt ˝ QE1pvq b 1 ´ p´1q|v|`1Htpvq ˝ QE1pvq b dt. +to the other side2, +pQE b id ` id b ddRq ˝ Hpvqptq “ pQE b id ` id b ddRq ˝ +´ +Jtpvq b 1 ´ p´1q|v|Htpvq b dt +¯ +“ QE ˝ Jtpvq b 1 ` p´1q|v| dJt +dt pvq b dt ´ p´1q|v|QE ˝ Htpvq b dt +By equating both sides we see that Jt and Ht satisfy the following condition +$ +& +% +Jt ˝ QE1pvq “ QE ˝ Jtpvq +dJt +dt pvq “ QE ˝ Htpvq ` Ht ˝ QE1pvq. +(2.10) +In addition, we have for v P E1 +´1 that +dJp0q +t +dt pvq “ Qp0q +E +˝ Hp0q +t +pvq ` Hp0q +t +˝ Qp0q +E1 +looooomooooon +“0 +pvq +and +Jp0q +t +pvq “ Φp0qpvq ` +ż t +a +ℓ1 ˝ Hp0q +s pvqds +“ Φp0qpvq ` ℓ1 ˝ +ż t +a +Hp0q +s pvqds +(2.11) +This implies that ρ ˝ Jp0q +t +pvq “ ρ ˝ Φp0qpvq “ ρ1pvq, since ρ ˝ ℓ1 “ 0. +Gluing homotopies as defined in Definition 2.3.8 is however not so easy. We can reformulate the +definition of homotopies between Lie 8-morphisms in the following manner. Before we do this, we +would like to fix some vocabulary. +Let us recall some facts on vector-valued functions. Let V be a vector space. Unless a topology on +V is chosen, the notion of V -valued continuous or differentiable or smooth function, and the concept +of a limit on an interval I “ ra, bs Ă R do not make any sense. However, we can always define the +following notion +2It should be understood that e.g, pQE b id ` id b ddRqpαptqv b 1q “ pQEpvq b α ` v b ddRαq ptq. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +35 +Definition 2.3.10. A vector-valued map γ : I ÝÑ V is said to be a piecewise rational path on I if +there exists a finite increasing sequence a “ t0 ď ¨ ¨ ¨ ď tN “ b of gluing points, such that for all +i “ 0, . . . , N ´ 1 the restriction γi of γ to rti, ti`1s is of the form +γiptq “ +nÿ +j“1 +βi +jptqvj, +for some n P N +where for every j “ 1, . . . , n, vj P V and βi +j : I Ñ R a real rational function on rti, ti`1s that has no +pole on rti, ti`1s. +1. We say that γ is continuous on I, if for all i “ 1, . . . , N ´ 1, γi and γi`1 coincide at the gluing +point ti`1. +2. When V is a space of linear maps between the vector spaces S and T, we shall say that a +V -valued map Ξ: I Ñ V is a piecewise rational (continuous) if the map pt P I ÞÑ Ξtpsqq is a +piecewise rational (continuous) T-valued path for all s P S. +3. We define the limit of γ at u P rti, ti`1s to be equal to +nÿ +j“1 +lim +tÑu βi +jptqvj +when, for every j “ 1, . . . , n, βi +j admits a limit a u. One can assume that tv1, v2, v3 . . . u is a +basis of V . +Let us recall the following. +Remark 2.3.11. It is a very classical fact that the integral of a piecewise-C1 function β : I ÝÑ R +on a compact interval I “ ra, bs Ă R which is subordinate to a subdivision a “ t0 ă ¨ ¨ ¨ ă tN “ b +admits primitives which are piecewise-C1 on the same subdivision a “ t0 ă ¨ ¨ ¨ ă tN “ b. +An +important fact is that continuous piecewise-C1 functions β : I ÝÑ R, i.e. +piecewise-C1 functions +which are also continuous (even at the junction points) admit piecewise continuous derivatives β1ptq, +and βpbq ´ βpaq “ +şb +a β1ptqdt. +Definition 2.3.10 has this important consequence. +Lemma 2.3.12. A piecewise rational continuous function γ : I Ñ V is differentiable at every point +which is not a gluing point, and the latter is again piecewise rational on I. +Conversely, every piecewise rational functions admit a piecewise rational continuous primitive, +unique up to a constant. +Proof. The derivative of γ can be defined in the usual way using the item 3 of Definition 2.3.10. +Likewise, for their primitives. The proof is then immediate. +We can now give the following definition. +Definition 2.3.13. A family pJtqtPI of co-algebra morphisms are said to be piecewise rational con- +tinuous if its Taylor coefficients Jpnq +t +are piecewise rational continuous for all n P N. +For such a family pJtqtPI, a family pHtqtPI made of Jt-co-derivations is said to be piecewise rational +if all its Taylor coefficients are. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +36 +We are now ready to define formulate the definition of homotopies as follows and extend Definition +3.53 in [LLS20] to the infinite rank case. +Definition 2.3.14. Let Φ and Ψ be Lie 8-algebroid morphisms from pE1, QE1, ρ1q to pE, QE, ρq. A +homotopy between or that joins Φ and Ψ is a pair pJt, HtqtPra,bs consisting of: +1. a piecewise rational continuous path t ÞÑ Jt valued in Lie 8-algebroid morphisms between S‚ +KpE1q +and S‚ +KpEq satisfying the boundary condition: +Φa “ Φ +and +Φb “ Ψ, +2. a piecewise rational path t ÞÑ Ht, with Ht a Jt-co-derivations of degree ´1 from S‚ +KpE1q to S‚ +KpEq, +such that the following equation: +dJt +dt “ QE ˝ Ht ` Ht ˝ QE1 +(2.12) +holds for every t Psa , br where it is defined (that is, not a gluing point for the Taylor coefficients). +More precisely, for every v P Sďn +K pE1q, +dJt +dt pvq “ QE ˝ Htpvq ` Ht ˝ QE1pvq +(2.13) +for all t which is not a gluing point of the Taylor coefficient of Jpkq +t +, Hpkq +t +for k “ 0, . . . , n. +When these conditions are satisfied, we say that Φ and Ψ are homotopy equivalent, and we write +Φ „ Ψ. +Remark 2.3.15. This clearly extends (2.10). +Remark 2.3.16. In the above definitions, it is not required that the gluing points of the various +Taylor coefficients Jpnq +t +or Hpnq +t +to be the same for all n P N0. +The following Proposition shows that the notion of homotopy given in Definition 2.3.14 implies +the usual notion of homotopy between chain maps (see Appendix B). +Proposition 2.3.17. Let Φ and Ψ be Lie 8-algebroid morphisms from pE1, QE1, ρ1q to pE, QE, ρq which +are homotopic. Then, +Ψ ´ Φ “ QE ˝ H ` H ˝ QE1 +(2.14) +for some O-linear map H : S‚ +KpE1q ÝÑ S‚ +KpEq of degree ´1. +Proof. Take pJt, HtqtPra,bs a homotopy between Φ and Ψ as in Definition 2.3.14. By definition, t ÞÑ Jt +is piecewise rational continuous, therefore it is continuous on ra, bs (even at the junctions points), we +can use Remark 2.3.11 and write +Φb ´ Φa “ +ż b +a +dJt +dt dt +Ψ ´ Φ “ +ż b +a +pQE ˝ Ht ` Ht ˝ QE1q dt +“ QE ˝ +ˆż b +a +Ht dt +˙ +` +ˆż b +a +Ht dt +˙ +˝ QE1. +Thus, the O-multilinear map H :“ +şb +a Ht dt satisfies Equation (2.14). + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +37 +Definition 2.3.18. We say that two Lie 8-algebroids pE1, QE1, ρ1q to pE, QE, ρq over O are homotopy +equivalent or homotopic if there exists two Lie 8-algebroid morphisms +pE1, QE1, ρ1q +Φ +� pE, QE, ρq +Ψ +� +such that Φ˝Ψ: pE, QE, ρq Ñ pE, QE, ρq and Ψ˝Φ: pE1, QE1, ρ1q Ñ pE1, QE1, ρ1q are homotopy equivalent +to the identity map of respective space. +The following proposition justifies Definition 2.3.14. +Proposition 2.3.19. Let Φ be a Lie 8-algebroid morphism from pE1, QE1, ρ1q to pE, QE, ρq. For all +t P ra , bs, let Hpnq +t +: Sn`1 +K +pE1q Ñ +E be a family O-multilinear piecewise rational maps indexed by +n P N0. Then, +1. There exists a unique piecewise rational continuous family of co-algebra morphisms Jt such that +(a) Ja “ Φ +(b) pJt, Htq is a solution of the differential equation (2.12), where Ht is the Jt-co-derivation +whose n-th Taylor coefficient is Hpnq +t +for all n ě 0. +2. Moreover, for all t P ra, bs, pJs, HsqsPra,ts is a homotopy that joins Φ and Jt. +Proof. Let us show item 1. We claim that equation (2.12) is a differential equation that can be solved +recursively. In polynomial-degree zero, it reads, +dJp0q +t +dt +“ Qp0q +E +˝ Hp0q +t +` Hp0q +t +˝ Qp0q +E1 +(2.15) +and +Jp0q +t +“ Φp0q ` +ż t +a +´ +Qp0q +E +˝ Hp0q +s +` Hp0q +s +˝ Qp0q +E1 +¯ +ds +(2.16) +is defined for all t P ra, bs. Also, d +dtJpn`1q +t +: Sn`2pE1q Ñ E is an algebraic expression of Qp0q +E , . . . , Qpn`1q +E +, +Qp0q +E1 , . . . , Qpn`1q +E1 +Jp0q +t +, . . . , Jpnq +t +, Hp0q +t +, . . . , Hpn`1q +t +. But Jpn`1q +t +does not appear in the pn ` 1q-th Taylor +coefficient of QE ˝Ht `Ht ˝QE1 by Equation (1.10). By Lemma 2.3.12, there exists a unique piecewise +rational continuous solution Jpn`1q +t +such that Jpn`1q +a +“ Φpn`1q. The construction of the Taylor coeffi- +cients of the co-algebra morphisms Jt then goes by recursion. Recursion formulas also show that Jt is +unique. +Let us show item 2. i.e. that Jt is a O-multilinear chain map for all t P ra, bs: For the same reason as +in Equation (2.11), Equation (2.16) implies that ρ ˝ Jp0q +t +|E1 “ ρ1. The function given by +Λkptq “ pQE ˝ Jt ´ Jt ˝ QE1qpkq +for all +t P ra, bs, k P N0 +are differentiable at all points t except for a finitely many t P ra, bs and are piecewise rational con- +tinuous. The map dJt +dt is a Lie 8-morphism because Q2 +E “ 0 and Q2 +E1 “ 0, hence Λ1ptq “ 0. By +continuity, Λkptq is constant over the interval ra, bs. Since Ja “ Φ is a Lie 8-algebroid morphism, we +have Λkpaq “ 0. Thus, Λkptq “ 0 and, +QE ˝ Jt “ Jt ˝ QE1, +for all t P ra, bs. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +38 +Let us show that homotopy in the sense above defines an equivalence relation „ between Lie +8-morphisms. We have the following lemma. +Lemma 2.3.20. A pair pJt, Htq is a homotopy between Lie 8-algebroid morphisms Ja and Jb if and +only if for all rational function, g: ra, bs Ñ rc, ds without poles on ra, bs, the pair pJgptq, g1ptqHgptqq is a +homotopy between Jgpaq and Jgpbq. +Proof. Let g: ra, bs Ñ rc, ds be a rational function without poles on ra, bs. A straightforward compu- +tation gives: +dJt +dt +“ +QE ˝ Ht ` Ht ˝ QE1 +(by definitionq +ñ +dJ +dt pgptqq +“ +QE ˝ Hgptq ` Hgptq ˝ QE1 +(by replacing t by gptq) +ñ +dJgptq +dt +“ +QE ˝ +` +g1ptqHgptq +˘ +` +` +g1ptqHgptq +˘ +˝ QE1 +(by multiplying both sides by g1ptq). +The last equation means that pJgptq, g1ptqHgptqq is a homotopy between Jgpaq and Jgpbq. The backward +implication is obvious, it suffices to consider a “ c, b “ d and g “ id. +Proposition 2.3.21. Homotopy between Lie 8-morphisms is an equivalence relation. In addition, +it is compatible with composition, that is, if Φ, Ψ: S‚ +KpE1q Ñ S‚ +KpEq are homotopic Lie 8-algebroid +morphisms and ˆΦ, ˆΨ: S‚ +KpEq Ñ S‚ +KpE2q are homotopic Lie 8-algebroid morphisms, then, so are their +compositions ˆΦ ˝ Φ and ˆΨ ˝ Ψ. +Proof. We first show that this notion of homotopy is an equivalence relation. Let Φ, Ψ and Ξ: S‚ +KpE1q ÝÑ +S‚ +KpEq be three Lie 8-morphisms of algebroids. +‚ Reflexivity: The pair pJt “ Φ, Ht “ 0qtPr0,1s defines a homotopy between Φ and Φ. +‚ Symmetry: Let pJt, HtqtPr0,1s be a homotopy between Φ to Ψ. By applying Lemma 2.3.20 with +gptq “ 1 ´ t, we obtain a homotopy between Ψ and Φ via the pair pΦ1´t, ´H1´tqtPr0,1s. +‚ Transitivity: Assume Φ„Ψ and Ψ„Ξ and let pJt, H1,tqtPr0, 1 +2 s be a homotopy between Φ and Ψ +and let p ¯Jt, H2,tqtPr 1 +2 ,1s be a homotopy between Ψ and Ξ. By gluing Jt and ¯Jt, respectively H1t +and H2,t we obtain a homotopy p ˜Jt, HtqtPr0,1s between Φ and Ξ. +We then show it is compatible with composition. Let us denote by pJt, Htq the homotopy between +Φ and Ψ, and p ˆJt, ˆHtq the homotopy between ˆΦ and ˆΨ . We obtain, +d ˆJt ˝ Jt +dt +“ d ˆJt +dt ˝ Jt ` ˆJt ˝ d ˆJt +dt +“ QE2 ˝ +´ +ˆHt ˝ Jt ` ˆJt ˝ Ht +¯ +` +´ +ˆHt ˝ Jt ` ˆJt ˝ Ht +¯ +˝ QE1. +Hence, ˆΦ ˝ Φ and ˆΨ ˝ Ψ are homotopic via the pair p ˆJt ˝ Jt, ˆHt ˝ Jt ` ˆJt ˝ Htq which is easily checked +to satisfy all axioms. This concludes the proof. +We conclude this section with a lemma that will be useful in the sequel. Notice that this is the +lemma that forces to extend Definition 2.3.8, for it would not be true anymore in the smooth setting. +Lemma 2.3.22. Let pJt, HtqtPrc,`8r be a homotopy such that for all n P N0 and for every t ě n, +Hpnq +t +“ 0. Then the n-th Taylor coefficient Jpnq +t +is constant on rn, `8r and the co-algebra morphism +J8 whose n-th Taylor coefficient is Jpnq +t +for any n P N0 and t P rn, `8r is a Lie 8-algebroid morphism. +Moreover, for g : ra, brÑ rc, `8r a rational function with no pole on ra, br and such that lim +tÑb gptq “ +`8, the pair pJgptq, g1ptqHgptqq is a homotopy between Jc and J8. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +39 +Proof. Since the n-th Taylor coefficient of the Jt-co-derivation dJpnq +t +dt +“ pQE ˝Ht `Ht ˝QE1qpnq depends +only on Hpiq +t +for i “ 0, . . . , n ´ 1, we have by assumption dJpnq +t +dt +“ 0 for all t ě n. As a consequence, +Jpnq +t +is constant on rn, `8r. It follows from Proposition 2.3.19 that J8 is a Lie 8-algebroid morphism +since for every n P N0 and t P rn, `8r +pQE ˝ J8 ´ J8 ˝ QE1qpnq “ +ÿ +i`j“n +pQpiq +E ˝ Jpiq +t +´ Jpjq +t +˝ Qpjq +E1 q +“ 0 +(since Jt is a Lie 8-algebroid morphism). +Let us prove the last part of the statement. By assumption, there exists a ď bn ď b such that for +all t P rbn, bs, we have gptq ě n, so that Jpnq +gptq “ Jpnq +8 +and g1ptqHpnq +gptq “ 0 on rbn, bs. The function Jpnq +t +(resp. Hpnq +t +) being piecewise rational continuous (resp. piecewise rational) on rc, ns, the same holds +for Jpnq +gptq (resp. g1ptqHpnq +gptq) on ra, bns. By gluing with a constant function J8 (resp. with 0), we see +that all Taylor coefficients of Jgptq (resp. g1ptqHgptq) are piecewise rational continuous (resp. piecewise +rational) with finitely many gluing points. This completes the proof. +Remark 2.3.23. Lemma 2.3.22 explains how to glue infinitely many homotopies, at least when for +a given n P N0, only finitely of them affects the n-th Taylor coefficient. This would not be possible +using only Definition 2.3.8. +2.3.3 +More technical lemmas and propositions +Let us state and prove these technical assertions for later use. +Let pE1, QE1, ρ1q and pE, QE, ρq be a Lie 8-algebroid over O. +Proposition 2.3.24. Let Φ: S‚ +KpE1q Ñ S‚ +KpEq be a Lie 8-algebroid morphism. For H: S‚ +KpE1q Ñ +S‚ +KpEq a O-multilinear Φ-co-derivation of degree k ď ´1, H ˝ QE1 ´ p´1qkQE ˝ H is a O-multilinear +Φ-co-derivation of degree k ` 1. +Proof. We first check that H ˝ QE ´ p´1qkQE1 ˝ H is a Φ-co-derivation: +∆ ˝ H ˝ QE1 “ pH b Φ ` Φ b Hq ˝ ∆1 ˝ Q1 +E, by definition of H +“ pH b Φ ` Φ b Hq ˝ pQE1 b id ` id b QE1q ˝ ∆1, by definition of Q1 +E +“ pH ˝ QE1 b Φ ` p´1qkΦ ˝ QE1 b H ` H b Φ ˝ QE1 ` Φ b H ˝ QE1q ˝ ∆1 +Subtracting a similar equation for p´1qk∆ ˝ QE ˝ H and using (2.6), one obtains the Φ-co-derivation +property for H ˝ QE1 ´ p´1qkQE1 ˝ H. We now check that H ˝ QE1 ´ p´1qkQE ˝ H is O-multilinear, +for which it suffices to check that its Taylor coefficients are O-multilinear by Lemma 1.2.17. +Let +x1, . . . , xn P E1 be homogeneous elements. Assume xi P E1 +´1 (if we have more elements of degree ´1 +the same reasoning holds). To verify O-multilinearity it suffices to check that for all f P O: +pr ˝ pH ˝ QE1 ´ p´1qkQE ˝ Hqpx1, . . . ,xi, . . . , fxj . . . , xnq “ +fpr ˝ pH ˝ QE1 ´ p´1qkQE ˝ Hqpx1, . . . , xi, . . . , xj . . . , xnq. +Only the terms where the 2-ary bracket with a degree ´1 element on one-side and f on the other side +may forbid f to go in front. There are two such terms: +ϵpxi, xj, xIijqHpn´1qpℓ1 +2pxi, fxjq, xIijq and ´ p´1qkϵpxi, xIiqℓ2pΦ0pxiq, fHpn´1qpxIiqq + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +40 +where xIi and xIij stand for the list x1, . . . , xn where xi and xi, xj are missing, respectively, and Hpn´1q +is the pn ´ 1q-th Taylor coefficient of H. Since ρ ˝ Φ0 “ ρ1, in both terms ρpxiqrfs appears, and these +two terms add up to zero. +Remark 2.3.25. If the degree of H is non-negative, then H ˝ QE1 ´ p´1qkQE ˝ H may not be O- +multilinear anymore, since there may exist extra terms where the anchor map appears, e.g. terms of +the form ℓ2pHpxIjq, fΦp0qpxjqq. +Lemma 2.3.26. Let Φ: S‚ +KpE1q Ñ S‚ +KpEq be a co-algebra morphism such that +1. Φ is O-multilinear, +2. ρ ˝ Φ0 “ ρ1 on E1, +If for every 3 n P N0, pΦ ˝ QE1 ´ QE ˝ Φqpiq “ 0 for each 0 ď i ď n, then the map SKpE1q Ñ SKpEq +given by: +pΦ ˝ QE1 ´ QE ˝ Φqpn`1q +1. is a Φp0q-co-derivation of degree `1, +2. is O-multilinear, +3. and the induced Φp0q-co-derivation +´Ä‚ E1, Qp0q +E1 +¯ +ÝÑ +´Ä‚ E, Qp0q +E +¯ +satisfies: +Qp0q +E +˝ pΦ ˝ QE1 ´ QE ˝ Φqpn`1q “ pQE ˝ Φ ´ Φ ˝ QE1qpn`1q ˝ Qp0q +E1 . +Proof. A straightforward computation yields: +∆pΦ ˝ QE1 ´ QE ˝ Φq “ pΦ b Φq ˝ ∆1 ˝ QE1 ´ pQE b id ` id b QEq ˝ ∆ ˝ Φ +“ ppΦ ˝ QE1 ´ QE ˝ Φq b Φ ` Φ b pΦ ˝ QE1 ´ QE ˝ Φqq ˝ ∆1. +Now, ∆ preserves polynomial-degree, i.e. ∆ : Sn +KpEq ÝÑ ‘i`j“nSi +KpEq b Sj +KpEq and so does ∆1. Taking +into account the assumption pΦ ˝ QE1 ´ QE ˝ Φqpiq “ 0 for every 0 ď i ď n, we obtain: +∆ ˝ pΦ ˝ QE1 ´ QE ˝ Φqpn`1q +“ +´ +pΦ ˝ QE1 ´ QE ˝ Φqpn`1q b Φp0q ` Φp0q b pΦ ˝ QE1 ´ QE ˝ Φqpn`1q¯ +˝ ∆1. +All the other terms disappear for polynomial-degree reasons Hence, pΦ ˝ QE1 ´ QE ˝ Φqpn`1q is a +Φp0q-co-derivation. +Let us prove that it is O-linear. It suffices to check O-linearity of TΦ :“ Φ ˝ QE1 ´ QE ˝ Φ. Let +us choose homogeneous elements x1, . . . , xN P E1 and let us assume that xi P E1 +´1 is the only term +of degree ´1: The proof in the case where there is more than one such an homogeneous element of +degree ´1 is identical. We choose j ‰ i and we compute TΦpx1, . . . , xi, . . . , fxj, . . . , xNq for some +f P O. The only terms in the previous expression which are maybe non-linear in f are those for which +the 2-ary brackets of a term containing fxj with xi or Φ0pxiq appear (since Φ and all other brackets +3Φ ˝ QE1 ´ QE ˝ Φ being a Φ-co-derivation, its component of polynomial-degree i is zero for 0 ď i ď n if only if its +i-th Taylor coefficient is zero for 0 ď i ď n. + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +41 +are O-linear). There are two such terms. The first one appears when we apply QE1 first, and then Φ: +this forces Φ pℓ1 +2pxi, fxjq, xIijq to appear, and the non-linear term is then: +ϵpσiqρ1pxiqrfs ΦpxIiq +(2.17) +with σi the permutation that let i goes in front and leave the remaining terms unchanged. There +is a second term that appears when one applies Φ first, then QE. +Since it is a co-morphism, +Φpx1 . . . xi . . . , fxj . . . xNq is the product of several terms among which only one is of degree ´1, +namely the term +ϵpx, σiqΦ0pxiqΦpfxIiq. +Applying QE to this term yields the non-linear term +ϵpσiqρpΦ0pxiqqrfs ΦpxIiq, +(2.18) +where Ii and Iij are as in Proposition 2.3.24. Since ρ ˝ Φ0 “ ρ1, we see that the terms (2.17) and +(2.18) containing an anchor add up to zero. +Let us check that pΦ ˝ QE1 ´ QE ˝ Φqpn`1q is a chain map, in the sense that it satisfies item 3). +Considering again TΦ :“ Φ ˝ QE1 ´ QE ˝ Φ, we have that T pkq +Φ +“ 0, for all k “ 0, . . . , n. +Since +TΦ ˝ QE1 “ QE ˝ TΦ, one has +0 “ pTΦ ˝ QE1 ` QE ˝ TΦqpn`1q “ T pn`1q +Φ +˝ Qp0q +E1 ` Qp0q +E +˝ T pn`1q +Φ +` +ÿ +i`j“n`1 +i,jě1 +´ +T piq +Φ ˝ Qpjq +E1 ` Qpjq +E +˝ T piq +Φ +¯ +loooooooooooooooomoooooooooooooooon +0 +By consequent, the O-linear map pΦ ˝ QE1 ´ QE ˝ Φqpn`1q satisfies item 3. +Lemma 2.3.27. Let Φ, Ξ : S‚ +KpE1q ÝÑ S‚ +KpEq be O-linear Lie 8-algebroid morphisms. Let n P N0. If +Ξpiq “ Φpiq for every 0 ď i ď n, then pΞ ´ Φqpn`1q : S‚ +KpE1q ÝÑ S‚ +KpEq +1. is a Φp0q-co-derivation +2. is O-multilinear +3. and the induced Φp0q-co-derivation +´Ä‚ E1, Qp0q +E1 +¯ +ÝÑ +´Ä‚ E, Qp0q +E +¯ +satisfies: +Qp0q +E +˝ pΞ ´ Φqpn`1q “ pΞ ´ Φqpn`1q ˝ Qp0q +E1 . +Proof. For all x1, . . . , xk P E1, one has: +∆pΞ ´ Φqpx1 d ¨ ¨ ¨ d xkq “ +kÿ +j“1 +ÿ +σPSpj,k´jq +ϵpσqΞpxσp1q d ¨ ¨ ¨ d xσpjqq b Ξpxσpj`1q d ¨ ¨ ¨ d xσpkqq +´ +kÿ +j“1 +ÿ +σPSpj,k´jq +ϵpσqpΦpxσp1q d ¨ ¨ ¨ d xσpjqq b Φpxσpj`1q d ¨ ¨ ¨ d xσpkqq +“ ppΞ ´ Φq b Φ ` Ξ b pΞ ´ Φqq ˝ ∆1px1 d ¨ ¨ ¨ d xkq. +Since ∆ has polynomial-degree 0 and pΞ ´ Φqpiq “ 0 for all 0 ď i ď n, we obtain +∆pΞ´Φqpn`1qpx1d¨ ¨ ¨dxkq “ +´ +pΞ ´ Φqpn`1q b Φp0q ` Φp0q b pΞ ´ Φqpn`1q¯ +˝∆1px1d¨ ¨ ¨dxkq. (2.19) + +CHAPTER 2. LIE 8-ALGEBROIDS AND THEIR MORPHISMS +42 +This proves the first item. Since both Φ and Ξ are O-multilinear, pΞ ´ Φqpn`1q is O-multilinear. This +proves the second item. Since, Φ and Ξ are Lie 8-morphisms: +pΞ ´ Φq ˝ QE1 ´ QE ˝ pΞ ´ Φq “ 0. +(2.20) +By looking at the component of polynomial-degree n`1, one obtains, pΞ´Φqpn`1q ˝Qp0q +E1 ´Qp0q +E ˝pΞ´ +Φqpn`1q “ 0. This proves the third item. +Conclusion: +This chapter recapitulates classical definition of Lie 8-algebroids, and in particular describes +it as co-derivation, which is not usual. +Morphisms and homotopies are described. +About +homotopies, two versions are given: one uses smooth maps, and is way easier (see Definition +2.3.8). Unfortunately, to allow infinitely many gluings, we have to introduce a more complicated +notion of homotopy, which uses continuous locally C1-maps (see Definition 2.3.14). + +CHAPTER 3 +Lie-Rinehart algebras and their morphisms +Except for Remark 3.2.1, this section is essentially a review of the literature on the subject, see, e.g. +[Hue04, Hue98]. +3.1 +Definitions +Lie-Rinehart algebras are the algebraic encoding of the notion of Lie algebroids over a manifold. We +owe this concept to [Rin63]. +Definition 3.1.1. A Lie-Rinehart algebra over an algebra O is a triple pA, r¨, ¨sA, ρAq with A an O- +module, r¨, ¨sA a Lie algebra bracket on A, and ρA : A ÝÑ DerpOq an O-linear Lie algebra morphism +called anchor map, satisfying the so-called Leibniz identity: +ra, fbsA “ ρApaqrfs b ` fra, bsA for all a, b P A, f P O. +1. Let η: O ÝÑ O1 be an algebra morphism. A Lie-Rinehart algebra morphism over η is a Lie +algebra morphism φ: A ÝÑ A1 such that for every a P A and f P O: +(a) φpfaq “ ηpfqφpaq +(b) ηpρApaqrfsq “ ρA1pφpaqrηpfqs, +When O “ O1 and η “ id, we say that φ is a Lie-Rinehart algebra morphism over O. +2. A submodule B Ď A is a said to be a Lie-Rinehart subalgebra of A if it carries a Lie-Rinehart +algebra structure over O whose Lie bracket and anchor map are the restriction of the bracket +r¨ , ¨sA and the anchor ρA respectively to B. +Lie-Rinehart algebras over O form a category that we denote by Lie-Rhart-alg/O +Remark 3.1.2. A Lie-Rinehart algebra is said to be a Lie algebroid if A is a projective O-module. +See Example 3.2.2 for the relation with usual Lie algebroid as vector bundles. +43 + +CHAPTER 3. LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS +44 +Remark 3.1.3. For every Lie 8-algebroid pE‚, ℓ‚, ρq over O, the quotient space +E´1 +ℓ1pE´2q comes equipped +with a natural Lie-Rinehart algebra over O. The 2-ary bracket ℓ2 goes to quotient to +E´1 +ℓ1pE´2q to define +a Lie algebra since for all x P E´2 and y P E´1 we have +ℓ2pℓ1pxq, yq “ ℓ1pℓ2px, yqq. +Also, the Jacobi identity holds, since +ℓ2pℓ2px, yq, yq` ö px, y, zq “ ´ℓ1pℓ3px, y, zqq +@x, y, z P E´1. +In addition, the anchor map ρ goes to quotient to a Lie algebra morphism +E´1 +ℓ1pE´2q Ñ DerpOq, since +ρ ˝ ℓ1 “ 0. This Lie-Rinehart algebra is called the basic Lie-Rinehart algebra of pE‚, ℓ‚, ρq. +To summarize, every Lie 8-algebroid induces a Lie-Rinehart algebra. The opposite direction, that +is, wondering whether a Lie-Rinehart algebra pA, r¨ , ¨sA , ρq over O is the basic Lie-Rinehart algebra of +some almost differential graded Lie algebroid or more generally a Lie 8-algebroid over O is part of the +questions that I discussed in this thesis (see Chapter 4). This extends the main results of [LLS20] from +locally real analytic finitely generated singular foliations (see Example 3.2.4) to arbitrary Lie-Rinehart +algebras. +Let us fix some vocabulary that will be used in the sequel. +Definition 3.1.4. We say that an almost differential graded Lie algebroid pE‚, ℓ1, ℓ2, ρq or a Lie +8-algebroid pE‚, ℓ‚, ρq over O +1. covers (through a hook π) a Lie-Rinehart algebra pA, r¨ , ¨sA, ρAq, if there exists a morphism of +brackets π: E´1 Ñ A such that π ˝ ρA “ ρ and πpE´1q “ A, +2. terminates in A through the hook π, when πpE´1q Ď A. +According to Remark 3.1.3, any Lie 8-algebroid pE‚, ℓ‚, ρq covers its basic Lie-Rinehart algebra +through the hook π which is given by the projection π: E´1 ÝÑ E´1{ℓ1pE´2q, since π respects the +brackets and πpE´1q “ E´1{ℓ1pE´2q. +Definition 3.1.5. Let pE1 +‚, ℓ1 +‚, ρ1q and pE‚, ℓ‚, ρq and be Lie 8-algebroids over O that terminate in +a Lie-Rinehart algebra pA, r¨ , ¨sA, ρAq through the hooks π1 and, π respectively. We say that a Lie +8-algebroid morphism Φ: S‚ +KpE1q Ñ S‚ +KpEq is over A, if π ˝ Φ0 “ π1. +We will need the following lemma. +Lemma 3.1.6. Let pE1 +‚, ℓ1 +‚, ρ1q and pE‚, ℓ‚, ρq be Lie 8-Lie algebroids that terminate in some Lie- +Rinehart algebra pA, r¨ , ¨sA , ρAq through hooks π1 and π. Let pJt, HtqtPra,bs be a homotopy that joins +Ja and Jb. If Ja is Lie 8-algebroid morphism that terminates at A (i.e., π ˝ Jp0q +a +|E1 “ π1), then so is +the 8-algebroid morphism Jt for all t P ra, bs. +Proof. This is a direct consequence of Equation (2.16), since Qp0q +E +|E “ ℓ1 : E´2 Ñ E´1 and Qp0q +E1 |E1 “ +ℓ1 +1 : E1 +´2 Ñ E1 +´1 are valued in the kernels of π and, π1 respectively. +Last, O-multilinearity of Jt follows from the O-multilinearity of QE ˝Ht`Ht˝QE1, which is granted +by Proposition 2.3.24. This completes the proof. + +CHAPTER 3. LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS +45 +3.2 +Algebraic and geometric examples +Example 3.2.1. For every commutative K-algebra O, the Lie algebra A “ DerpOq of derivations of +a commutative algebra O is a Lie-Rinehart algebra over O, with the identity as an anchor map. This +Lie-Rinehart algebra is a terminal object in the category Lie-Rhart-alg/O: for every Lie-Rinehart +pA, r¨ , ¨sA , ρAq the anchor A +ρA +ÝÑ DerpOq is obviously a Lie-Rinehart algebra morphism over O. +In particular, vector fields on a smooth or Stein manifold or an affine variety are examples of +Lie-Rinehart algebras over their respective natural algebras of functions. +Example 3.2.2. Let M be a smooth manifold. By Serre-Swan Theorem, Lie algebroids over M are +precisely Lie-Rinehart algebras over C8pMq of the form pΓpAq, r¨, ¨s, ρq where A is a vector bundle +over M and ρ : A Ñ TM is a vector bundle morphism. +Example 3.2.3. Sections of a Lie algebroid that have values in the kernel of the anchor map form a +Lie-Rinehart algebra KerpρAq for which the anchor map is zero. +The next two examples are part of our the main source of motivation to study Lie-Rinehart +algebras. The second is more general than the first one. +Example 3.2.4. Singular foliations. [AS09, AZ14, Cer79, Deb01, LLS20, LGLR22] A singular folia- +tion on a smooth, real analytic, or complex manifold M or Zariski open subset U Ď Cd is a subsheaf +F Ď XpMq that fulfills the following conditions +• Stability under Lie bracket: rF, Fs Ď F. +• Locally finitely generateness: every m P M admits an open neighborhood U together with +a finite number of vector fields X1, . . . , Xr P XpUq such that for every open subset V Ď U the +vector fields X1|V, . . . , Xr|V generates F on V as a C8pVq-module. +There are several other ways to define singular foliations on a manifold M [AZ13, Cer79, Daz85, +Deb01]. All these definitions have in common to define then as Lie-Rinehart sub-algebras F of the +Lie-Rinehart algebra XpMq of vector fields on M (or compactly supported vector fields XcpMq on M). +Here are some important consequences of the above definition. +1. Singular foliation admits leaves: there exists a partition of M into submanifolds called leaves +such that for all m P M, the image of the evaluation map F Ñ TmM is the tangent space of +the leaf through m. When F coincides with the space of vector fields tangent to all leaves at all +points, we shall speak of a “Stefan-Sussman singular foliation”. +2. Singular foliations are self-preserving : the flow of vector fields in F, whenever defined, preserves +F [AS09, GY18]. +Notice that in Remark 2.1.13, F :“ ρpΓpE´1qq is a singular foliation in the sense above, called the +basic singular foliation of pE, pℓkqkě1, ρq. We say, then the Lie 8-algebroid pE, pℓkqkě1, ρq is over F. +Example 3.2.5. [AZ18, Zam18, ZA18] Let pA, r¨ , ¨sA , ρAq be a Lie algebroid over a manifold M. A +singular subalgebroid B of A, is a C8pMq-submodule of ΓpAq that is locally finitely generated and +involutive i.e. stable under the Lie bracket r¨ , ¨sA. This notion is a generalization of singular foliations +in the sense of Example 3.2.4, since singular foliations on M are singular subalgebroids of TM. + +CHAPTER 3. LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS +46 +A singular subalgebroid B is an example of Lie-Rinehart algebra: its bracket and anchor are the +restrictions of r¨ , ¨sA and ρA to B respectively. +Example 3.2.6. For a singular foliation F on a manifold M, consider S :“ tX P XpMq | rX, Fs Ď Fu +(i.e. infinitesimal symmetries of F) and +C :“ tf P C8pMq | Y rfs “ 0, for all Y P Fu +(that can be thought of as functions constant along the leaves of F). The quotient S +F is Lie-Rinehart +algebra over C. +Example 3.2.7 (Poisson manifold). [LGPV13] We recall that a Poisson manifold is a manifold M +together with a biderivation t¨ , ¨u on its algebra of smooth functions C8pMq that satisfies Jacobi’s +identity, i.e. +1. pC8pMq, t¨ , ¨uq is Lie algebra. +2. The biderivation t¨ , ¨u is compatible with the product of functions in the following sense: for all +f, g, h P C8pMq +tf.g, hu “ f.tg, hu ` g.tf, hu. +This is equivalent to giving a bivector field π P Γp^2TMq such that the Schouten-Nijenhuis bracket +with itself vanishes, that is, rπ, πsSN “ 0. +For every a Poisson manifold pM, πq, the operator δπ : Γp^‚TMq Ñ Γp^‚`1TMq, P ÞÑ ´rP, πsSN +defines a complex +¨ ¨ ¨ +� Γp^p´1TMq +δp´1 +π +� Γp^pTMq +δp +π � Γp^p`1TMq +� ¨ ¨ ¨ , +since rπ, πsSN “ 0 implies δπ ˝ δπ “ 0. For every p P N0, the quotient Hp +πpMq :“ +ker δp +π +Imδp´1 +π +is called +the p-th Poisson cohomology of pM, πq. We define A :“ H1 +πpMq to be the first Poisson cohomology +of π and O :“ H0 +πpMq “ Caspπq to be the algebra of Casimir functions. The bracket of vector fields +makes A a Lie-Rinehart algebra over Caspπq. +3.2.1 +Basic constructions +Let pA, r¨ , ¨sA , ρAq a Lie-Rinehart algebra over an algebra O and I Ă O be an ideal. The submodule +IA is a Lie-Rinehart subalgebra of A and its anchor is given by the restriction of ρA over IA Ă A. +This follows easily from +rfa, gbsA “ fgra, bsA ` fρApaqrgs b ´ gρApbqrfs a for all a, b P A, f, g P I. +1. Restriction. Consider a Lie-Rinehart algebra pA, r¨, ¨sA, ρAq over O. For every Lie-Rinehart ideal +I Ă O, i.e. any ideal such that +ρApAqrIs Ă I +the quotient space A{IA inherits a natural Lie-Rinehart algebra structure over O{I. We call +this Lie-Rinehart algebra the restriction w.r.t the Lie-Rinehart ideal I. In the context of affine +varieties, when I is the ideal of functions vanishing on an affine sub-variety W, we shall denote +A +IA by i˚ +W A. + +CHAPTER 3. LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS +47 +2. Localization. Localizing w.r.t a multiplicative subset S Ă O (i.e. S Ď Ozt0u is closed under +multiplication and 1 P S) is a very powerful tool in commutative algebra and in algebraic +geometry to deal with "global" problems by reducing them to "local" ones. Let us recall the +construction (e.g. see [Sta22], Section 10.9 or [And] for details). +(a) Let V be a O-module. +The localization of V at S is defined as follows: consider the +equivalence relation on S ˆ V that is given by +ps, vq „ ps1, v1q ðñ Du P S, ups1v ´ sv1q “ 0. +The class of an element ps, vq P S ˆ V by v +s, and by S´1V :“ S ˆ V{„ the set of equivalence +classes. +In particular, the localization of O at S is the localization of O as a O-module. In that +case, S´1O is a K-algebra together with the addition and multiplication defined by +f +s ` g +s1 :“ fs1 ` sg +ss1 +and +f +s +g +s1 :“ fg +ss1 . +Similarly, S´1V is a S´1O-module with addition and scalar product multiplication defined +in an obvious way. Also, we have S´1V » S´1O bO V. +(b) A derivation D P DerpOq admits a unique extension to a derivation S´1D P DerpS´1Oq +which given for ps, fq P S ˆ O by the classical formula +pS´1Dq +ˆf +s +˙ +:“ Dpfqs ´ Dpsqf +s2 +. +Let pA, r¨ , ¨sA , ρq be a Lie-Rinehart algebra over O. The localization module S´1A “ S´1ObOA +comes equipped with a natural structure of Lie-Rinehart algebra over the localization algebra +S´1O. The new anchor map is defined by +a +s P S´1A ÞÑ 1 +sS´1pρApaqq P DerpS´1Oq, +and the new Lie algebra bracket is given by +„1 +sa, 1 +ub +ȷ +S´1A +“ 1 +sura, bsA ´ ρApaqrus +su2 +b ` ρApbqrss +s2u +a +for a, b P A, ps, uq P S2. The localization map A ãÑ S´1A is a Lie-Rinehart algebra morphism +over the localization map O ãÑ S´1O. +Since localization exists, the notion of sheaf of Lie- +Rinehart algebras [Vil20] over a projective variety, or a scheme, makes sense. +3. Algebra extension. Assume that the algebra O has no zero divisor, and let O be its field of +fraction. For any subalgebra ˜O with O Ă ˜O Ă O such that ρpaq is for any a P A valued in +derivations of O that preserves ˜O, there is a natural Lie-Rinehart algebra structure over ˜O on +the space ˜O bO A. +4. Blow-up at the origin. Let us consider a particular case of the previous construction, when O is +the algebra Crx1, . . . , xNs. If the anchor map of a Lie-Rinehart algebra A over O takes values in + +CHAPTER 3. LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS +48 +vector fields on CN vanishing at the origin, then for all i “ 1, . . . , N, the polynomial algebra OUi +generated by x1 +xi , . . . , xi´1 +xi , xi, xi`1 +xi , . . . , xN +xi satisfies the previous condition, and OUi bO A comes +equipped with a Lie-Rinehart algebra. Geometrically, this operation corresponds to taking the +blow-up of CN at the origin, then looking at the i-th natural chart Ui on this blow-up: OUi are +the polynomial functions on Ui. The family OUi bO A (for i “ 1, . . . , N) is therefore an atlas +for a sheaf of Lie-Rinehart algebras (in the sense of [Vil20]) on the blow-up of CN at the origin, +referred to as the blow-up of A at the origin. +3.2.2 +On free resolutions of length ď 2 and Lie-Rinehart algebras +In this section, we discuss the case when a Lie-Rinehart algebra A admits a free resolution of length +1 and 2. In those cases, we claim that there are Lie algebra-like structures on them. See B.1 for the +notion of free resolution of modules. These results will be soon generalized. +I start with the following remark (owed to Marco Zambon). +Remark 3.2.8. Any Lie-Rinehart algebra pA, r¨ , ¨sA , ρAq is the image of an almost differential graded +Lie algebroid pE´1, ρq concentrated in degree ´1: to see this, let tai P A | i P Iu be a set of generators +of A (take all elements of A if necessary). There exists elements uk +ij P O, such that for given indices +i, j, the coefficient uk +ij is zero except for finitely many indices k, together with +rai, ajsA “ +ÿ +kPI +uk +ijak +@i, j P I. +(3.1) +The coefficients uk +ij can be chosen to satisfy the skew-symmetry condition uk +ij “ ´uk +ji: by skew- +symmetry of the bracket r¨ , ¨sA on has +rai, ajsA “ 1 +2 prai, ajsA ´ raj, aisAq +“ +ÿ +kPI +1 +2puk +ij ´ uk +jiqak. +Thus, one can replace uk +ij by 1 +2puk +ij ´ uk +jiq if necessary. +Now, choose E´1 to be the free O-module generated by the symbols peiqiPI together with the +surjective map +π: E´1 Ñ A, ei ÞÑ ai. +We now define: +1. an anchor map by ρpeiq :“ ρApai), for all i P I, i.e. ρ “ ρA ˝ π, +2. a skew-symmetric operation r¨ , ¨sE´1 on E as follows: +rei, ejsE´1 “ +ÿ +kPI +uk +ijek for all i, j P I. +We extend by O-linearity and Leibniz identity. By construction pE´1, r¨ , ¨sE´1, ρ) is an almost Lie +algebroid over O whose image through the anchor map is A. +Remark 3.2.9. In general, this bracket does not satisfy the Jacobi identity. If E´1 » A, this bracket +is a Lie algebroid. + +CHAPTER 3. LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS +49 +On free resolutions of length 1 +Lie-Rinehart algebra A admits a free resolution of length 1 if and only if A is free. In that case, +the almost Lie algebroid bracket r ¨, ¨sE´1 is a Lie algebroid bracket. In conclusion: free resolutions of +length 1 admit a Lie algebroid structure. +Free resolutions of length 2 +Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra that admits a free resolution of length 2, namely an +exact sequence of the form +0 +� E´2 +ℓ1 +� E´1 +ρ +� � +π +� � A +ρA� DerpOq +(3.2) +with E´1, E´2 free modules. +Since Equation (3.2) is a free resolution of A, the map π is in particular surjective. By Remark 3.2.8, +E´1 can be endowed with an almost Lie algebroid bracket, such that π is a morphism of brackets. We +can extend this bracket to sections of degree ´2 to obtain an almost differential graded Lie algebroid. +Let us compute it: +Let pe1 +iqiPI1 and pejqiPI be a basis of E´2 and E´1, respectively. For all i, j, k P I Y I1 we have +1. +πprℓ1e1 +i, ejsE´1q “ rπ ˝ ℓ1pe1 +iq, πpejqs “ 0, (since π ˝ ℓ1 ” 0). +In other words, rℓ1pe1 +iq, ejsE´1 P ker π. By exactness of the complex (3.2) there exists an element +∇e1 +iej P E´2 such that +πp∇e1 +iejq “ rℓ1pe1 +iq, ejsE´1. +(3.3) +Equation (3.3) allows defining a bilinear map: +E´1 b E´2 +Ñ +E´2 +px, yq +ÞÑ +∇xy +by extending the ∇e1 +iej’s by linearity and Leibniz identity with the understanding that the anchor +map ρ vanishes on E´2 in order to have +(a) ℓ1p∇xyq “ rℓ1pxq, ysE´1, @x E´2, y P E´1, +(b) for all f P O: ∇xfy “ f∇xy ` ρpxqrfs y and ∇fxy “ f∇xy, for all x P E´1, y P E´2, +2. Remember that +Jacpei, ej, ekq :“ rei, rej, eks2s2 ` rej, rek, eis2s2 ` rek, rei, ejs2s2 P ker π. +By using again exactness of the complex (3.2) there is an element that we denote by rei, ej, eksE´1 P +E´2 that satisfies +ℓ1 +` +rei, ej, eksE´1 +˘ +“ Jacpei, ej, ekq. +(3.4) + +CHAPTER 3. LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS +50 +Thus, we can define a skew-symmetric trilinear map: +r¨ , ¨ , ¨sE´1 : E´1 ^ E´1 ^ E´1 ÝÑ E´2 +such that +ℓ1prx, y, zsE´1q “ rx, ry, zs2s2 ` ry, rz, xs2s2 ` rz, rx, ys2s2, @x, y, z P E´1. +The following Proposition concludes the discussion above. +Proposition 3.2.10. Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra that admits a free resolution of +length 2 as in (3.2) +1. E´1 admits an almost Lie algebroid structure pE´1, r¨ , ¨sE´1, ρq. +2. There is a bilinear map: +E´1 b E´2 +Ñ +E´2 +px, yq +ÞÑ +∇xy +and a skew-symmetric trilinear map: +r¨ , ¨ , ¨sE´1 : E´1 ^ E´1 ^ E´1 ÝÑ E´2 +such that for all f P O: +(a) ∇xfy “ f∇xy ` ρpxqrfs y and ∇fxy “ f∇xy, for all x P E´1, y P E´2, +(b) rfx, y, zsE´1 “ frx, y, zsE´1 for all x, y, z P E´1, +such that the 2-ary bracket on E´1 ‘ E´2 defined by: +rx, ys2 “ +$ +’ +’ +’ +’ +& +’ +’ +’ +’ +% +rx, ysE´1 +for +x, y P E´1 +∇xy +for +x P E´1, y P E´2 +∇yx +for +x P E´2, y P E´1 +0 +for +x, y P E´2 +together with the 3-ary bracket on E´1 ‘ E´2 defined by rx, y, zs3 “ rx, y, zsE´1 if x, y, z P E´1 +and zero otherwise, satisfies +(a) for all x P E´2, y P E´1, +ℓ1prx, ys2q ` rℓ1pxq, ys2 “ 0, +(3.5) +(b) for all x, y, z P E´1 +ℓ1prx, y, zs3q ` rx, ry, zs2s2 ` ry, rz, xs2s2 ` rz, rx, ys2s2 “ 0 +(c) for all x, y P E´1 and z P E´2 +rx, y, ℓ1pzqs3 ` rx, ry, zs2s2 ` ry, rz, xs2s2 ` rz, rx, ys2s2 “ 0. + +CHAPTER 3. LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS +51 +The structure pE‚, ℓ1, ρ, r¨ , ¨s2, r¨ , ¨, ¨s3q described in Proposition 3.2.10 is called Lie 2-algebroid +[BC03, Lea17, SZ11], i.e. a Lie 8-algebroid with E´i “ 0 for i ě 3. +A generalization of this construction on free resolutions of higher (even infinite) length is the object +of the next Chapter. +Conclusion: +We describe Lie-Rinehart algebras, give examples and several constructions. Section 3.2.2 is a +pedagogical section, which presents an elementary case of the general case that we will study. + +CHAPTER 4 +Main results of Part I +4.1 +Presentation of the problem +In order to understand the geometry of affine varieties or some similar problems related to singular +foliations using the Lie algebra of their vector fields which is in fact Lie-Rinehart algebras, have mo- +tivated us to understand Lie-Rinehart algebras in general from another point of view. +We have seen in the Chapter 3, Remark 3.1.3 that any Lie 8-algebroid over O induces a Lie- +Rinehart algebra which we call its basic Lie-Rinehart algebra. In this chapter, we are investigating +the opposite direction, i.e., we study the following questions: given a Lie-Rinehart algebra A over O, +can we find a Lie 8-algebroid over O whose basic Lie-Rinehart algebra is A? Also, in case where it +exists, do we have uniqueness? +Now let us formalize that in a categorical language. +We denote by Lie-8-alg-oids/O the quotient category where +1. the objects are Lie 8-algebroids over O, +2. arrows are homotopy equivalence classes of morphisms of Lie 8-algebroids over O. +and by Lie-Rhart-alg/O the category of Lie-Rinehart algebra over O. We want to study the functor, +Lie-8-alg-oids/O +F +ÝÑ +Lie-Rhart-alg/O +D!? +ÐÝ +The question now turn out to be: when does F admit a left/right inverse? +In the next section, we present in detail the main results related to this question. +52 + +CHAPTER 4. MAIN RESULTS OF PART I +53 +4.2 +Main results +An existence Theorem +The results below appeared in my first article [LGL22b] entitled "Lie-Rinehart algebra » acyclic Lie +8-algebroid" co-written with my supervisor C. Laurent-Gengoux. +This section extends the main results of [LLS20] from locally real analytic finitely generated sin- +gular foliations to arbitrary Lie-Rinehart algebras. +Here is our first main result. It states that universal Lie 8-algebroids over a given Lie-Rinehart +algebra exist. It extends Theorem 2.8 in [LLS20]. We are convinced that it may be deduced using the +methods of semi-models categories as in Theorem 4.2 in [FJO18], but does not follow from a simple +homotopy transfer argument. +Theorem 4.2.1. Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O. Any resolution of A by free +O-modules +¨ ¨ ¨ +ℓ1 +ÝÑ E´3 +ℓ1 +ÝÑ E´2 +ℓ1 +ÝÑ E´1 +π +ÝÑ A +(4.1) +comes equipped with a Lie 8-algebroid structure whose unary bracket is ℓ1 and that covers A through +the hook π. +Since any module admits free resolutions (see Proposition B.2.5), Theorem 4.2.1 implies that: +Corollary 4.2.2. Any Lie-Rinehart algebra A over O is the basic Lie-Rinehart algebra of an acyclic +Lie 8-algebroid over O. +While proving Theorem 4.2.1, we will see that if E´1 can be equipped with a Lie algebroid bracket +(i.e. a bracket whose Jacobiator is zero), then all k-ary brackets of the universal Lie 8-algebroid +structure may be chosen to be zero on E´1: +Proposition 4.2.3. Let pE, ℓ1, πq be a free resolution of a Lie-Rinehart algebra A. If E´1 admits a +Lie algebroid bracket r¨, ¨s such that π : E´1 Ñ A is a Lie-Rinehart morphism, then there exists a +structure of universal Lie 8-algebroid pE‚, ℓ‚, ρq that covers A whose 2-ary bracket coincides with r¨, ¨s +on E´1 and such that for every k ě 3 the k-ary bracket ℓk vanishes on Äk E´1. +Universality of Theorem 4.2.1 and its corollaries +Here is our second main result. It is related to Proposition 2.1.4 in [FJO18] (but morphisms are not +the same), and extends Theorem 2.9 in [LLS20]. +Theorem 4.2.4. Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O. Given, +a) a Lie 8-algebroid pE1 +‚, ℓ1 +‚, ρ1q that covers A through a hook π1, and +b) any acyclic Lie 8-algebroid pE‚, ℓ‚, ρq that covers A through a hook π, +then +1. there exists a morphism of Lie 8-algebroids from pE1 +‚, ℓ1 +‚, ρ1q to pE‚, ℓ‚, ρq over A. + +CHAPTER 4. MAIN RESULTS OF PART I +54 +2. and any two such morphisms are homotopic. +Here is an immediate corollary of Theorem 4.2.4. +Corollary 4.2.5. Any two acyclic Lie 8-algebroids that cover a given Lie-Rinehart algebra are ho- +motopy equivalent. This homotopy equivalence, moreover, is unique up to homotopy. +We will prove that the morphism that appears in Theorem 4.2.4 can be made trivial upon choosing +a “big enough” universal Lie 8-algebroid: +Proposition 4.2.6. Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O. Given a Lie 8-algebroid +structure pE1 +‚, ℓ1 +‚, ρ1q that terminates in A through a hook π1, then there exist an acyclic Lie 8-algebroid +pE‚, ℓ‚, ρq that covers A through a hook π such that +1. E contains E1 as a subcomplex, +2. the Lie 8-algebroid morphism from E1 to E announced in Theorem 4.2.4 can be chosen to be the +inclusion map E1 ãÑ E (i.e. a Lie 8-morphism where the only non-vanishing Taylor coefficient +is the inclusion E1 ãÑ E). +The following Corollary follows immediately from Proposition 4.2.6: +Corollary 4.2.7. Let A be a Lie-Rinehart algebra over O and B be a Lie-Rinehart subalgebra of A. +Any universal Lie 8-algebroid of B can be contained in a universal Lie 8-algebroid of A. +Induced Lie 8-algebroids structures on TorOpA, O{Iq. +Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O. +Definition 4.2.8. We say that an ideal I Ă O is a Lie-Rinehart ideal of A if ρApaqrIs Ă I for all +a P A. +Remark 4.2.9. For any Lie-Rinehart ideal IO of A, +1. we have rIA, AsA Ă IA. Therefore, the quotient space A{IA comes equipped with a natural +Lie-Rinehart algebra structure over O{I. +2. For pE‚, ℓ‚, ρq an acyclic Lie 8-algebroid that covers A, the quotient space E‚{I » O{I bO E‚ +comes equipped with an induced Lie 8-algebroid structure: the n-ary brackets for n ‰ 2 go +to quotient by linearity, while for n “ 2, the 2-ary bracket goes to the quotient in view of +the relation ρEpE´1qrIs Ă I. Also, π goes to the quotient to a Lie-Rinehart algebra morphism +E´1{I Ñ A{IA. +Definition 4.2.10. Let A be a Lie-Rinehart algebra over O. For every Lie-Rinehart ideal I Ă O, we +call Lie 8-algebroid of I the quotient Lie 8-algebroid E‚{I, with pE‚, ℓ‚, ρq a universal Lie 8-algebroid +that covers A. +Remark 4.2.11. The complexes on which the Lie 8-algebroids of the ideal I are defined compute +TorOpA, O{Iq by construction (see B.2). + +CHAPTER 4. MAIN RESULTS OF PART I +55 +Moreover, for any two universal Lie 8-algebroids of A, defined on E, E1 the homotopy equivalences +Φ : E1 Ñ E and Ψ : E Ñ E1, whose existence is granted by Corollary 4.2.5, go to the quotient and +induce an homotopy equivalences between O{I bO E‚ » E‚{I and O{I bO E1 +‚ » E1 +‚{I. The following +corollary is then an obvious consequence of Theorem 4.2.4. +Corollary 4.2.12. Let A be a Lie-Rinehart algebra over O. Let I Ă O be a Lie-Rinehart ideal. Then +any two Lie 8-algebroids of I are homotopy equivalent, and there is a distinguished class of homotopy +equivalences between them. +Taking under account Remark 4.2.11, here is an alternative manner to restate this corollary. +Corollary 4.2.13. Let A be a Lie-Rinehart algebra over O. Let I Ă O be a Lie-Rinehart ideal. Then +the complex computing Tor‚ +OpA, O{Iq comes equipped with a natural Lie 8-algebroid structure over +O{I, and any two such structures are homotopy equivalent in a unique up to homotopy manner. +When, in addition to being a Lie-Rinehart ideal, I is a maximal ideal, then K :“ O{I is a field and +Lie 8-algebroids of I are a homotopy equivalence class of Lie 8-algebras. In particular their common +cohomologies, which is easily seen to be identified to Tor‚ +OpA, Kq comes equipped with a graded +Lie algebra structure. In particular, Tor´1 +O pA, Kq is a Lie algebra, Tor´2 +O pA, Kq is a representation +of this algebra, and the 3-ary bracket defines a class in the third Chevalley-Eilenberg cohomology of +Tor´1 +O pA, Kq valued in Tor´2 +O pA, Kq. This class does not depend on any choice made in its construction +by the previous corollaries, and trivially extends the class called NMRLA-class in [LLS20]. If it is +not zero, then there is no Lie algebroid of rank r equipped with a surjective Lie-Rinehart algebra +morphism onto A, where r is the rank of A as a module over O. All these considerations can be +obtained by repeating verbatim Section 4.5.1 in [LLS20] (where non-trivial examples are given). +A categorical approach of the results +Theorem 4.2.4 means that acyclic Lie 8-algebroids that covers Lie-Rinehart algebra pA, r¨ , ¨sA , ρAq +are terminal objects in the subcategory of Lie-8-alg-oids/O whose objects are Lie 8-algebroids that +terminate in A. Whence, acyclic Lie 8-algebroids over O that cover pA, r¨ , ¨sA , ρAq are deserved to +be called "universal 8-algebroids of A". From now on, we call them by this name. +Let us re-state Corollary 4.2.5 differently. By associating to any Lie 8-algebroid its basic Lie-Rinehart +algebra one obtains therefore a natural functor: +• from the category Lie-8-alg-oids/O, +• to the category Lie-Rhart-alg/O +Theorem 4.2.1 gives a right inverse of this functor. In particular, this functor becomes an equivalence +of categories when restricted to homotopy equivalence classes of acyclic Lie 8-algebroids over O, i.e: +Corollary 4.2.14. There is an equivalence of categories between: +(i) Lie-Rinehart algebras over O, +(ii) acyclic Lie 8-algebroids over O. + +CHAPTER 4. MAIN RESULTS OF PART I +56 +This corollary justifies the title of the first part of the thesis. +Remark 4.2.15. In the language of categories, Corollary 4.2.12 means that there exists a functor from +Lie-Rinehart ideals of a Lie-Rinehart algebra over O, to the category of Lie 8-algebroids, mapping a +Lie-Rinehart ideal I to an equivalence class of Lie 8-algebroids over O{I. +4.3 +Proof of main results +4.3.1 +A crucial bi-complex: PagepnqpE1, Eq +Description of PagepnqpE1, Eq +Let V be an O-module, and let pE, d, πq and pE1, d1, π1q be complexes of projective O-modules that +terminates at V: +¨ ¨ ¨ dÝÑ E´2 +dÝÑ E´1 +πÝÑ V, +¨ ¨ ¨ d1 +ÝÑ E1 +´2 +d1 +ÝÑ E1 +´1 +π1 +ÝÑ V. +(4.2) +For every k ě 1, the pk ` 1q-th graded symmetric power Äk`1 E1 of E1 over O is a projective +O-module, and comes with a natural grading induced by the grading on E1. +Definition 4.3.1. Let k P N0. We call page number k of pE, d, πq and pE, d1, π1q the bicomplex of +O-modules on the upper left quadrant Z´ ˆ N0 defined by: +PagepkqpE1, Eqj,m :“ HomO +˜ +k`1 +ä +E1 +|´k´m´1 , Ej +¸ +, +for m ě 0 and j ď ´1 +(4.3) +PagepkqpE1, Eq0,m :“ HomO +˜ +k`1 +ä +E1 +|´k´m´1 , V +¸ +, +for m ě 0, +(4.4) +together with the vertical differential Dv defined for any one of the two O-modules (4.3) or (4.4) by +Dv : PagepkqpE1, Eqj,m +ÝÑ +PagepkqpE1, Eqj,m`1 +Φ +ÞÝÑ +DvpΦq: Äk`1 E1 +ÝÑ Ej +x1 d . . . d xk`1 +ÞÑ Φ ˝ d1 px1 d ¨ ¨ ¨ d xk`1q +where d1 acts as an O-derivation on x1 d . . . d xk`1 P Äk E1 (and is 0 on E1 +´1). +The horizontal +differential, Dh : PagepkqpE1, Eqj,m ÝÑ PagepkqpE1, Eqj`1,m, is given by +Φ ÞÑ d ˝ Φ or Φ ÞÑ π ˝ Φ +depending on whether Φ is of type (4.3) with j ď ´2 or the type (4.3) with j “ ´1. It is zero on +elements of type (4.4). We denote by +´ +Pagepkq +‚ pE1, Eq, D +¯ +its associated total complex. When E1 “ E +we shall write Pagepkq +‚ pEq instead of Pagepkq +‚ pE, Eq. + +CHAPTER 4. MAIN RESULTS OF PART I +57 +The following diagram recapitulates the whole picture of Pagepkq +‚ pE1, Eq: +... +... +... +Ò +Ò +Ò +¨ ¨ ¨ +Ñ +HomO +´Äk`1 E1 |´k´3, E´2 +¯ +Dh +Ñ +HomO +´Äk`1 E1 |´k´3, E´1 +¯ +Dh +Ñ +HomO +´Äk`1 E1 |´k´3, V +¯ +Ñ +0 +Dv Ò +Dv Ò +Dv Ò +¨ ¨ ¨ +Ñ +HomO +´Äk`1 E1 |´k´2, E´2 +¯ +Dh +Ñ +HomO +´Äk`1 E1 |´k´2, E´1 +¯ +Dh +Ñ +HomO +´Äk`1 E1 |´k´2, V +¯ +Ñ +0 +Dv Ò +Dv Ò +Dv Ò +¨ ¨ ¨ +Ñ +HomO +´Äk`1 E1 |´k´1, E´2 +¯ +Dh +Ñ +HomO +´Äk`1 E1 |´k´1, E´1 +¯ +Dh +Ñ +HomO +´Äk`1 E1 |´k´1, V +¯ +Ñ +0 +Ò +Ò +Ò +0 +0 +0 +"-2 column" +"-1 column" +"last column" +(4.5) +For later use, we spell out the meaning of being D-closed. +Remark 4.3.2. An element P P Pagepkq +j pE1, Eq in ‘iě1HomO +´Äk`1 E1 |´j´i, E´i +¯ +is D-closed if and +only if: +1. the component P´1 : Äk`1 E1 |´j´1 Ñ E´1 is valued in the kernel of π : E´1 Ñ V, +2. the following diagram commutes: +Äk`1 E1 |´j´i +p´1qjd1 +� +P´i +� +Äk`1 E1 |´j´i`1 +P´i`1 +� +E´i +d +� E´i`1 +with P´i being the component of P in HomO +´Äk`1 E1 |´j´i, E´i +¯ +. +For pE, dq “ pE1, d1q, the second condition above also reads rd, PsRN “ 0. +Here is an important +technical result. +Proposition 4.3.3. Let pE, d, πq be a resolution of V in the category of O-modules. Then, for every +k ě 0, +1. the cohomology of the complex pPagepkq +‚ pEq, Dq for the total differential D‚ :“ Dh ´ p´1q‚Dv is +zero in all degrees; +2. Moreover, a D-closed element whose component on the “last column” of the diagram above is +zero is the image through D of some element whose two last components are also zero. +3. More generally, for all n ě 1, for a D-closed element P P Pagepkq +j pEq of the form +à +iěn +HomO +˜ +k`1 +ä +E1 +|´j´i, E´i +¸ +, +one has P “ DpRq and R P Pagepkq +j´1pEq can be chosen in À +iěn`1 HomO +´Äk`1 E1 |´j´i`1, E´i +¯ +. +Proof. Since Äk E1|j`m´k is a projective O-module for all pj, mq P Z´ˆN0, and pE, d, πq is a resolution, +all the lines of the above bicomplex are exact. This proves the first item. The second and the third +are obtained by diagram chasing (see Appendix B.2 for more details). + +CHAPTER 4. MAIN RESULTS OF PART I +58 +Interpretation of PagepnqpE1, Eq in our context +We denote by Qp0q +E +and Qp0q +E1 the differentials of polynomial-degree 0 on Ä‚ E and Ä‚ E1 induced by d +and d1. Whence, pHomOpÄ‚ E1, Ä‚ Eq, Bq with +B: H ÞÑ Qp0q +E +˝ H ´ p´1q|H|H ˝ Qp0q +E1 +is a complex of O-modules (see Lemma B.1.13). +Lemma 4.3.4. Let φ: pE1, d1q Ñ pE, dq be a chain map, and let Φp0q : Ä‚ E1 Ñ Ä‚ E be its extension +to a co-algebra morphism, namely: +Φp0qpx1 ¨ ¨ ¨ ¨ ¨ xnq :“ φpx1q ¨ ¨ ¨ ¨ ¨ φpxnq. +1. We have, Φp0q ˝ Qp0q +E1 “ Qp0q +E +˝ Φp0q. +2. Φp0q-co-derivations form a sub-complex of pHomOpÄ‚ E1, Ä‚ Eq, Bq. +Proof. The first item can be easily checked. The second item follows exactly the same pattern as +Lemma 2.3.24. +The following proposition is very important. +Proposition 4.3.5. For every k P N0, and φ: pE1, d1q Ñ pE, dq be a chain map as in Lemma 4.3.4. The +sub-complex of Φp0q-co-derivations of polynomial-degree k is isomorphic to the complex z +Page +pkqpE1, Eq +obtained from PagepkqpE1, Eq by crossing its “last column”, see diagram (4.5). +Proof. The chain isomorphism δ: HomOpÄ‚ E1, Ä‚ Eq Ñ HomOpÄ‚ E1, Eq consists in mapping a +Φp0q-co-derivation H of polynomial-degree k and degree j to its unique Taylor coefficient Hk P +Pagepkq +j pE1, Eq, i.e. δpHq “ pr ˝ H. Let us check that this map is indeed a chain map: for every +x “ x1 d ¨ ¨ ¨ d xk`1 P Äk`1 E1 one has, +δ ˝ pQp0q +E +˝ H ´ p´1q|H|H ˝ Qp0q +E1 qpxq +“ ℓ1 ˝ Hkpxqp´1q|H|Hk +˜k`1 +ÿ +i“1 +´p´1qp|x1|`¨¨¨`|xi´1|q|xi|ℓ1 +1pxiq d x1 d ¨ ¨ ¨ d xk`1 +¸ +“ DhpHkqpxq ´ p´1q|Hk|DvpHkqpxq, +by definition of Dh and Dv +“ DpHkqpxq +“ D ˝ δpHqpxq, +by definition of δ. +Here is another type of interpretation for z +Page +‚pE1, Eq involving the Richardson-Nijenhuis bracket. +Proposition 4.3.6. [FN56] When E1 “ E, z +Page +‚pE1, Eq with no 0-column is the bi-graded complex of +exterior forms on E and the differential D of z +Page +‚pE1, Eq is Dp¨q “ rd, ¨ sRN. +We finish the section with the following lemma that will be important to prove Proposition 4.2.6. +It uses the consequence of the cone construction (see Appendix B). + +CHAPTER 4. MAIN RESULTS OF PART I +59 +Lemma 4.3.7. Let pR, dR, πRq be an arbitrary complex of projective O-module that terminates in a +O-module V. There exists a projective resolution pE, dE, πEq of V, which contains pR, dR, πRq as a +sub-complex. Moreover, we can assume that R admits a projective submodule in E in direct sum. +Proof. Resolutions of an O-modules V are universal objects in the category of complexes of projective +O-modules. In particular, for every projective resolution pF, dF, πFq of V, there exist a (unique up to +homotopy) chain map: +φ: pR, dR, πRq Ñ pF, dF, πFq. +We apply the cone construction (see, e.g. [Cha14], Section 1.5) to: +1. the complex pR, dR, πRq +2. the direct sum of the complexes pR, dRq and pF, dFq namely, +` +R ‘ F, dR ‘ dF, πR ‘ πF˘ +3. the chain map obtained by mapping any x P R to px, φpxqq P R ‘ F. +The differential is given by +dEpx, y, zq “ p´dRx, dRy ´ x, dFz ´ ϕpxqq +(4.6) +for all px, y, zq P E´i “ R´i`1‘R´i‘F´i, i ě 2. Since the chain given in item 3 is a quasi-isomorphism, +its cone is an exact complex. We truncate the latter at degree ´1 without destroying its exactness by +replacing the cone differential at degree ´1 as follows: πE : R´1 ‘ F´1 Ñ V, pr, eq ÞÑ πFpeq ´ πRprq. +For a visual description, see Equation (4.7) below: the resolution of V described in Lemma 4.3.7 is +defined by: +¨ ¨ ¨ +� F´3 +dF +� F´2 +dF +� F´1 +πF +� V +¨ ¨ ¨ +� R´3 +dR +� R´2 +dR +� R´1 +πR +� +¨ ¨ ¨ +� R´2 +id +� +dR +� +φ +� +R´1 +id +� +φ +� +(4.7) +The proof of the exactness of this complex is left to the reader. +The henceforth defined complex pE, dE, πEq is a resolution of V, and obviously contains pR, dR, πRq as +a sub-chain complex of O-modules. +Let pE, dE, πEq be a free resolution of V and pR, dR, πRq a subcomplex of projective O-modules, as +in Lemma 4.3.7. We say that P P Pagepkq +j pEq of the form ‘iěnHomO +´Äk`1 E |´j´i, E´i +¯ +preserves R +if Äk`1 R |´j´i is mapped by P to R´i for all possible indices. In such case, it defines by restriction +to Ä‚ R an element ι˚ +RP in the graded O-module Pagepkq +j pRq :“ ‘iěnHomO +´Äk`1 R |´j´i, R´i +¯ +. +For the sake of clarity, let us denote by DE and DR the respective differentials of the bi-complexes +Pagepkq +j pEq and Pagepkq +j pRq and by DE +h, DR +h and DR +v , DR +v the horizontal differential resp. vertical +differential, of their associated bi-complexes. Also, ι˚ +RP stands for the restriction of P P Pagepkq +j pEq +to Ä‚ R (a priori it is not valued in R but in E). +Lemma 4.3.8. Let pE, dE, πEq be a free resolution of V. Let R Ă E be a subcomplex made of free +sub-O-modules such that there exists a graded free O-module V such that E “ R ‘ V. + +CHAPTER 4. MAIN RESULTS OF PART I +60 +1. For every k ě 0, a DE-cocycle P P Pagepkq +j pEq which preserves R is the image through DE of +some element Q P Pagepkq +j´1pEq which preserves R if and only if its restriction ι˚ +RP P Pagepkq +j pRq +is a DR-coboundary. +2. In particular, if the restriction of dE and πE to R makes it a resolution of πEpR´1q Ă V, then +any DE-cocycle P P Pagepkq +j pEq which preserves R is the image through DE of some element +Q P Pagepkq +j´1pEq which preserves R. +Proof. Let us decompose the element P P Pagepkq +j pEq as P “ ř +iě1 Pi with, for all i ě 1, Pi in +HomO +´Äk`1 E |´j´i, E´i +¯ +. Assume P P Pagepkq +j pEq is a DE-cocycle which preserves R. +Let us prove one direction of item 1. If P is the image through DE of some element Q P Pagepkq +j´1pEq +which preserves R, then DR pι˚ +RQq “ ι˚ +RDEpQq “ ι˚ +RP, with ι˚ +RQ P Pagepkq +j´1pRq. Thus, the restriction +ι˚ +RP P Pagepkq +j pRq of P is a DR-coboundary. +Conversely, let us assume that ι˚ +RP P Pagepkq +j pRq is a DR-coboundary, i.e. ι˚ +RP “ DRQR for some +QR P Pagepkq +j´1pRq. Take ˆQ P Pagepkq +j´1pEq any extension of QR (e.g. define ˆQ to be 0 as soon as one +element in V is applied to it). Then P ´ DEp ˆQq : Äk`1 E ÝÑ E is zero on Äk`1 R. We have to check +that it is a DE-coboundary of a map with the same property. Put κ “ P ´ DEp ˆQq. By Proposition +4.3.3, item 1, there exists τ P Pagepkq +j´1pEq such that DEpτq “ κ. The equation DEpτq “ κ is equivalent +to the datum of a collection of equations +DE +v pτiq ` DE +hpτi`1q “ κi`1, i ě 1, +and +DE +hpτ1q “ κ1, +(4.8) +with, τi P HomO +´Äk`1 E |´j´i`1, E´i +¯ +and κi P HomO +´Äk`1 E |´j´i, E´i +¯ +for every i ě 1. Since +ι˚ +Rκ1 “ 0, we have that DE +hpι˚ +Rτ1q “ ι˚ +R +` +DE +hpτ1q +˘ +“ 0, (with the understanding that ι˚ +Rτ1|V ” 0). Using +the exactness of the horizontal differential DE +h, there exists C1 P PagepkqpEq such that DE +hpC1q “ ι˚ +Rτ1. +We now change τ1 to τ 1 +1 and τ2 to τ 1 +2 by putting τ 1 +1 :“ τ1 ´ ι˚ +Rτ1 and τ 1 +2 :“ τ2 ` DE +v pC1q. One can easily +check that Equation (4.8) still holds under these changes, i.e., +DE +v pτ 1 +1q ` DE +hpτ 1 +2q “ κ2 +and +DE +hpτ 1 +2q ` DE +v pτ3q “ κ3. +We can therefore choose τ such that ι˚ +Rτ1 “ 0. We then iterate this procedure, which allows us to +choose τ P Pagepkq +j´1pEq such that ι˚ +Rτ “ 0 and DEpτq “ κ. By construction, Q :“ τ ` ˆQ preserves R, +while ι˚ +RQ “ QR, and DEpQq “ P. The second item follows from the first one. +4.3.2 +Proof on the existence +In this section, we prove Theorem 4.2.1. +Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra. Consider pE, d “ ℓ1, πq a resolution of A by free +O-modules: such resolutions always exist, see Proposition B.2.5. To start, we define a binary bracket +ℓ2. The pair pd, ℓ2q will obey the axioms of the object that we now introduce. +Definition 4.3.9. [Lav17] An almost differential graded Lie algebroid of a Lie-Rinehart algebra +pA, ρA, r¨ , ¨sAq is a complex +¨ ¨ ¨ +d +ÝÑ E´3 +d +ÝÑ E´2 +d +ÝÑ E´1 +π +ÝÑ A +of projective O-modules equipped a graded almost differential graded Lie algebroid pE‚, ℓ1, ℓ2, ρq over +O such that + +CHAPTER 4. MAIN RESULTS OF PART I +61 +1. ρ “ ρA ˝ π: E´1 ÝÑ DerpOq, +2. π is a morphism, i.e. for all x, y P E´1 +πpℓ2px, yqq “ rπpxq, πpyqsA. +We start by proving this lemma. +Lemma 4.3.10. Every free resolution pE, d, πq of a Lie-Rinehart algebra pA, r¨ , ¨sA, ρAq comes equipped +with a binary bracket ℓ2 that makes it an almost differential graded Lie algebroid of A. +Proof. For all k ě 1, let us denote by pep´kq +i +qiPIk a family of generators of the free O-module E´k. By +construction tai “ πpep´1q +i +q P A | i P I1u is a set of generators of A. In particular, there exists elements +uk +ij P O, such that for given indices i, j, the coefficient uk +ij is zero except for finitely many indices k, +and satisfying the skew-symmetry condition uk +ij “ ´uk +ji together with +rai, ajsA “ +ÿ +kPI +uk +ijak +@i, j P I1 +(4.9) +We now define: +1. an anchor map by ρpep´1q +i +q “ ρApai) for all i P I, +2. a degree `1 graded symmetric operation ˜ℓ2 on E as follows: +(a) ˜ℓ2 +´ +ep´1q +i +, ep´1q +j +¯ +“ ř +kPI uk +ijep´1q +k +for all i, j P I´1. +(b) ˜ℓ2 +´ +ep´kq +i +, ep´lq +j +¯ +“ 0 for all i P Ik, j P Il with k ě 2 or l ě 2. +(c) we extend ˜ℓ2 to E using O-bilinearity and Leibniz identity with respect to the anchor ρ. +By construction, ˜ℓ2 satisfies the Leibniz identity with respect to the anchor ρE. Also, ρ ˝ d “ 0 on +E´2. The map defined for all homogeneous x, y P E by +rd, ˜ℓ2sRNpx, yq “ d ˝ ˜ℓ2 px, yq ` ˜ℓ2 pdx, yq ` p´1q|x|˜ℓ2 px, dyq , +is a graded symmetric degree `2 operation pE bEq‚ ÝÑ E‚`2, and rd, ˜ℓ2sRN|E´1 “ 0. Let us check that +it is O-bilinear, i.e. for all f P O, x, y P E: +rd, ˜ℓ2sRNpx, fyq ´ frd, ˜ℓ2sRNpx, yq “ 0. +1. if x P E´1, this quantity is zero in view of +rd, ˜ℓ2sRNpx, fyq “ frd, ˜ℓ2spx, yq ` dρpxqrfs y ´ ρpxqrfs dy +loooooooooooooomoooooooooooooon +“0 +2. if x P E´2, one has +rd, ˜ℓ2sRNpx, fyq ´ frd, ˜ℓ2spx, yq “ ˜ℓ2pdx, fyq ´ f ˜ℓ2pdx, yq “ ρpdxqpfq y “ 0 +since ρ ˝ d “ ρA ˝ π ˝ d “ 0, +3. if x P E´i with i ě 3, it is obvious by O-linearity of ˜ℓ2 on the involved spaces. + +CHAPTER 4. MAIN RESULTS OF PART I +62 +As a consequence, rd, ˜ℓ2sRN is a degree `2 element in the total complex Pagep1qpEq. By construction +rd, ˜ℓ2sRN has no component on the last column. Since πprd, ˜ℓ2sRN|E´1 q “ 0 and also rd, rd, ˜ℓ2sRNsRN|Eď´2 “ +0, the O-bilinear operator rd, ˜ℓ2sRN is D-closed in Pagep1qpEq. +By virtue of the first item of Proposition 4.3.3, the operator rd, ˜ℓ2sRN is then a D-coboundary, so +there exists τ2 P ‘jě2HomO +´Ä2 E|´j´1, E´j +¯ +such as Dpτ2q “ ´rd, ˜ℓ2sRN. Upon replacing ˜ℓ2 by ˜ℓ2 `τ2 +we obtain a 2-ary bracket ℓ2 of degree +1 which satisfies all items of Definition 4.3.9. +Proof (of Theorem 4.2.1). Lemma 4.3.10 gives the existence of an almost differential graded Lie alge- +broid with differential ℓ1 “ d and binary bracket ℓ2. We have to construct now the higher brackets ℓk +for k ě 3. +Step 1: Construction of the 3-ary bracket ℓ3. (Its construction being different from the one +of the higher brackets, we put it apart). We first notice that the graded Jacobiator defined for all +x, y, z P E by +Jacpx, y, zq :“ ℓ2pℓ2px, yq, zq ` p´1q|y||z|ℓ2pℓ2px, zq, yq ` p´1q|x||y|`|x||z|ℓ2pℓ2py, zq, xq +is O-linear in each variable, hence is a degree `2 element in À +jě1 HomOpÄ3 E |´j´2, E´jq Ă Pagep2qpEq. +For degree reason, its component on the last column of diagram (4.5) is zero, i.e. +it belongs to +z +Page +p1qpEq. +Let us check that it is D-closed: for this purpose we have to check that both conditions in Lemma +4.3.2 hold: +1. Since π is a morphism from pE´1, ℓ2q to pA, r¨, ¨sA, ρAq, and since r¨, ¨sA satisfies the Jacobi +identity, one has for all x, y, z P E´1: +Jacpx, y, zq P ker π. +2. Furthermore, a direct computation of rJac, dsRN gives in view item 2 of Definition 2.1.3: +dJacpx, y, zq “ Jacpdx, y, zq ` p´1q|x|Jacpx, dy, zq ` p´1q|x|`|y|Jacpx, y, dzq +for all x, y, z P E. +Thus, DpJacq “ 0. By Proposition 4.3.3, item 2, Jac is a D-coboundary, and, more precisely, there +exists an element ℓ3 “ ř +jě2 ℓj +3 P z +Page +p2q +1 pEq with ℓj +3 P HompÄ3 E |´j´1, E´jq such that +Dpℓ3q “ ´Jac +i.e. +rd, ℓ3sRN “ ´Jac. +(4.10) +We choose the 3-ary bracket to be ℓ3. +Step 2: Recursive construction of the k-ary brackets ℓk for k ě 4. Let us recapitulate: ℓ1 “ d +, ℓ2 and ℓ3 are constructed and the lowest polynomial-degree terms of rℓ1 ` ℓ2 ` ℓ3, ℓ1 ` ℓ2 ` ℓ3sRN +satisfy +1. rℓ1, ℓ1sRN “ 0 (since d2 “ 0), +2. rℓ1, ℓ2sRN “ 0 (since d “ ℓ1 and ℓ2 define an almost Lie algebroid structure). +3. rℓ2, ℓ2sRN `2rℓ3, ℓ1sRN “ 2pJac`rℓ3, ℓ1sRNq “ 0 by definition of ℓ3, and because rℓ2, ℓ2sRN “ 2Jac. + +CHAPTER 4. MAIN RESULTS OF PART I +63 +However, the following term of degree `2 and polynomial-degree 3 may not be equal to zero: +rℓ3, ℓ2sRN P +à +HomO +˜ +4 +ä +Ej`1, E´j +¸ +“ z +Page +p3q +1 pEq. +(4.11) +Let us check that this term is indeed a O-multilinear map: For x1 P E´1, x2, x3, x4 P Eď´2 and f P O, +the only terms of pℓ3 ˝ ℓ2 ` ℓ2 ˝ ℓ3qpx1, fx2, x3, x4q where the anchor shows up are: +$ +’ +’ +’ +& +’ +’ +’ +% +ℓ3pℓ2px1, fx2q, x3, x4q +“ ρpx1qrfsℓ3px2, x3, x4q ` fpℓ3pℓ2px1, x2q, x3, x4qq +p´1q|x2|`|x3|`|x4|ℓ2pfℓ3px2, x3, x4q, x1q +“ ´ρpx1qrfsℓ3px2, x3, x4q +`fpp´1q|x2|`|x3|`|x4|ℓ2pfℓ3px2, x3, x4q, x1qq +The terms containing the anchor map add up to zero. When there are more elements in E´1, the +computation follows the same line. Moreover, by graded Jacobi identity of the Richardson-Nijenhuis +bracket: +rrℓ1 ` ℓ2 ` ℓ3, ℓ1 ` ℓ2 ` ℓ3sRN, ℓ1 ` ℓ2 ` ℓ3sRN “ 0 +The term of polynomial-degree 4 in the previous expression gives rrℓ3, ℓ2sRN, ℓ1sRN “ 0. Hence, by +Proposition 4.3.6, rℓ3, ℓ2sRN is a D-cocycle in the complex Pagep3qpEq, whose components on the last +column and the column ´1 are zero. It is therefore a coboundary by Proposition 4.3.3 item 3: we can +continue a step further and define ℓ4 P ‘jě3Hom +´Ä4 E|´j´1, E´j +¯ +such that: +´ rℓ2, ℓ3sRN “ rℓ1, ℓ4sRN “ rd, ℓ4sRN . +(4.12) +We choose the 4-ary bracket to be ℓ4. We now proceed by recursion. We assume that we have +constructed all the k-ary brackets, ℓk such as : +rd, ℓksRN “ ´ +ÿ +i`j“k`1 +iďj +rℓi, ℓjsRN “ ´1 +2 +ÿ +i`j“k`1 +i,jě1 +rℓi, ℓjsRN +(4.13) +for every k “ 1, . . . , n with n ě 4. The (n`1)-ary bracket is constructed as follows. First, the operator +ř +i`j“k`1 +i,jě1 +rℓi, ℓjsRN is checked to be O-linear as before. Now, we have +ÿ +i`j“n`2 +i,jě1 +“ +d, rℓi, ℓjsRN +‰ +RN “ ´2 +ÿ +i`j“n`2 +i,jě1 +“ +ℓi, rd, ℓjsRN +‰ +RN +(by graded Jacobi identity). +Since ℓj satisfies Equation (4.13) up to order n, we obtain +ÿ +i`j“n`2 +i,jě1 +“ +d, rℓi, ℓjsRN +‰ +RN “ +ÿ +i`j`k“n`3 +i,j,kě1 +“ +ℓi, rℓj, ℓksRN +‰ +RN “ 0, +where we used the graded Jacobi identity of the Nijenhuis-Richardson bracket in the last step. There- +fore, ř +i`j“n`2 +i,jě1 +rℓi, ℓjsRN, seen as an element in Pagepi`j´2qpEq by Remark 4.3.6, is a cocycle and for +degree reason it has no element on the last column, and the columns ´1, . . . , 3 ´ n in 4.5. The third +item of Proposition 4.3.3 gives the existence of an pn ` 1q-ary bracket ℓn`1 such as +rd, ℓn`1sRN “ ´ +ÿ +i`j“n`2 +iďj +rℓi, ℓjsRN . +This completes the proof. + +CHAPTER 4. MAIN RESULTS OF PART I +64 +Proof of Proposition 4.2.3 and Proposition 4.2.6 +Proof (of Proposition 4.2.3). This is a consequence of Proposition 4.2.1 and the third item of the +Proposition 4.3.3: If the component of Jac on the column ´1 is zero, we can choose ℓ3 with no +component on the last column and in column ´1 (see Proposition 4.3.3), i.e. the restriction of ℓ3 +to Ä3 E´1 is zero. Then ℓ3 has no component on the last column, the column ´1 and the column +´2. so rℓ2, ℓ3sRN has no component in the last column, ´1 and ´2 columns as well. Hence, ℓ4 can +be chosen with no component on column ´1, ´2 and ´3 by the third item of Proposition 4.3.3. The +proof continues by recursion. +We finish this section with a proof of Proposition 4.2.6. +Proof (of Proposition 4.2.6). We prove this Proposition in two steps. +1. Lemma 4.3.7 guarantees the existence a free resolution pE, d, πq of the Lie-Rinehart algebra A +such that E contains E1 and such that there exists a graded free module V with E1 ‘ V “ E. +2. Let DE and DE1 be as in the proof of Lemma 4.3.8. We construct the n-ary brackets on E by +extending the ones of pE1, pℓ1 +kqkě1, ρE1, π1q in the following way: +(a) We first construct an almost Lie algebroid bracket ˜ℓ2 on E´1 that extends the 2-ary bracket +of E1 +´1. Since the 2-ary bracket is determined by its value on a basis, the existence of a free +module V´1 such that E1 +´1 ‘ V´1 “ E´1 allows to construct ˜ℓ2 on E such that its restriction +to E1 is ℓ1 +2 and such that it satisfies the Leibniz identity. +As in the proof of Theorem 4.2.1 (to be more precise: Lemma 4.3.10), we see that r˜ℓ2, dEsRN +is O-linear, hence belongs to Pagep2q +2 pEq and is a DE-cocycle. Since E1 is a Lie 8-algebroid, +its restriction to Ä2 E1 is zero. +Lemma 4.3.8 allows to change ˜ℓ2 to an 2-ary bracket +ℓ2 :“ ˜ℓ2 `τ2 with τ2 “ 0 on Ä2 E1. Hence, ℓ2 defines a graded almost Lie algebroid bracket, +whose restriction to E1 is still ℓ1 +2. +(b) Since ℓ2 is an extension of ℓ1 +2, its Jacobiator Jac P Pagep2q +2 pEq of the 2-ary bracket ℓ2 +preserves E1. Also, its restriction ι˚ +E1Jac P Pagep2q +2 pE1q is the Jacobiator of ℓ1 +2, and the latter is +the DE1-coboundary of ℓ1 +3 in view of the higher Jacobi identity of E1. Since Jac P Pagep2q +2 pEq +is a DE-cocycle, Lemma 4.3.8 assures that Jac is the image through DE of some element +ℓ3 P Pagep2q +1 pEq which preserves E1 and whose restriction to Ä3 E1 is ℓ1 +3. The proof continues +by recursion: at the n-th step, we use Lemma 4.3.8 to construct an n-ary bracket for E that +extends the n-ary bracket of E1. +By construction, the inclusion map ι: E1 ãÑ E is a morphism for the n-ary brackets for all n ě 1. +4.3.3 +Proof of universality +Before proving the universal character of the construction, we need to do some preparations. +Let us prove the following lemma, +Lemma 4.3.11. Let Ψ, Ξ : S‚ +KpE1q Ñ S‚ +KpEq be O-linear Lie 8-algebroid morphisms. Let n P N0. If +Ξpiq “ Ψpiq for every 0 ď i ď n, there exists +1. a Lie 8-morphism of algebroids J1 : SKpE1q Ñ SKpEq + +CHAPTER 4. MAIN RESULTS OF PART I +65 +2. and a homotopy pJt, Htqr0,1s joining Ψ and J1, +such that +1. the components of polynomial-degree less or equal to n of Ht vanish, +2. Jpiq +1 +“ Ξpiq for every 0 ď i ď n ` 1. +Proof. Let us consider pΞ ´ Ψqpn`1q : Ä‚ E1 ÝÑ Ä‚ E. By assumption, pΞ ´ Ψqpiq “ 0 for all i ď n, +so that in view of Lemma 2.3.27 +1. pΞ ´ Ψqpn`1q is a Ψp0q-co-derivation. +2. Proposition 4.3.5 means that the map the restriction of the map pΞ ´ Ψqpn`1q to Än`2 E1 +corresponds to a closed element of degree 0 in Pagepn`1qpE1, Eq equipped with differentials ℓ1, ℓ1 +1. +Proposition 4.3.3 implies that there exists O-linear map Hn`1 : Än`2pE1q ÝÑ E, a degree ´1 and of +polynomial-degree n ` 1, i.e. an element in Pagepn`1qpE1, Eq, such that, +pΞ ´ Ψqpn`1q “ Qp0q +E +˝ Hn`1 ` Hn`1 ˝ Qp0q +E1 . +(4.14) +We denote its extension to a Ψp0q-co-derivation of degree ´1 by Hpn`1q. We now consider the following +differential equation for t P r0, 1s: +dJt +dt “ QE ˝ Ht ` Ht ˝ QE1, +and +J0 “ Ψ, +(4.15) +where Ht is the unique Jt-co-derivation of degree ´1 whose unique non-zero Taylor coefficient is Hn`1. +The existence of a solution for the differential equation (4.15) is granted by Proposition 2.3.19. By +considering the component of polynomial-degree 1, . . . , n, n ` 1 in Equation (4.15), we find +$ +& +% +dJpiq +t +dt +“ +0 +for i “ 0, . . . , n , +dJpn`1q +t +dt +“ +Qp0q +E +˝ Hpn`1q ` Hpn`1q ˝ Qp0q +E1 “ pΞ ´ Ψqpn`1q +Hence: +# +Jpiq +t +“ +Ψpiq +for i “ 0, . . . , n , +Jpn`1q +t +“ +Φpn`1q ` tpΞ ´ Ψqpn`1q. +Therefore, applying t “ 1 to the previous relation, one finds +# +Jpiq +t +“ +Ψpiq +for i “ 0, . . . , n , +Jpn`1q +1 +“ +Ψpn`1q ` pΞ ´ Ψqpn`1q “ Ξpn`1q +This completes the proof. +Construction of the Lie 8-morphism +Proof (of Theorem 4.2.4). Let us prove item 1. We construct the Taylor coefficients of the Lie 8- +algebroid Φ by recursion. +The Taylor coefficient of polynomial-degree 0 is obtained out of classical properties of projective +resolutions of O-modules. Given any complex pE1, ρ1, ℓ1 +1, π1q which terminates in A through π, for + +CHAPTER 4. MAIN RESULTS OF PART I +66 +every free resolution pE, ρ, ℓ1, πq of A, there exists a chain map Φp0q : pE1, ℓ1 +1q Ñ pE, ℓ1q as in Equation +(4.2), and any two such chain maps are homotopic. +We still denote by Φp0q its extension to an +polynomial-degree 0 co-morphism Ä‚ E1 Ñ Ä‚ E. +To construct the second Taylor coefficient, let us consider the map: +S2 +KpE1q +Ñ +E +px, yq +ÞÑ +Φp0q ˝ ℓ1 +2px, yq ´ ℓ2pΦp0qpxq, Φp0qpyqq. +(4.16) +This map is in fact O-bililinear, i.e. belongs to HomOpÄ2 E1, Eq, hence to Pagep1qpE1, Eq, see Equation +(4.5). Let us check that it is a D-cocycle: +A. If either one of the homogeneous elements x P E1 or y P E1 is not of degree ´1, a straightforward +computation gives: +ℓ1 ˝ +´ +Φp0q ˝ ℓ1 +2px, yq ´ ℓ2 +´ +Φp0qpxq, Φp0qpyq +¯¯ +“ Φp0q ˝ ℓ1 +1 ˝ ℓ1 +2px, yq ` ℓ2 +´ +Φp0q ˝ ℓ1 +1pxq, Φp0qpyq +¯ +` p´1q|x|ℓ2 +´ +Φp0qpxq, Φp0q ˝ ℓ1 +1pyq +¯ +“ +´ +Φp0q ˝ ℓ1 +2 ´ ℓ2 +´ +Φp0q, Φp0q¯¯ +˝ ℓ1 +1px d yq. +B. If both x, y P E1 are of degree ´1: +π +´ +Φp0qℓ1 +2px, yq ´ ℓ2pΦp0qx, Φp0qyq +¯ +“ +π1 ˝ ℓ1 +2px, yq ´ π ˝ ℓ2 +´ +Φp0qx, Φp0qy +¯ +“ +rπ1pxq, π1pyqs ´ +” +π +´ +Φp0qx +¯ +, π +´ +Φp0qx +¯ı +“ +rπ1pxq, π1pyqs ´ rπ1pxq, π1pyqs +“ +0. +By Proposition 4.3.3 item 2), there exists Φp1q P HomO +´Ä2 E1, E +¯ +, of degree 0, so that +Φp0q ˝ ℓ1 +2px, yq ` Φp1q ˝ ℓ1 +1px d yq “ ℓ1 ˝ Φp1qpx, yq ` ℓ2pΦp0qpxq, Φp0qpyqq +for all x, y P E1. +(4.17) +Φp0q is a chain map and Relation (4.17) can be rewritten in terms of QE and QE1 as follows +# +Qp0q +E +˝ Φp0q +“ +Φp0q ˝ Qp0q +E1 +Qp0q +E +˝ Φp1q ´ Φp1q ˝ Qp0q +E1 +“ +Φp0q ˝ Qp1q +E1 ´ Qp1q +E +˝ Φp0q +(4.18) +The construction of the morphism Φ announced in Theorem 4.2.4 is then done by recursion. The +recursion assumption is that we have already defined a O-multilinear co-morphism Φ : S‚ +KpE1q Ñ S‚ +KpEq +with +pΦ ˝ QE1 ´ QE ˝ Φqpkq “ 0 +for all +0 ď k ď n. +The co-morphism Φ : S‚ +KpE1q Ñ S‚ +KpEq with Taylor coefficients Φp0q and Φp1q satisfies the recursion +assumption for n “ 1. +Assume now that we have a co-morphism Φ that satisfies this assumption for some n P N, and +consider the map TΦ :“ Φ ˝ QE1 ´ QE ˝ Φ. +Lemma 2.3.26 implies that T pn`1q +Φ +is a O-multilinear +Φp0q-co-derivation, and that it corresponds to a D-closed element1 in Pagepn`1qpE1, Eq. Since it has no +1The following remark is crucial. Under the assumptions of Lemma 2.3.26, pΦ˝QE1 ´QE ˝Φqpn`1q corresponds to a D- +closed element of degree `1 in the bi-complex Pagepn`1qpE1, Eq through the chain isomorphism described in Proposition +4.3.5. Here, E, E1 are equipped with the differentials ℓ1, ℓ1 +1 which are the restriction of the components Qp0q +E , Qp0q +E1 . + +CHAPTER 4. MAIN RESULTS OF PART I +67 +component on the last column for degree reason, Proposition 4.3.3 implies that T pn`1q +Φ +is a coboundary: +That is to say that there is a Φp0q-co-derivation Θ P Pagepn`1qpE1, Eq (of polynomial-degree n ` 1 and +degree 0) which can be seen as a map Θ : Än`2pE1q Ñ E such that: +T pn`1q +Φ +“ Qp0q +E +˝ Θ ´ Θ ˝ Qp0q +E1 . +Consider now the co-morphism ˜Φ whose Taylor coefficients are those of Φ in polynomial-degree 0, . . . , n +and Φpn`1q ` Θ in polynomial-degree n ` 1: +˜Φpiq :“ +$ +& +% +Φpiq +if 0 ď i ď n, +Φpn`1q ` Θ +if i “ n ` 1 +(4.19) +This is easily seen to satisfy the recursion relation for n`1. This concludes the recursion. The Taylor +coefficients obtained by recursion define a Lie 8-algebroid Φ: S‚ +KpE1q ÝÑ S‚ +KpEq which is compatible +by construction with the hooks π, π1. +By continuing this procedure, we construct a Lie 8-morphism from S‚ +KpE1q to S‚ +KpEq. This proves +the first item of Theorem 4.2.4. +Construction of a homotopy that joins two such morphisms +Let us prove the second item in Theorem 4.2.4. Notice that in the proof of the existence of the Lie +8-morphism between S‚ +KpE1q and S‚ +KpEq obtained in the first item, we made many choices, since we +have chosen a coboundary at each step of the recursion. +Let Φ, Ψ be two such Lie 8-morphisms between S‚ +KpE1q and S‚ +KpEq. +The polynomial-degree 0 +component of the co-morphisms Φ and Ψ restricted to E1 are chain maps: +¨ ¨ ¨ +� E1 +´2 +� +h +� +Φp0q +� +Ψp0q +� +E1 +´1 +Φp0q +� +Ψp0q +� +h +� +π1 +� +A +¨ ¨ ¨ +� E´2 +� E´1 +π +� � +which are homotopy equivalent in the usual sense because pE, ℓ1q is a projective resolution of A: said +differently, there exists a degree ´1 O-linear map h: E1 Ñ E such that +Ψp0q ´ Φp0q “ ℓ1 ˝ h ` h ˝ ℓ1 +1 +on E1. +(4.20) +Let us consider the following differential equation: +$ +& +% +dJt +dt “ QE ˝ HtpJtq ` HtpJtq ˝ QE1, +for t P r0, 1s +J0 “ Φ. +(4.21) +with HtpJtq being a Jt-co-derivation of degree ´1 whose Taylor coefficient of polynomial-degree 0 is +h. This equation does admit solutions in view of Proposition 2.3.19. +By looking at the component polynomial-degree 0 of Equation (4.21) on E1, one has: +dJp0q +t +dt +“ ℓ1 ˝ h ` h ˝ ℓ1 +1 +“ Ψp0q ´ Φp0q. + +CHAPTER 4. MAIN RESULTS OF PART I +68 +Hence, Jp0q +t +“ Φp0q ` t +` +Ψp0q ´ Φp0q˘ +is a solution such that Jp0q +1 +“ Ψp0q. +By construction, J1 is +homotopic to Φ via the pair pJt, Htq over r0, 1s, and its polynomial-degree 0 Taylor coefficient coincides +with the Taylor coefficient of Ψ. +From there, the construction goes by recursion using Lemma 4.3.11. Indeed, this lemma allows +constructing recursively a sequence of Lie 8-algebroids morphism pΨnqně0 and homotopies pJn,t, Hn,tq +(with t P rn, n ` 1s) between Ψn and Ψn`1 such that: Hpiq +n,t is zero for t ě n and i ‰ n ` 1. By +Lemma 4.3.11, all these homotopies are compatible with the hooks. These homotopies are glued in +a homotopy pJt, Htqr0,`8r such that for every n P N0, the components of polynomial-degree n of the +Lie 8-algebroids morphism Jpnq +t +are constant and equal to Ψpnq for t ě n. By Lemma 2.3.22, these +homotopies can be glued to a homotopy on r0, 1s. Explicitly, since t ÞÑ +t +1´t maps r0, 1r to r0, `8r +and by Lemma 2.3.22, the pair +´ +J +t +1´t , +1 +p1´tq2 Hk, +t +1´t +¯ +is a homotopy between Φ and Ψ. This proves the +second item of the Theorem 4.2.4. +4.4 +Examples of universal Lie 8-algebroids of Lie-Rinehart algebras +4.4.1 +New constructions from old ones +In this section, we explain how to construct universal Lie 8-algebroids of some Lie-Rinehart algebra +which is derived from a second one through one of natural constructions as in Section 3.2.1 (localization, +germification, restriction), when a universal Lie 8-algebroid of the latter is already known. +Localization +Localization is an useful algebraic tool, specially in algebraic geometry. When O is an algebra of +functions, it corresponds to study local properties of a space, or germs of functions. +Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O. Let S Ă O be a multiplicative-closed subset +containing no zero divisor. +We recall from item 2, Example 3.2.1 that the localization S´1A – +A bO S´1O of A at S comes equipped with a natural structure of Lie-Rinehart algebra over the +localization algebra S´1O. Recall that for ϕ: E ÝÑ T a homomorphism of O-modules, there is a +well-defined homomorphism of O-modules, +ϕ b id: E bO S´1O ÝÑ T bO S´1O, ϕ b id px b f +s q :“ ϕpxq b f +s +that can be considered as a S´1O-module homomorphism +S´1ϕ: S´1E ÝÑ S´1T with S´1ϕ +´x +s +¯ +:“ ϕpxq +s +, x P E, pf, sq P O ˆ S, +called the localization of ϕ. +Given a Lie 8-algebroid structure pE‚, ℓ‚, ρq that covers A through π. The triplet pE1 +‚, ℓ1 +‚, ρ1q is a Lie +8-algebroid structure that covers S´1A through the hook π1 where +1. E1 “ S´1E; + +CHAPTER 4. MAIN RESULTS OF PART I +69 +2. The anchor map ρ1 is defined by +ρ1 : S´1E´1 +ÝÑ +DerpS´1Oq +x +s +ÞÝÑ +ρ1 ` x +s +˘ +: +S´1O +ÝÑ +S´1O +f +u +ÞÝÑ +1 +s ¨ +´ +ρpxqrfsu´fρpxqrus +u2 +¯ +for x P E, f P O, ps, uq P S ˆ S; +3. ℓ1 +k “ S´1ℓk, for all k P Nzt2u; +4. The binary bracket is more complicated because of the anchor map: we set +ℓ1 +2 +ˆ1 +s x, 1 +uy +˙ +“ 1 +suℓ2px, yq ´ ρpxqrus +su2 +y ` ρpyqrss +s2u +x +for x, y P E, ps, uq P S2 (with the understanding that ρ ” 0 on E´i with i ě 2); +5. π1 “ S´1π. +One can check that these operations above are well-defined and for all z P S´1E´1 the map ρ1pzq is +indeed a derivation on S´1O. The previously defined structure is also a Lie 8-algebroid that we call +localization of the Lie 8-algebroid pE‚, ℓ‚, ρq with respect to S. +Proposition 4.4.1. Let S Ă O be a multiplicative subset containing no zero divisor. The localization +of a universal Lie 8-algebroid of a Lie-Rinehart algebra A is a universal Lie 8-algebroid of S´1A. +Proof. The object ppE1 +‚, ℓ1 +‚, ρ1qq described above is also a Lie 8-algebroid terminating in S´1A through +π1. It is universal because localization preserves exact sequences [And]. +Restriction +When OY is the ring of functions of an affine variety Y (see Section 5.1), to every subvariety X Ă Y +corresponds its zero locus, which is an ideal IX Ă OY . A Lie 8-algebroid or a Lie-Rinehart algebra +over OY may not restrict to a Lie-Rinehart algebra over OX: it only does so when one can quotient all +brackets by IX, which geometrically means that the anchor map takes values in vector fields tangent +to X. We can then “restrict”, i.e. replace OY by OY {IX. This operation has already been defined in +Section 4.2, and here is an immediate consequence of Corollary 4.2.13: +Proposition 4.4.2. Let I Ă O be a Lie-Rinehart ideal, i.e. an ideal such that ρApAqrIs Ă I. The +quotient of a universal Lie 8-algebroid of A with respect to an ideal I is a Lie 8-algebroid that +terminates in A{IA. It is universal if and only if pp E´i +IE´i qiě1, ¯ℓ1, πq is exact, i.e. if Tor‚ +OpA, O{Iq “ 0. +Remark 4.4.3. Note that the anchor map ρ: E´1 Ñ DerpOq goes to quotient to +E´1 +IE´1 Ñ DerpOq +as an O-linear map, but needs the extra condition ρpE´1qrIs Ă I to induce an O{I- linear map +E´1 +IE´1 Ñ DerpO{Iq. +Germification +Let W Ď CN be an affine variety and OW its coordinates ring (see Section 5.1). For a P W, consider +OW,a the local ring at a. Note that OW,a » pOW qma [Har77], where ma “ tf P OW | fpaq “ 0u +and pOW qma is the localization w.r.t the complement of ma, Proposition 4.4.1 implies the following +statement: + +CHAPTER 4. MAIN RESULTS OF PART I +70 +Proposition 4.4.4. Let W be an affine variety with functions OW . For every point a P W and any +Lie-Rinehart A over OW , the germ at a of the universal Lie 8-algebroid of A is the universal Lie +8-algebroid of the germ of A at a. +Therefore, the germ at a of a Lie-Rinehart algebra or a Lie 8-algebroid is simply its localization w.r.t +the complement of ma. +4.4.2 +Sections vanishing on a codimension 1 subvariety +Let pA, r¨ , ¨s , ρAq be an arbitrary Lie-Rinehart algebra over O. For any ideal I Ă O, IA is also a +Lie-Rinehart algebra (see Example 3.2.1). When O are functions on a variety M, I are functions +vanishing on a subvariety X and A is a O-module of sections over M, IA corresponds geometrically +to sections vanishing along X. It is not an easy task. In codimension 1, i.e. when I is generated by +one element, the construction can be done by hand. +Proposition 4.4.5. Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over a commutative algebra O. Let +pE, ℓk “ t¨ ¨ ¨ ukě1, ρq be a Lie 8-algebroid that terminates in A through a hook π. For any element +χ P O, the O-module A1 “ χA Ď A is closed under the Lie bracket, so the triple pχA, r¨, ¨sA, ρAq is a +Lie-Rinehart algebra over O. A Lie 8-algebroid pE1 “ E, ℓ1 +k “ t¨ ¨ ¨ u1 +kě1, ρ1q hooked in χA through π1 +can be defined as follows: +1. The brackets are given by +(a) t¨u1 +1 “ t¨u1, +(b) the 2-ary bracket: +tx, yu1 +2 :“ χtx, yu2 ` ρpxqrχs y ` p´1q|x||y|ρpyqrχs x, +(4.22) +for all x, y P E‚, with the understanding that ρ “ 0 on Eď´2, +(c) t¨ ¨ ¨ u1 +k “ χk´1t¨ ¨ ¨ uk for all k ě 3, +2. ρE1 “ χρ, +3. π1 “ χπ. +Proof. We leave it to the reader. +Proposition 4.4.6. If χ is not a zero-divisor in O, and pE, t¨ ¨ ¨ ukě1, ρ, πq is a universal Lie 8- +algebroid of A, then the Lie 8-structure described in the four items of 4.4.1 is the universal Lie +8-algebroid of χA. +Proof. If χ is not a zero-divisor in A (i.e. if a ÞÑ χa is an injective endomorphism of A), then the +kernel of π1 coincides with the kernel of π, i.e. with the image of t¨u1 “ t¨u1, so that pE, ℓ1, χπq is a +resolution of χA. + +CHAPTER 4. MAIN RESULTS OF PART I +71 +4.4.3 +Algebra extension +Recall that for O a unital algebra with no zero divisor, derivations of O induce derivations of its field +of fractions O. +Proposition 4.4.7. Let O be an unital algebra with no zero divisor, O its field of fractions, and ˜O an +algebra with O Ă ˜O Ă O. For every Lie-Rinehart algebra A over O whose anchor map takes values +in derivations of O preserving ˜O, then +1. any Lie 8-algebroid structure pE, pℓkqkě1, ρ, πq that terminates at A extends for all i “ 0, . . . , n +to a Lie 8-algebroid structure on ˜O bO E, +2. and this extension ˜O bO E is a Lie 8-algebroid that terminates at the Lie-Rinehart algebra +˜O bO A. +Proof. Since they are O-linear, the hook π, the anchor ρ, and the brackets ℓk for k ‰ 2 are extended +to ˜O-linear maps. Since the image of ρ is the image of ρA, it is made of derivations preserving O, +which is easily seen to allow an extension of ℓ2 to ˜O bO E using the Leibniz identity. +Remark 4.4.8. Of course, the Lie 8-algebroid structure obtained on ˜O bO E is not in general the +universal Lie 8-algebroid of ˜O bO A, because the complex p ˜O bO E, ℓ1, πq may not be a resolution of +˜O bO A (see Example 4.4.10). +Remark 4.4.9. Since any module over a field is projective, any Lie-Rinehart algebra over a field is +a Lie algebroid. If we choose ˜O “ O therefore, the Lie-Rinehart algebra O bO A is a Lie algebroid, +so is homotopy equivalent to any of its universal Lie 8-algebroid. Unless A is a Lie algebroid itself, +the Lie 8-algebroid in Proposition 4.4.7 will not be homotopy equivalent to a Lie 8-algebroid whose +underlying complex is of length one, and is therefore not a universal Lie 8-algebroid of O bO A. +4.4.4 +Blow-up +Let us investigate the behavior of the universal Lie 8-algebroids under blow-up. +We recall that the blowup of CN`1 at the origin is given by, B0pCN`1q “ tpx, ℓq P CN`1 ˆ PN | x P ℓu +together with the map +π: B0pCN`1q ÝÑ CN`1 +px, ℓq ÞÑ x. +For O “ Crz0, . . . , zNs the coordinate ring of CN`1, the blow-up of CN`1 at the origin is covered +by affine charts: in the i-th affine chart Ui, the coordinate ring is +OUi “ Crz0{zi, . . . , zi, . . . , zN{zis. +By Remark 3.2.1, and Proposition 4.4.7 for any Lie-Rinehart algebra A whose anchor map takes values +in vector fields vanishing at 0 P CN`1, we obtain a Lie 8-algebroid of OUi bO A that we call blow-up of +at 0 in the chart Ui. Proposition 4.4.7 then says that the blow-up at 0 of the universal Lie 8-algebroid +of A, in each chart, is a Lie 8-algebroid that terminates in the blow-up of A (as defined in remark +3.2.1). It may not be the universal one, see Example 4.4.10. + +CHAPTER 4. MAIN RESULTS OF PART I +72 +Example 4.4.10 (Universal Lie-8-algebroids and blow-up: a counter example). Consider the poly- +nomial function in N ` 1 variables ϕ “ řN +i“0 z3 +i . Let us consider the singular foliation Fϕ Ă XpCNq as +in Example 4.4.6. Its generators are ∆ij :“ z2 +i +B +Bzj ´ z2 +j +B +Bzi , for 0 ď i ă j ď N. +Let us consider its blow-up in the chart UN. Geometrically speaking, Ă +Fϕ “ OUN bO Fϕ is the +OUN -module generated by the blown-up vector fields r∆ij “ zN +´ +z2 +i +B +Bzj ´ z2 +j +B +Bzi +¯ +, j ‰ N, and r∆iN “ +zN +´ +zNz2 +i +B +BzN ´ +B +Bzi ´ z2 +i +řN´1 +j“0 zj B +Bzj +¯ +of the vector fields ∆ij, for i, j P t0, . . . , Nu, w.r.t affine chart +UN. The vector fields r∆ij, j ‰ N, belong to the OUN -module generated by the vector fields r∆iN, +(explicitly r∆ij “ z2 +j r∆iN ´z2 +i r∆jN). Said differently, the vector fields r∆iN, i “ 0, . . . N´1 are generators +of Ă +Fϕ. Since they are independent, the singular foliation Ă +Fϕ is a free OUN -module. A universal Lie +8-algebroid for it is therefore concentrated in degree ´1 and is given by the OUN -module generated +by some set pei, i “ 0 . . . N ´ 1q, equipped with the Lie bracket: +rei, ejs :“ 2zN +` +z2 +i ej ´ z2 +j ei +˘ +(4.23) +together with the anchor map ρ which assigns ei to r∆iN, for i “ 1 . . . N ´ 1. +On the other hand, the blow-up of the universal Lie 8-algebroid of Fϕ is not homotopy equivalent +to a Lie algebroid. Let us show this point. In this case, the Lie 8-algebroid is given in Section 4.4.6: +notice that E´i ‰ 0 for i “ 1, . . . , N and that ℓ1|0 “ 0. The pull-back Lie 8-algebroid p ˜E, p˜ℓkqkě1, ˜ρ, ˜πq +verifies by construction that ˜E´i ‰ 0 for i “ 1, . . . , N and that ˜ℓ1|x “ 0 for every x in the inverse +image of zero, and such a complex can not be homotopy equivalent to a complex of length 1. In other +words, OUi bO ¨ is not an exact functor in this case (which is a classical fact in algebraic geometry). +This example tells us that the blow-up of the universal Lie 8-algebroid of an affine variety W may +not be the universal Lie 8-algebroid of its blow-up, (even locally). +4.4.5 +Universal Lie 8-algebroids of some singular foliations +Singular foliation is defined in Section 3.2. The coming example uses the notion "multi-derivation", it +is useful to recall the definition and write down some basic operations on them. We refer the reader +to Section 3.1 of [LGPV13] for more details on this notion. +Definition 4.4.11. A skew symmetric k-multilinear map P P HomKpOk, Oq is a said to a k-multi- +derivation of O if P is a derivation in each of its argument, i.e., for every i P t1, . . . , ku and for all +f, g P O we have +Prf1, . . . , +fg +loomoon +i´th slot +. . . , fks “ Prf1, . . . , +f +loomoon +i´th slot +. . . , fks g ` f Prf1, . . . , +g +loomoon +i´th slot +. . . , fks +(4.24) +When O happen to be the coordinate ring of some affine variety W, k-multi-derivations of O are +called k-multi-vector fields on W, and denote by XkpWq the module of k-multi-vector fields on W. In +general, it is denoted by XkpOq. +Remark 4.4.12. By skew-symmetry argument, it is enough that the relation (4.24) holds for the first +slot, i.e. for i “ 1. + +CHAPTER 4. MAIN RESULTS OF PART I +73 +Lemma 4.4.13. The graded module X‚pOq :“ À +kě0 XkpOq comes equipped with graded algebra struc- +ture whose product ^ is defined as follows, +pP ^ Rqrf1, . . . , fk`ls :“ +ÿ +σPSpk,lq +ϵpσqPrfσp1q, . . . , fσpkqsRrfσpk`1q, . . . , fσpk`lqs, +for P P XkpOq, R P XlpOq and f1, . . . , fk`l P O. By convention X0pOq “ O. Also, f ^ X :“ fX for +all f P O. +Lemma 4.4.14. For every f P O, the contraction by f, ιf : XkpOq Ñ Xk´1pOq, @ k ě 1, which is +defined as follows2 +P ÞÑ ιfpPqrf1, . . . , fk´1s :“ Prf, f1, . . . , fk´1s, +for all +f1, . . . , fk´1 P O +(4.25) +satisfies +• ιf ˝ ιf “ 0, +• for all P P XkpOq and R P XlpOq, +ιfpP ^ Rq “ ιfpPq ^ R ` p´1qkP ^ ιfpRq. +(4.26) +Proof. The first item is true by skew-symmetry of multi-derivations. Now let us prove the second +item. For f1, . . . , fk`l´1 P O, we have +ιfpP ^ Rqrf1, . . . , fk`l´1s “ pP ^ Rqrf, f1, ¨ ¨ ¨ , fk`l´1s +“ +ÿ +σPSpk´1,lq +ϵpσqPrf, fσp1q, . . . , fσpk´1qsRrfσpkq, . . . , fσpk`l´1qs` +p´1qk +ÿ +σPSpk,l´1q +ϵpσq Prfσp1q, . . . , fσpkqsRrf, fσpk`1q, . . . , fσpk`l´1qq +looooooooooooooooooooooooooooomooooooooooooooooooooooooooooon +here, the inversion number with f is k +“ ιfpPq ^ R ` p´1qkP ^ ιfpRq. +We have used the fact that for every σ P Spk, lq, one has that either σp1q “ 1 or σpk ` 1q “ 1. +Remark 4.4.15. Notice the following: +• For all f P O and X P DerpOq, one has ιfpXq “ Xrfs. +• For any two derivation X, Y P DerpOq we have +pX ^ Y qrf, gs “ XrfsY rgs ´ Y rgsXrfs. +• For a family of derivations X1, . . . Xk P DerpOq, the k-multi-derivation X1 ^ ¨ ¨ ¨ ^ Xk applied to +f1, ¨ ¨ ¨ , fk P O is equal to the determinant +�������� +X1rf1s +¨ ¨ ¨ +X1rfks +... +... +Xkrf1s +¨ ¨ ¨ +Xkrfks +�������� +. +• When O is the algebra of polynomial functions in d variables Krx1, . . . , xds, a k-multi-derivation +P admits the coordinate expression +P “ +ÿ +1ďi1㨨¨ăikďd +Prxi1, . . . , xiks B +Bxi1 +^ ¨ ¨ ¨ ^ +B +Bxik +. +2It should be understood that ιfpOq :“ 0. + +CHAPTER 4. MAIN RESULTS OF PART I +74 +Lie derivative +For P P XkpOq and X P DerpOq, the Lie derivative LXP P XkpOq of P along X is defined as +pLXPqrf1, ¨ ¨ ¨ , fks :“ X rPrf1, ¨ ¨ ¨ , fkss ´ +kÿ +j“1 +Prf1, . . . , Xrfjs, . . . , fks. +(4.27) +There is another important operation on multi-derivations, the so-called "Schouten bracket" which +is a generalization of the commutator of derivations, also Lie derivative of multi-derivation along a +vector field. We will not recall how it is defined here, since we do not really make use of it. We refer +the reader to, e.g [LGPV13] for this notion. +4.4.6 +Vector fields annihilating a Koszul function ϕ +This universal Lie 8-algebroid was already described in Section 3.7 of [LLS20], where the brackets +were simply checked to satisfy the higher Jacobi identities - with many computations left to the reader. +Here, we give a theoretical explanation of the construction presented in [LLS20]. +Let O be the algebra of all polynomials on V :“ Cd. A function ϕ P O is said to be a Koszul +polynomial, if the Koszul complex +. . . +ιϕ +ÝÑ X3pV q +ιϕ +ÝÑ X2pV q +ιϕ +ÝÑ XpV q +ιϕ +ÝÑ O ÝÑ 0 +(4.28) +is exact in all degree, except in degree 0. By virtue of a theorem of Koszul [Eis95], see [Hid89] Theorem +16.5 piq, ϕ is Koszul if +´ +Bϕ +Bx1 , ¨ ¨ ¨ , Bϕ +Bxd +¯ +is a regular sequence. +From now on, we choose ϕ a Koszul function, and consider the Lie-Rinehart algebra (which is a +singular foliation) +Fϕ :“ tX P XpV q : Xrϕs “ 0u “ Kerpιϕq: XpV q +ιϕ +ÝÑ O. +(4.29) +The Koszul complex (4.28) truncated of its degree 0 term gives a free resolution pE, d, ρq of Fϕ, with +E´i :“ Xi`1pV q, d :“ ιϕ, and ρ :“ ´ιϕ. +Remark 4.4.16. Exactness of the Koszul complex implies in particular that Fϕ is generated by the +vector fields: +" Bϕ +Bxi +B +Bxj +´ Bϕ +Bxj +Bϕ +Bxi +, | 1 ď i ă j ď d +* +. +(4.30) +In [LLS20], this resolution is equipped with a Lie 8-algebroid structure, whose brackets we now +recall. +Proposition 4.4.17. A universal Lie 8-algebroid of Fϕ Ă XpV q is given on the free resolution +` +E´‚ “ X‚`1pV q, d “ ιϕ, ρ “ ´ιϕ +˘ +by defining the following n-ary brackets: +tBI1, ¨ ¨ ¨ , BInun :“ +ÿ +i1PI1,...,inPIn +ϵpi1, . . . , inqϕi1¨¨¨inBIi1 +1 ‚¨¨¨‚Iin +n ; +(4.31) +and the anchor map given for all i, j P t1, . . . , nu by +ρ +ˆ B +Bxi +^ +B +Bxj +˙ +:“ Bϕ +Bxj +B +Bxi +´ Bϕ +Bxi +B +Bxj +. +(4.32) +Above, for every multi-index J “ tj1, . . . , jnu Ď t1, . . . , du of length n, BJ stands for the n-vector +field +B +Bxj1 ^ ¨ ¨ ¨ ^ +B +Bxjn and ϕj1¨¨¨jn :“ +Bnϕ +Bxj1¨¨¨Bxjn . +Also, I1 ‚ ¨ ¨ ¨ ‚ In is a multi-index obtained by + +CHAPTER 4. MAIN RESULTS OF PART I +75 +concatenation of n multi-indices I1, . . . , In. For every i1 P I1, . . . , in P In, ϵpi1, . . . , inq is the signature +of the permutation which brings i1, . . . , in to the first n slots of I1 ‚ ¨ ¨ ¨ ‚ In. Last, for is P Is, we define +Iis +s :“ Iszis. +To understand this structure, let us first define a sequence of degree `1 graded symmetric poly- +derivations on X‚pV q (by convention, i-vector fields are of degree ´i ` 1) by: +tBi1, . . . , Biku1 +k :“ +Bkϕ +Bxi1 ¨ ¨ ¨ Bxik +. +(4.33) +We extend them to a graded poly-derivation of X‚pV q. +Lemma 4.4.18. The poly-derivations pt¨ ¨ ¨ u1 +kqkě1 are O-multilinear and equip X‚pV q with a (graded +symmetric) Poisson 8-algebra structure. Also, t¨u1 +1 “ ιϕ. +Proof. For degree reason, tF, X1, . . . , Xk´1u1 +k “ 0 for all X1, . . . , Xk´1 P X‚pV q and all F P X0pV q “ O. +This implies the required O-multilinearity. +It is clear that the higher Jacobi identities hold since +brackets of generators tδi1, ¨ ¨ ¨ , δinu1 are elements in O, and all brackets are zero when applied an +element in O. +Proof (of Proposition 4.4.17). The brackets introduced in Proposition 4.4.17 are modifications of the +Poisson 8-algebra described in Lemma 4.4.18. By construction, t¨ ¨ ¨ u +1 +n “ t¨ ¨ ¨ un when all arguments +are generators of the form BI for some I Ă t1 . . . , nu of cardinal ě 2. By O-multilinearity, this implies +t¨ ¨ ¨ u +1 +n “ t¨ ¨ ¨ un when n ě 3, or when n “ 2 and no argument is a bivector-field, or when n “ 1 and +the argument is not a bivector field. As a consequence, all higher Jacobi identities hold when applied +to n-vector fields with n ě 3. +Let us see what happens when one of the arguments is a bivector field, i.e. in the case where +we deal with at least an element of degree ´1. Let us assume that there is one such element, i.e. +Q1 “ Bi ^ Bj, Q2 “ BI2, . . . , Qn “ BIn with |Ij| ě 3, j “ 2, . . . , n. Then, in view of the higher Jacobi +identity for the Poisson 8-brackets pt¨ ¨ ¨ u1 +kqkě1 gives: +0 “ +ÿ +2ďkďn´2 +ÿ +σPSk,n´k +ϵpσq +!␣ +Qσp1q, . . . , Qσpkq +(1 +k , Qσpk`1q, . . . , Qσpnq +)1 +n´k`1 +(4.34) +` +ÿ +σPSn´1,1,σpnq‰1 +ϵpσq +!␣ +Qσp1q, . . . , Qσpn´1q +(1 +n´1 , Qσpnq +)1 +2 +(4.35) +` +ÿ +σPS1,n´1,σp1q‰1 +ϵpσq +!␣ +Qσp1q +(1 +1 , Qσp2q . . . , Qσpnq +)1 +n +(4.36) +` p´1q +řn +k“2|BIk| ␣ +tQ2, . . . , Qnu1 +n´1 , Q1 +(1 +2 +(4.37) +` +␣ +tQ1u1 +1 , Qσp2q . . . , Qσpnq +(1 +n . +(4.38) +In lines (4.34)-(4.35)-(4.36) above, we have t¨ ¨ ¨ u1 “ t¨ ¨ ¨ u for all the terms involved. This is not the +case for (4.37)-(4.38). Indeed: +! +tBI2, ¨ ¨ ¨ , BInu +1 +n´1 , Bi ^ Bj +)1 +2 “ +␣ +tBI2, ¨ ¨ ¨ , BInun´1 , Bi ^ Bj +( +2 +´ +ÿ +i2PI2,...,inPIn +ϵpi2, . . . , inqρpBi ^ Bjqrϕi2¨¨¨ins BIi2 +2 ‚¨¨¨‚Iin +n + +CHAPTER 4. MAIN RESULTS OF PART I +76 +and +␣ +tBi ^ Bju1 +1 , BI2 . . . , BIn +(1 +n “ p´1q +řn +k“2|BIk|`1 ` +ϕi tBj, BI2 . . . , BInu1 +n ´ ϕj tBi, BI2 . . . , BInu1 +n +˘ +“ ´p´1q +řn +k“2|BIk| +ÿ +i2PI2,...,inPIn +ϵpi2, . . . , inqιϕpBi ^ Bjqrϕi2¨¨¨ins BIi2 +2 ‚¨¨¨‚Iin +n +“ p´1q +řn +k“2|BIk| +ÿ +i2PI2,...,inPIn +ϵpi2, . . . , inqρpBi ^ Bjqrϕi2¨¨¨ins BIi2 +2 ‚¨¨¨‚Iin +n +since ρ “ ´ιϕ. Hence, the quantities in lines (4.38) and (4.37) add up, when we re-write them in +terms of the new brackets t¨ ¨ ¨ uk, to yield precisely the higher Jacobi identity for this new bracket. It +is then not difficult to see this is still the case if there is more than one bivector field, by using many +times the same computations. +4.4.7 +Restriction to ϕ “ 0 of vector fields annihilating ϕ +We keep the convention and notations of the previous section. Let us consider the restriction i˚ +W Fϕ of +the Lie-Rinehart algebra Fϕ to the zero-locus W of a Koszul polynomial ϕ. Since all vector fields in +Fϕ are tangent to W, this restriction is now a Lie-Rinehart algebra over OW “ +O +Oϕ, see Section 3.2.1 +(1). +Proposition 4.4.19. Let ϕ be a Koszul Polynomial. The restriction of the universal Lie 8-algebroid +of Proposition 4.4.17 to the zero-locus W of ϕ is a universal Lie 8-algebroid of the Lie-Rinehart +algebra i˚ +W Fϕ. +Since the image of its anchor map are vector fields tangent to W, it is clear that the universal Lie +8-algebroid of Proposition 4.4.17 restricts to W. To prove Proposition 4.4.19, it suffices to check that +the restriction i˚ +W XpV q to W of the Koszul complex is still exact, except in degree 0. +Lemma 4.4.20. The restriction to the zero locus W of ϕ of the Koszul complex (4.28), namely the +complex, +. . . +ιϕ +ÝÑ i˚ +W X3pV q +ιϕ +ÝÑ i˚ +W X2pV q +ιϕ +ÝÑ i˚ +W F +is a free resolution of i˚ +W F in the category of OW -modules. +Proof. Let k ě 2 be an integer. For a given P P i˚ +W XkpV q, the relation ιϕP “ 0 means that for any +˜P P XkpV q extending P, there exists U P Xk´1pV q such that ιϕ ˜P “ ϕU. By exactness of the Koszul +complex (4.28), one has, U “ ιϕ ˜Q for some ˜Q P XkpV q. Hence, ˜P ´ ϕ ˜Q is an extension of the bivector +field P such that ιϕp ˜P ´ ϕ ˜Qq “ 0. Using one more time exactness of the Koszul complex (4.28), we +construct ˜R P Xk`1pV q such that ˜P “ ϕQ ` ιϕ ˜R . Thus, P “ i˚ +W ˜P “ ιϕi˚ +W ˜R “ ιϕR. +4.4.8 +Vector fields vanishing on subsets of a vector space +Let O be the algebra of smooth or holomorphic or polynomial or formal functions on Kd, and I Ă O be +an ideal. Then IDerpOq, i.e. vector fields of the form: řd +i“1 fi B +Bxi , with f1, . . . , fd P I, is a Lie-Rinehart +algebra (It is also a singular foliation). +Remark 4.4.21. Geometrically, when I corresponds to functions vanishing on a sub-variety N Ă Kn, +IDerpOq must be interpreted as vector fields vanishing along N. + +CHAPTER 4. MAIN RESULTS OF PART I +77 +Let us describe a Lie 8-algebroid that terminates at IDerpOq, then discuss when it is universal. +Let pϕiqiPI be generators of I. Consider the free graded algebra K “ OrpµiqiPIs generated by variables +pµiqiPI of degree ´1. The degree ´1 derivation B :“ ř +iPI ϕi B +Bµi squares to zero. The O-module K´j +of elements degree j in K‚ is made of all sums ř +i1,...,ijPI fi1...ijµi1 ¨ ¨ ¨ µij with fi1...ij P O. Consider the +complex of free O-modules +¨ ¨ ¨ BbOid +ÝÑ K´2 bO DerpOq BbOid +ÝÑ K´1 bO DerpOq +(4.39) +Proposition 4.4.22. The complex (4.39) comes equipped with a Lie 8-algebroid structure that ter- +minates in IDerpOq through the anchor map given by µi B +Bxj ÞÑ ϕi B +Bxj for all i P I, and j P 1, . . . , d. +Proof. First, one defines a O-linear Poisson-8-algebra structure on the free algebra generated by +pµiqiPI (in degree ´1) and +´ +B +Bxj +¯d +j“1 (in degree 0) and 1 by: +" +µi, +B +Bxj1 +, . . . , +B +Bxjr +*1 +r`1 +:“ +Brϕi +Bxi1 . . . Bxir +(4.40) +all other brackets of generators being equal to 0. Since the brackets of generators take values in O, +and since an n-ary bracket where an element of O appears is zero, this is easily seen to be a Poisson +8-structure. The general formula is +␣ +µI1 bO Bxa1, . . . , µIn bO Bxan +(1 +n :“ +ÿ +j “ 1, . . . , n +ij P Ij +ϵ +Bn´1ϕij +Bxa1 ¨ ¨ ¨ x +Bxaj ¨ ¨ ¨ Bxan +µI1 ¨ ¨ ¨ µij +Ij ¨ ¨ ¨ µInbOBxaj , +(4.41) +where µJ “ µj1 . . . µjs for every list J “ tj1, . . . , jsu, where ϵ is the Koszul sign, and where for a list J +containing j, Jj stands for the list J from which the element j is crossed out, as in Equation (4.31). +The O-module generated by µi1 ¨ ¨ ¨ µik bO Bxa , i.e. the complex (4.39) is easily seen to be stable +under the brackets t¨ ¨ ¨ u1 +k for all k ě 1, so that we can define on K bO DerpOq a sequence of brackets +pℓk “ t¨ ¨ ¨ ukqkě1 by letting them coincide with the previous brackets on the generators, i.e. t¨ ¨ ¨ un +is given by Equation (4.41) for all n ě 1. The brackets are then extended by derivation, O-linearity +or Leibniz identity with respect to the given anchor map, depending on the degree. In particular, +t¨ ¨ ¨ u1 +k “ t¨ ¨ ¨ uk for k ě 2. For k “ 1, t¨ ¨ ¨ u1 +1 “ t¨ ¨ ¨ u1 on ‘iě2K´i bO DerpOq. For k “ 2, we still have +t¨ ¨ ¨ u1 +2 “ t¨ ¨ ¨ u2 on ‘i,jě2K´i d K´j bO DerpOq. +Let us verify that all required axioms are satisfied. For n “ 2, Equation (4.41) specializes to: +ℓ2pµi bO Bxa, µj bO Bxbq “ Bϕj +Bxa +µi bO Bxb ´ Bϕi +Bxb +µj bO Bxa +which proves that the anchor map is a morphism when compared with the relation: +rϕiBxa , ϕjBxbs “ Bϕj +Bxa +ϕi Bxb ´ Bϕi +Bxb +ϕj Bxa. +The higher Jacobi identities are checked on generators as follows: +1. When there are no degree ´1 generators, it follows from the higher Jacobi identities of the +Poisson 8-structure (4.40) and the O-multilinearity of all Lie 8-algebroid brackets involved. + +CHAPTER 4. MAIN RESULTS OF PART I +78 +2. When generators of degree ´1 are involved, the higher Jacobi identities are obtained by doing +the same procedure as in the proof of Proposition 4.4.18, that is, we first consider the higher +Jacobi identities for the Poisson 8-structure (4.40), and we put aside the terms where t¨u1 is +applied to these degree ´1 generators. We then check that the latter terms are exactly the terms +coming from an anchor map when the 2-ary bracket is applied to generators of degree ´1 and +the pn ´ 1q-ary brackets of the remaining generators. +4.4.9 +Vector fields vanishing on a complete intersection +Proposition 4.4.23. Let W Ă Cn be an affine variety defined by a regular sequence ϕ1, . . . , ϕk P O. +Then, the Lie 8-algebroid described in Proposition 4.4.22 is the universal Lie 8-algebroid of the +singular foliation of vector fields vanishing along W. +Proof. In the notation of the proof of Proposition 4.4.22, K‚ equipped with the derivation B “ +řk +i“1 ϕi +B +Bµk is a free O-resolution of the ideal IW of functions vanishing along W, since ϕ1, . . . , ϕk +is a regular sequence. Since XpCdq is a flat O-module, the sequence +¨ ¨ ¨ +BbOid +� K´2 bO XpCdq +BbOid +� K´1 bO XpCdq +BbOid +� IW XpCdq. +(4.42) +is a free O-resolution of the singular foliation IW XpCdq. The Lie 8-algebroid structure of Proposition +4.4.22 is therefore universal. +Example 4.4.24. As a special case of the Proposition 4.4.23, let us consider a complete intersection +defined by one function, i.e. an affine variety W whose ideal xϕy is generated by a regular polynomial +ϕ P CrX1, . . . , Xds. One has a free resolution of the space of vector fields vanishing on W given as +follows: +0 +� Oµ bO XpCdq +ϕ B +Bµ bOid +� IW XpCd q, +where µ is a degree ´1 variable, so that µ2 “ 0. The universal Lie 8-algebroid structure over that +resolution is given on the set of generators by : +tµ bO Bxa, µ bO Bxbu2 :“ Bϕ +Bxa +µ bO Bxb ´ Bϕ +Bxb +µ bO Bxa +and t¨ ¨ ¨ uk :“ 0 for every k ě 3. It is a Lie algebroid structure. Notice that this construction could +be also be recovered using Section 4.4.2. + +CHAPTER 4. MAIN RESULTS OF PART I +79 +Conclusion: +This chapter described the 1-1 correspondence "Lie-Rinehart algebras ÐÑ Lie 8-algebroids +on acyclic complexes". It extends greatly [LLS20] for singular foliations. The functor "ÐÝ" +consists in the 1-truncation of the Lie 8-agebroid structure. The converse functor consists in +taking any free resolution, and constructing the brackets by recursion. +We prove that it is unique by proving it is universal. Notice that we need the "complicated" +notion of homotopy given in Definition 2.3.14. +Last, some examples of [LLS20] are conceptually understood, and new examples are given. +Some algebraic constructions (blow-up, localization, germs, quotient) are also given. +Obstruction classes to the existence of a Lie algebroid with a surjective morphism onto the +Lie-Rinehart algebra are also described. +We never assume finite rank here! + +Part II +Geometric Applications +80 + +CHAPTER 5 +Universal Lie 8-algebroids of affine varieties +In this chapter, we apply the results of Chapter 4 to answer some elementary but open questions +that have to do with algebraic geometry, such as the interaction between the singularities of an affine +variety and its Lie algebra of vector fields. +Notice that the Theorem 2.1 of Chapter 4.2 only says that it is possible to associate a Lie 8- +algebroid structure to an affine variety by considering the Lie-Rinehart algebra made of its vector +fields. But the construction can be extremely complicated, see e.g. Section 5.3. We only have an +existence theorem. This leads to the natural question: +Question. How is the geometry of an affine variety related to its universal Lie 8-algebroid? +For instance, in view of the construction of Section 3.10. +It is also relevant to ask about the +effect of blow-ups on this construction. We will see in Example 4.4.10 that blow-ups may change a +universal Lie 8-algebroid to a Lie algebroid. But this example does not tell us really how the higher +brackets disappear under the effect of blow-ups. One of the important question we may ask is about +the description of “the big theorem” of Hironaka [Já07] in terms of the universal Lie 8-algebroids +obtained at each step while resolving singularities. This question remains open, but there are several +other problems about which we are able to make some progresses. Those are quite modest, but, at +least, we want to have the question clarified. Application to "blowup-up" will be discusses in Section +7.2. +5.1 +Background on affine varieties and some constructions +We recall definitions and some main properties of the notion of affine variety in order to fix notations. +Our main references for this chapter are [Har77, Eis95, LB15]. +In this chapter we will sometimes see Cd as the d-dimensional affine space which is commonly +denoted by Ad +C, forget about its vector space structure, but here we will not make any notational +distinctions. +The latter is equipped with the Zariski topology that is, the topology whose closed +subsets are the zero set of some ideal I Ď Crx1, . . . , xds “: O, i.e. +which are of the form ta “ +81 + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +82 +pa1, . . . , adq P Cd | fpaq “ 0, @f P Iu. One can check that these subsets indeed define a topology on Cd +[KES00, Har77, LB15]. +Definition 5.1.1. An affine variety W Ď Cd is a the zero locus of an ideal I Ď O, i.e., W :“ ZpIq :“ +ta “ pa1, . . . , adq P Cd | fpaq “ 0, @f P Iu. It admits a topology, induced by the Zariski topology in +Cd. +Remark 5.1.2. In Definition 5.1.1, we do not exclude irreducible varieties, e.g. W “ tpx, yq P C2 | +xy “ 0u is an affine variety. +The following facts and remarks are important. For W Ď Cd an affine variety in the notation of +Definition 5.1.1, +1. we denote by IW the vanishing ideal of W, namely, +IW “ tf P O | fpxq “ 0, @ x P Wu. +In general, we have IW ‰ I because the vanishing ideal of the affine variety W “ tx2 “ 0u is +the ideal IW “ xxy ‰ xx2y. Notice that IW can be defined for any arbitrary subset W Ă Cd. +It is easy to check that +(a) for S Ď T Ď O, one has ZpSq Ě ZpTq, +(b) for U Ď V Ď Cd, one has IU Ě IV . +2. The ideal IW is larger than I. Hilbert’s Nullstellensatz theorem (e.g. see Theorem 1.6 of [Eis95]) +claims that IW “ +? +I :“ tf P O | fN P I, for some N P Nu. +3. We have, W “ ZpIW q: if x P W, then by definition of IW , one has fpxq “ 0 for all f P IW . +Whence, W Ď ZpIW q. Conversely, it is clear that I Ď IW . This fact proves the other inclusion. +4. Also, by Noetheriality of the polynomial ring O, the ideal IW Ă O is generated by a finite +number of generators. In the sequel, we shall define an affine variety W as the zero locus of an +ideal generated by a finite set of polynomials ϕ1, . . . , ϕr P O. +5. The Zariski closure V of a subset V Ď Cd is equal to ZpIV q: by definition of IV , one has +V Ď ZpIV q. Conversely, V Ď V “ ZpIq for some ideal I Ď O. By item 1.(b), IV Ď IV . In +particular, I Ď IV . By item 1.(a), this implies ZpIV q Ď ZpIq “ V . +Definition 5.1.3. Let W Ď Cd be an affine variety. +1. A function F : W Ñ C is said to be a polynomial if Fpxq “ fpxq, @x P W, for some element +f P O. The set CrWs of polynomial functions on W is, under the restriction map f P O ÞÑ f|W , +isomorphic the quotient O{IW “: OW , called the coordinate ring of W. +2. Elements of the Lie algebra of C-linear derivations, DerpOW q “: XpWq, of OW are called vector +fields on W. +Remark 5.1.4. Notice that + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +83 +1. the coordinate ring OW of an affine variety besides being a ring is also a vector space over C, +hence it is a C-algebra. This algebra is generated by the images ¯x1, . . . , ¯xd in OW through the +projection map of the coordinate functions x1, . . . , xd P O. +For each a P W, the kernel of the evaluation map eva : OW Ñ C, F ÞÑ Fpaq, is the maximal +ideal ma :“ kerpevaq made of all polynomial functions on W that vanish at a. +2. W “ Cd is an affine variety with IW “ t0u, and OW “ O. +3. W “ tau Ď Cd is an affine variety with IW “ px1 ´ x1paq, . . . , xd ´ xdpaqq the maximal ideal of +a, and OW “ C. +4. For an affine variety W Ă Cd with corresponding ideal IW , we have +DerpOW q » tX P XpCdq | XrIW s Ă IW u +IW XpCdq +. +Lemma 5.1.5. For every affine variety W Ď Cd, the ring OW is Noetherian. +Proof. Let I be an ideal of OW . Denote by p: O Ñ O{IW be the quotient map. Then p´1pIq is also +an ideal of O. By Noetheriality of O, p´1pIq is finitely generated. In particular, I “ ppp´1pIqq is +finitely generated. This shows that OW is Noetherian. +Germs +Here we mention the notion of local rings. We refer the reader to [Cha14, Hid89, Eis95] for more details. +Let W Ď CN be an affine variety and OW its coordinates ring. We recall for U Ď W an open +subset, a function f : U ÝÑ C is said to be regular at a P U if there exists g, h P OW with hpaq ‰ 0 +such that f “ g +h in a neighborhood of a, namely there exists an open set V Ă U that contains a +such that f|V “ g +h|V . A function germ at a point a P W is an equivalence class pfqa of pairs pU, fq +with a P U Ă W an open subset containing a, and f : U ÝÑ C is regular at a, under the relation +equivalence: pU, fq „ pV, gq if f|W “ g|W on an open subset W Ď U XV . The set of equivalence classes +of the above equivalence relation inherits naturally an associative C-algebra, that is called germs of +regular functions at a and is denoted by OW,a. Also, a function germ pfqa at a P W has a well-defined +value at a, given by the image of any representative pU, fq at a, namely pfqapaq :“ fpaq. Since the +map +OW,a ÝÑ pOW qmW,a, pU, fq ÞÑ f|U “ g +h +with g, h P OW and h does not vanish on U, is a bijection. One has, OW,a » pOW qmW,a [Har77]. Here +mW,a “ tf P OW | fpaq “ 0u and pOW qmW,a is the localization w.r.t the complement of mW,a. It is +important to notice that OW,a is a local ring. We denote the unique maximal ideal of OW,a again by +mW,a. Also, It is worth it to notice that OW,a isomorphic to the quotient Oa{Ia of the local ring Oa +of Cd at a by the ideal Ia which is spanned by the ideal IW in Oa. +The Zariski tangent space +Let a P W Ď Cd a point of an affine variety W. There are several equivalent descriptions of the +tangent space of W variety at the point a. Here we define it as pointwise derivations of the local ring + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +84 +OW,a, see [Har77, vIR94] or Appendix B.2 of [LGPV13], for more details on this topic. +A pointwise derivation of OW at a is a C-linear map +δa : OW,a Ñ C +satisfying the following Leibniz identity, δapFGq “ δapFqGpaq ` FpaqδapGq. In particular, if a regular +function f is constant in a neighborhood of a then its germ pfqa at a satisfies δappfqaq “ 0, since +δap1 ¨ 1q “ δap1q ¨ 1 ` 1 ¨ δap1q “ 2δap1q +δap1q “ 0. +It is not hard to check that the set of all pointwise derivations of OW at a is a C-vector space. +Remark 5.1.6. Assume now that W “ Cd. Denote by pe1, . . . , edq the canonical basis of Cd. For +every i P t1, . . . , du +peiqa : OW,a Ñ C, pfqa ÞÑ lim +tÑ0 +fpa ` teiq ´ fpaq +t +“: Bf +Bxi +paq, +is a well-defined pointwise derivation of O at a. One can show that (see e.g [LGPV13], Appendix B.2) +any pointwise derivation δa of O at a P Cd has the form +δa “ +dÿ +i“1 +δappxiqaq peiqa, +(5.1) +i.e., +δapfqa “ +dÿ +i“1 +Bf +Bxi +paq δapxiqa. +(5.2) +Hence, pointwise derivations peiqa, i “ 1, . . . , d form a basis for the vector space of pointwise derivation +of O at a. +Definition 5.1.7. The Zariski tangent space TaW of W Ď Cd at a P W is the vector space of all +pointwise derivations of OW at a. +Proposition 5.1.8. [vIR94] For a P W, one has TaW » +´ +mW,a{m2 +W,a +¯˚ +. +The vector space mW,a{m2 +W,a is called the cotangent space to W at a. +Remark 5.1.9. Notice that, +1. by Remark 5.1.6, we have TaCd » Cd. +2. the tangent space TaW of W Ď Cd at a P W can be seen as pointwise derivations δa of O at a +of the form (5.1) such that δappfqaq “ 0 for all f P IW . From this point of view, one has +TaW » +# +pv1, . . . , vdq P Cd +ˇˇˇˇˇ +dÿ +i“1 +vi +Bf +Bxi +paq “ 0, @f P IW ++ +. + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +85 +Singularities of an affine variety W +In this section we recall some definitions and some facts on singularities on affine varieties and fix +some notations. We refer the reader to [Har77, vIR94] for the full theory. +There are various equivalent ways to define the dimension of an affine variety W, we refer the +reader to Page 4 of [Har77] also to the Chapter 11 of [Joe92] for more details. The dimension dim W +of W is defined to be the maximal length d of the chains W0 Ă W1 Ă ¨ ¨ ¨ Ă Wd of distinct nonempty +irreducible sub-varieties of W. Notice that a chain of irreducible sub-varieties corresponds to a chain +of prime ideals in OW , by Noetheriality it must be of finite length. +Definition 5.1.10. A point a P W is said to be regular if dim TaW “ dim W. Otherwise, we say that +a is singular. The set of regular points of W is denoted by Wreg, and the singular ones by Wsing. +We say that W is regular if Wsing “ H. +Proposition 5.1.11. [Har77, vIR94]We have the following +1. For every a P W, dim TaW ě dim W +2. There is an open dense open subset of W such that the map a ÞÑ dim TaW is constant. In +particular, +• regular points of W form an open dense subset of W, +• and singular points a (closed) proper sub-variety of W. +Remark 5.1.12. In particular, a P W is a singular point of W if only if dim TaW ą dim W. That is, +Wsing “ ta P W | dim TaW ą dim Wu. +Local coordinates at a point +Let a P W Ď Cd. We recall that (see e.g [vIR94]) that a family of elements t1, . . . , tr P OW,a are called +local coordinates of W at a, if they vanish at a (i.e. ti P mW,a for i “ 1, . . . , r), and if the classes of +t1, . . . , tr P mW,a{m2 +W,a form a basis. +Example 5.1.13. If a “ pa1, . . . , adq P Cd, then x1 ´ a1, . . . , xd ´ ad are local coordinates of Cd at a. +Remark 5.1.14. Notice that: +Local coordinates at a P Cd generate the maximal ideal ma of Oa. Indeed, let t1, . . . , td P Oa be +local coordinates at a. By applying Nakayama Lemma (B.2.11) to R “ Oa Ě ma and V “ ma: +the basis t1, . . . , td P ma{m2 +a lifts to a (minimal) generating set t1, . . . , td for ma. +The following proposition explains how an affine variety looks around a regular point (see [Hau14] +Proposition 3.5, also [dJP00]). +Proposition 5.1.15. Let W Ď Cd be affine variety of codimension k, (i.e., k “ d ´ dim W). Then, +W is regular at a point a P W if and only if there exist local coordinates y1, . . . , yd of Cd at a such +that W is locally of the form +y1 “ ¨ ¨ ¨ “ yk “ 0. + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +86 +5.1.1 +Three main constructions +Consider an affine variety W which is given by an ideal IW Ă O, with O the algebra of polynomials +in d-variables, and OW “ O{IW the algebra of functions on W. There are three natural Lie-Rinehart +algebras associated to W: +1. The OW -module XpWq of vector fields on W (i.e. derivations of OW ) is a Lie-Rinehart algebra +over OW ; its anchor map is the identity. +2. The O-module XW pCdq of vector fields on Cd tangent to W (i.e. derivations of O preserving +IW ) is a Lie-Rinehart algebra with respect to the O-module structure; its anchor map is the +inclusion XW pCdq ãÑ XpCdq. +3. The O-module IW XpCdq of vector fields on Cd vanishing at every point of W (i.e. IW -valued +derivations of O) is a Lie-Rinehart algebra with respect to the O-module structure; its anchor +map is again the inclusion IW XpCdq ãÑ XpCdq. +These three Lie-Rinehart algebras are related: +1. There is an inclusion IW XpCdq Ă XW pCdq +2. the restriction of XW pCdq to W coincides with XpWq. Let us justifies this. Every vector field +on W extends to Cd: to see that, let δ P XpWq, we have δpxi ` IW q “ fi ` IW for some +fi P Crx1, . . . , xds, i “ 1, . . . , d. We define the vector field +rδ :“ +dÿ +i“1 +fi +B +Bxi +on Cd. The vector field rδ restricts to δ on W, since for every f P Crx1, . . . , xns, +rδpfq ` IW “ δpf ` IW q. +In particular, rδpIW q Ă IW . +Note that the Lie-Rinehart algebras XW pCdq, IW XpCdq Ă XpCdq are finitely generated as O-modules, +since O is Noetherian (Proposition B.2.7). Whence, these Lie-Rinehart algebras are singular foliations +on the complex manifold Cd in the sense of Example 3.2.4. +Remark 5.1.16. What happens if we take a look at the evaluation map at some point a P Cd? Any +vector field X “ +dÿ +i“1 +Xrxis B +Bxi +P XW pCdq tangent to W induces a pointwise derivation of OW at a P W +as follows: +1. We extend X by localization at the maximal ideal ma to a derivation pXq P DerpOW,aq. +2. We define X|appfqaq :“ +dÿ +i“1 +Xrxispaqpeiqappfqaq P TaW. +X|a is well-defined, since XrIW s Ă IW for all f P IW , in particular X|appfqaq “ 0 for all f P IW . +In fact, we do not need to localize X to define X|a :“ pXrx1spaq, . . . , Xrxdspaqq P TaW ãÑ Cd. + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +87 +Lemma 5.1.17. The image of the map +Eva : XW pCdq Ñ Cd, X ÞÑ X|a +(5.3) +is denoted by TaXW pCdq +1. if a P W, then TaXW pCdq Ď TaW. +2. If a R W, then TaXW pCdq “ Cd. +3. If W is a complete intersection (i.e. IW “ pϕ1, . . . , ϕrq and dim W “ d ´ r), then TaXW pCdq “ +TaW for all a P Wreg. +4. More generally, for an arbitrary affine variety W, TaXW pCdq “ TaW for all a P Wreg. +Proof. Item 1. is given by the construction in item 2. of Remark 5.1.16. Let us show item 2: let +pv1, . . . , vdq P Cd. For a R W, there exists ϕ P IW such that ϕpaq ‰ 0. The vector field +X “ +ϕ +ϕpaq +dÿ +i“1 +vi +B +Bxi +belongs to IW XpCdq Ă XW pCdq “ Cd and Xpaq “ pv1, . . . , vdq. +Now we prove item 3. Let pv1, . . . , vdq P TaW “ kerpJpaqq, with J :“ +´ +Bϕi +Bxj +¯ +i,j. By assumption, we +have rkpJpaqq “ r. Thus, J admits pr, rq-minor µ such that µpaq ‰ 0. We can assume that µ is the +determinant of the first r-columns of J. Consider the vector fields +Hj :“ +������������ +B +Bx1 +¨ ¨ ¨ +B +Bxr +B +Bxj +Bϕ1 +Bx1 +¨ ¨ ¨ +Bϕ1 +Bxr +Bϕ1 +Bxj +... +... +... +Bϕr +Bx1 +¨ ¨ ¨ +Bϕr +Bxr +Bϕr +Bxj +������������ +, for j P tr ` 1, . . . , du, +understood as the cofactor expansion along the first row. Since for each i P t1, . . . , ru, Hjrϕis has two +repetitive lines, therefore it vanishes. Therefore Hj’s are tangent to W. We claim that the following +vector fields does the job, namely +X “ p´1qr +dÿ +j“r`1 +vj +µpaqHj. +(5.4) +Indeed, if we denote by µ1, . . . , µr the minors associated to the partials +B +Bx1 , . . . , +B +Bxr , respectively, then +for every j the decomposition of Hj reads +Hj “ +rÿ +i“1 +p´1qi`1µi +B +Bxi +` p´1qrµ B +Bxj +. +Hence, +X “ +dÿ +j“r`1 +vj +µpaqµ B +Bxj +` p´1qr +µpaq +˜ rÿ +i“1 +dÿ +j“1 +p´1qi`1vjµi +B +Bxi +¸ + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +88 +But, by developing the minors µipaq’s along their last columns, it takes the form, +µipaq “ ˘Bϕ1 +Bxj +paqC1 ˘ ¨ ¨ ¨ ˘ Bϕr´1 +Bxj +paqCr´1. +Notice that the determinants C1, . . . , Cr´1 are the same for each Hj. Thus, +dÿ +j“1 +vjµipaq “ +r´1 +ÿ +s“1 +˘ +˜ dÿ +j“1 +vj +Bϕs +Bxj +paq +¸ +looooooooomooooooooon +“0 +Cs +“ 0. +Item 4. is obtained as follows: the local ring at a is by definition the localization Oa of Crx1 . . . , xds +with respect to the multiplicative set of all polynomials that do not vanish at a. By Proposition 5.1.15 +a P W is a regular point if and only if there exists "local coordinates" y1, . . . , yd P Oa such that W is +locally of the form +y1 “ ¨ ¨ ¨ “ yk “ 0, +i.e. the localization of IW is generated by these variables. Hence, the tangent space at m is the vector +space, spant B +Byi |m, i ě k ` 1u. Therefore, for v P TaW the local vector field +X “ +dim W +ÿ +i“1 +vi +B +Byk`i +maps Oa to Oa, in particular it maps O to Oa and we have XrIW s Ă pIW qma. +Therefore, for +every, i P t1, . . . , du there exists a polynomial function gi that does not vanish at a such that +giY rxis P Crx1, . . . , xds. Hence, the vector field ˆX “ +g1¨¨¨gr +g1paq¨¨¨grpaqX is tangent to W satisfies ˆXpaq “ v +and ˆXrIW s Ă IW . +This concludes the proof. +Remark 5.1.18. We may not have equality in item 1. in Lemma 5.1.17. To see this, consider the +cups, +W “ tpx, yq | C2 | x3 ´ y2 “ 0u. +It is clear that the tangent space T0W of W at 0 P W is the whole space C2. But the vector fields in +XW pC2q vanish at zero, since it is spanned as a Crx, ys-module by the Hamiltonian 2y B +Bx ` 3x2 B +By and +the weighted Euler vector field 2x B +Bx ` 3y B +By (see Proposition 5.3.1). +The following lemma shows that the vector fields that are tangent to W are also tangent to every +strata of the stratification that consists of by taking the singular locus Wsing of the singular locus of +W then the singular locus pWsingqsing of the singular locus Wsing and so on.... We obtain a sequence +of inclusions of the form +W Ą pWsingqsing +looooomooooon +“:W1 +Ą ppWsingqsingqsing +looooooooomooooooooon +“:W2 +Ą ¨ ¨ ¨ . +(5.5) +Lemma 5.1.19. We have the following inclusions +XW pCdq Ď XW1pCdq Ď ¨ ¨ ¨ Ď XWipCdq Ď ¨ ¨ ¨ +(5.6) + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +89 +Proof. Let us prove that if X P XpCdq is such that XrIW s Ă IW then XrIWsings Ă IWsing, where +IWsing is the ideal of functions on the singular part of W. Since IWsing is obtained by considering the +minors of order k “ d ´ dim W of k elements chosen into the generators ϕ1, . . . , ϕr. That is, Wsing is +given the ideal +A +ϕ1, ¨ ¨ ¨ ϕr, Prϕi1, ¨ ¨ ¨ , ϕiks, P P XkpCdq, for integers 1 ď i1 ă ¨ ¨ ¨ ă ik ď r +E +(5.7) +Let us explain why the vector fields that tangent to W are also tangent to its singular locus. +For a vector field X P XW pCdq one has by Formula (4.27) that, +X rPrϕi1, ¨ ¨ ¨ , ϕikss “ pLXPqrϕi1, ¨ ¨ ¨ , ϕiks ` +kÿ +j“1 +Prϕi1, . . . , Xrϕijs, . . . , ϕiks. +(5.8) +Notice that pLXPqrϕi1, ¨ ¨ ¨ , ϕiks P Ising since pLXPq P XkpCdq. On the other hand, for every j there +exists polynomial functions f1, . . . , fr such that Xrϕijs “ řr +i“1 flϕl. Since P is a multi-derivation, +one has, +Prϕi1, . . . , Xrϕijs, . . . , ϕiks “ +rÿ +l“1 +ϕlPrϕi1, . . . , fl, . . . , ϕiks` +rÿ +i“1 +flPrϕi1, . . . , ϕl, . . . , ϕiks +It is now clear that the RHS of the equation (5.8) is in the ideal Ising. The proof goes by recursion. +Here is a direct consequence of Lemma 5.1.19. +Theorem 5.1.20. Every vector field X P XpWq is tangent to the stratification (5.5), i.e. X P XpWiq +for each i ě 1. +The coming example shows that the inclusions (5.6) may be strict. This Example can also be +found in the problem list [LLG22] of the lecture on singular foliations, Poisson 2022 [LGLR22]. +Example 5.1.21. Let W “ tpx, y, zq P C3 | xypx ` yqpx ` yzq “ 0u Ă C3. + +Staatq +Stnoto +Lin 9 +Saotas +ARCHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +90 +The straight line x “ y “ 0 is a strata of the previous affine variety W. Any vector field tangent to +W is tangent to this straight line. Let us show that it has to vanish at every point of this straight line. +If not, its flow at time t would map a point p0, 0, z0q to a point p0, 0, z1q with z1 ‰ z0. Its differential +then induce a linear automorphism of the normal bundle of that straight line that has to preserve the +straight lines x “ 0, y “ 0, x ` y “ 0. Since a linear endomorphism of C2 preserving three straight +lines has to be a multiple of the identity map, this differential cannot map the straight line x ` z0y to +the straight line x ` z1y. +On their universal Lie 8-algebroids +Let W be an affine variety. The following is a direct consequence of the results of Chapter 4. +Proposition 5.1.22. Let W be an affine variety as above. +1. XpWq admits a universal Lie 8-algebroid made of free OW -modules of finite rank. +2. IW XpCdq and XW pCdq admit universal Lie 8-algebroids made of finitely many free O-modules +of finite rank. +Proof. Classical theorems of commutative algebras allows equipping the three Lie-Rinehart algebras +above with resolutions of a certain type: +1. Since O is a Noetherian regular ring, XW pCdq and IW XpCdq admit free resolutions by finitely +generated O-modules. By Hilbert Syzygy Theorem B.2.8, those can be chosen to be of finite +length. +2. Since OW is a Noetherian ring, XpWq admits a free resolution by finitely generated OW -modules +(by Proposition B.2.7). +In view of Theorem 4.2.1, these three resolutions do admit Lie 8-algebroid structures over their respec- +tive algebras, and those are universal Lie 8-algebroids. We denote by UOW pXpWqq, UOpXW pCdqq and +UOpIW XpCdqq the universal Lie 8-algebroids associated to the three Lie-Rinehart algebras above. +Remark 5.1.23. There exist natural Lie 8-algebroid morphisms between these structures: +1. Since IW XpCdq Ă XW pCdq, Theorem 4.2.4 implies the existence of a unique up to homotopy +Lie 8-algebroid morphism Ψ: UOpIW XpCdqq ÝÑ UOpXW pCdqq. In view of Proposition 4.2.6, +the morphism Ψ may be represented by the inclusion map for well-chosen representation of +UOpIW XpCdqq and UOpXW pCdqq. +2. The anchor map of UOpXW pCdqq is tangent to W, hence the restriction i˚ +W UOpXW pCdqq to W +of UOpXW pCdqq exists, and is a Lie 8-algebroid over XpWq. In general, it does not need to +be a universal one, but Theorem 4.2.4 implies the existence of a unique up to homotopy Lie +8-algebroid morphism: +Φ: i˚ +W UOpXW pCdqq ÝÑ UOW pXpWqq. +Notice that this morphism is in general not a Lie 8-quasi-isomorphism. + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +91 +5.2 +Universal Lie 8-algebroid of an affine variety +We give the following definition, +Definition 5.2.1. A Lie 8-algebroid of an affine variety W is the homotopy class of Lie-8-algebroid +associated to the Lie-Rinehart algebra XpWq over OW . +Example 5.2.2 (Vector fields on hyperelliptic curves). We follow the notations of [BF18]. Let us +consider on C2 the hyperelliptic curve H given by the equation y2 “ 2hpxq, where h is a monic +polynomial of odd degree 2ν ` 1 ě 3. +Let OH “ +Crx,ys +xy2´2hpxqy be the coordinate ring of H. +Let +XpHq “ DerpOHq be the Lie-Rinehart algebra of derivations of OH, i.e. of vector fields on H. As an +OH-module, XpHq is the submodule +" +f B +Bx ` g B +By | f, g P OH, yg ´ h1pxqf “ 0 +* +Ă OH +B +Bx ‘ OH +B +By. +The curve H is non-singular if and only if gcdphpxq, h1pxqq = 1. In that case, the Lie algebra of vector +fields XpHq is a free OH-module of rank 1 generated by the vector field X “ y B +Bx `h1pxq B +By. A universal +Lie 8-algebroid is given by the Lie-Rinehart algebra E´1 “ OHX Ă OH B +Bx ‘ OH B +By. In the singular +case, i.e., gcdphpxq, h1pxqq “ dpxq ‰ 1, the OH -module XpHq is not free and has two generators, +X “ y B +Bx ` h1pxq B +By and Y “ 2hpxq +dpxq +B +Bx ` y h1pxq +dpxq +B +By with a relation yX “ dpxqY (see [BF18] for more +details). A free resolution is described as follows: E´1 is the OH-module, generated by two elements +that we denote by τ, µ. The anchor is defined then by +ρpτq “ X, +ρpµq “ Y. +Then we choose E´2 to be the OH-module given by the generator η, and we set E´i “ 0 for i ě 3. +The differential map ℓ1 “ d is chosen to be zero, except on degree ´2 where it is the OH-linear map +d: E´2 ÝÑ E´1 given by +dpηq “ yτ ´ dpxqµ. +Let us now describe a universal Lie 8-algebroid structure: The 2-ary bracket is defined on generators +of E´1 by +tτ, µu2 “ h1pxq +dpxq τ ´ yd1pxq +dpxq µ. +Then we extend this bracket to the whole space E´1 by OH-linearity, skew-symmetric and Leibniz +identity. Notice that tdη, τu2 “ tdη, µu2 “ 0, thus, one can define the 2-bracket by tη, τu2 “ tη, µu2 “ +0. We extend all brackets using Leibniz identity. All k-ary brackets are zero for k ě 3. +Lie 8-algebra of an affine variety at a point: Let UOW pXpWqq “ pE, ℓ‚, ρq be a universal Lie +algebroid of W. Let us choose a P W. As stated in Section 4.4.1, the Lie 8-algebroid structure of W +restricts at a (i.e. goes to the quotient with respect to the maximal ideal ma) if and only if ρrmas Ď ma, +(i.e. ma is a Lie-Rinehart ideal). +Define the OW -submodule +Skerapρq :“ te P E´1 | ρpeqrOs Ď mau Ď E´1. + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +92 +The k-ary bracket ℓk, k ě 1 and ρ restrict to the exact complex +¨ ¨ ¨ +� E´3 +ℓ1 +� E´2 +ℓ1 � Skerapρq +ρ +� DerpOW q . +(5.9) +Let us check that ℓ2 is well-defined: for all e, e1 P Skerapρq, +ρpℓ2pe, e1qqrOs “ rρpeq, ρpe1qspOq, +(since ρ is morphism of brackets) +Ă ma, +(by definition of the commutator r¨ , ¨s). +The latter Lie 8-algebroid goes to quotient to a Lie 8-algebroid +ˆ +p‘iě2 +E´i +maE´i +q ‘ kerapρq, ¯ℓ‚, ρ +˙ +(5.10) +over OW {ma, where kerapρq :“ +Skerapρq +maSkerapρq. Since this quotient is the base field C, we obtain in fact a +Lie 8-algebra. +Remark 5.2.3. In particular, if a P W is an isolated singular point, then E´1 “ Skerapρq. In that +case, UOW pXpWqq is a universal of the Lie-Rinehart algebra Aa “ tδ P XpWq | δrmas Ă mau “ XpWq. +By Section 4.4.1 again, (5.10) is a Lie 8-algebra on TorOW pXpWq, Cq. +Lie 8-algebroids on minimal resolutions +The germ at a of the Lie-Rinehart algebra of vector fields on W is easily checked to coincide with the +Lie-Rinehart algebra of derivations of OW,a. +Here is an immediate consequence of Proposition 4.4.4. +Proposition 5.2.4. Let W be an affine variety. For every a P W, the germ at a of the universal Lie +8-algebroid of W is the universal Lie 8-algebroid of DerpOW,aq. +To describe this structure, let us start with the following Lemma. +Lemma 5.2.5. Let a be a point of an affine variety W. The universal Lie 8-algebroid of DerpOW,aq +can be constructed on a resolution ppEa +´iqiě1, ℓ1, πq, with Ea +´i free OW,a-modules of finite rank for all +i ě 1, which is minimal in the sense that ℓ1pE´i´1q Ă maE´i for all i ě 1. +Proof. Since Noetherian property is stable by localization, the ring OW,a is a Noetherian local ring. +Proposition 8.2 in [May] assures that OW,a bOW DerpOW q admits a free minimal resolution by free +finitely generated OW,a-modules. Since OW,a is a local ring with maximal idea ma, we can assume +that this resolution is minimal. In view of Theorem 4.2.1, there exists a Lie 8-algebroid structure +over this resolution, and the latter is an universal of DerpOW,aq. +By Theorem 4.2.1, a resolution of DerpOW,aq as in Lemma 5.2.5 comes equipped with a universal +Lie 8-algebroid structure for DerpOW,aq. +The quotient with respect to ma is a Lie 8-algebra of +the isolated singular point a with trivial 1-ary bracket. Using Corollary 4.2.13 and its subsequent +discussion, we can prove the next statement. + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +93 +Proposition 5.2.6. For any universal Lie 8-algebroid structure on a resolution of DerpOW,aq as in +Lemma 5.2.5, the quotient with respect to the ideal ma is a representative of the Lie 8-algebra of the +isolated singular point a, with trivial 1-ary bracket, on a graded vector space canonically isomorphic +to TorOW pXW , Cq (C being a OW -module through evaluation at a). +In particular, its 2-ary bracket is a graded Lie bracket on TorOW pXW , Cq which does not depend +on any choice made in the construction, and its 3-ary bracket is a Chevalley-Eilenberg cocycle whose +class is also canonical. +5.3 +Some examples of universal Lie-algebroids over an affine variety +5.3.1 +Vector fields tangent to W: a codimension one example +The zero-set in W Ă V “ Cd of a weight homogeneous polynomial function ϕ P CrX1, ¨ ¨ ¨ , Xds +admitting only an isolated singularity at the origin is one of the simplest possible example of a non- +smooth affine variety W Ă V “ Cd. We give in this section a description of UOpXpWqq. Although our +description is not complete, it will show how complex the universal Lie 8-algebroids may be, even for +simple objects. +First, let us describe XpWq “ DerpOW q. We denote by ω1, . . . , ωd the weights of the variables +x1, . . . , xd and by |ϕ| the weighted degree of ϕ. Recall that, by definition, OW :“ CrX1,¨¨¨ ,Xds +xϕy +. +Proposition 5.3.1. As a OW -module, XpWq :“ DerpOW q is generated by the restrictions to W of +the following vector fields in Cd: +1. the weighted Euler vector field ÝÑ +E :“ řd +i“1 ωixiBxi +2. the dpd ´ 1q{2 vector fields given by: +Xij :“ Bϕ +Bxi +Bxj ´ Bϕ +Bxj +Bxi with 1 ď i ă j ď d. +We start with a lemma. Recall that a homogenous function ϕ with an isolated singularity at 0 is +a Koszul function (see Example 4.4.6), so that the Koszul complex, i.e. the complex of poly-vector +fields on V “ Cd, equipped with ιϕ: +¨ ¨ ¨ +ιϕ +ÝÑ X2pCdq +ιϕ +ÝÑ X1pCdq +ιϕ +ÝÑ O +has no cohomology except in degree 0. We denote it by pX‚, ιϕq. +Lemma 5.3.2. If P P Xi`1pCdq, Q P XipCdq satisfy ιϕpPq “ ϕQ, then there exists R P Xi`2pCdq such +that +P “ 1 +|ϕ| +ÝÑ +E ^ Q ` ιϕpRq +Proof. This follows from the easily checked fact that ιϕpQq “ 0, so that +ιϕ +ˆ 1 +|ϕ| +ÝÑ +E ^ Q +˙ +“ ϕQ. +This implies that +ιϕ +ˆ +P ´ 1 +|ϕ| +ÝÑ +E ^ Q +˙ +“ 0 +and the existence of R now follows from the exactness of the Koszul complex. + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +94 +Proof (of Proposition 5.3.1). Any X P XpWq is the restriction to W of a vector field ˜X in V tangent +to W, i.e. that satisfies ˜Xrϕs “ fϕ for some f P O. By Lemma 5.3.2 the vector field ˜X can be written +as ˜X “ +f +|ϕ| +ÝÑ +E ` ιϕpPq, for some bivector field P P X2pCdq. The restriction to W of the first (resp the +second) term is tangent to W and is of the first (resp. second) type described in Proposition 5.3.1. +This completes the proof. +We now intend to construct a free resolution of XpWq in the category of OW . Let us denote by +^ÝÑ +E : XipV q Ñ Xi`1pV q the map ω ÞÑ ω ^ ÝÑ +E . The map ^ÝÑ +E is a chain map from the restriction i˚ +W X‚ +to W of the Koszul complex, shifted by one to itself. More precisely, ^ÝÑ +E in the diagram below is a +chain map: +¨ ¨ ¨ +ιϕ� i˚ +W X2pV q +ιϕ +� +^ÝÑ +E +� +i˚ +W XpV q +iϕ +� +^ÝÑ +E +� +OW +^ÝÑ +E +� +¨ ¨ ¨ +ιϕ� i˚ +W X3pV q +ιϕ +� i˚ +W X2pV q +ιϕ +� XpWq +Lemma 5.3.3. The chain map ¨ ^ ÝÑ +E is a quasi-isomorphism. +Proof. It is quite clear for i ě 1 since the complex pE1r1s, ιϕq has no cohomology. +Let us check +bijectivity in the last column. Surjectivity follows from Proposition 5.3.1. To check injectivity consider +a function f P O such that +fÝÑ +E “ ιϕpπq ` ϕQ, +(5.11) +for some bivector field π P X2pV q and vector field Q P XpV q. Upon taking ιϕ to both sides, Equation +(5.11) implies, +f|ϕ|ϕ “ ϕιϕpQq. +Hence, we obtain f “ +1 +|ϕ|ιϕpQq. +The lemma 5.3.3 implies that the mapping cone construction provides a free resolution of the +OW -module of vector fields of XpWq on W namely: +˜ +E´i “ i˚ +W Xi´1pV q ‘ i˚ +W Xi`1pV q, i ě 1; D “ +˜ +´ιϕ +0 +´ ^ ÝÑ +E +ιϕ +¸ +, π +¸ +. +(5.12) +In degree ´1 we read E´1 “ OW ‘i˚ +W X2pV q and π is defined on the generators of E´1 as πp1‘0q :“ ÝÑ +E +and +πp0 ‘ Bxi ^ Bxjq :“ Bϕ +Bxi +Bxj ´ Bϕ +Bxj +Bxi, for all, 1 ď i ă j ď d. +Let us now describe some of the k-ary brackets: +Proposition 5.3.4. The Lie-Rinehart algebra XpWq of vector fields on W admits a universal Lie +8-algebroid whose +1. underlying complex is (5.12) (which is a free resolution of XpWq), +2. 1-ary bracket is given by the resolution (5.12). +3. anchor map ρ :“ π. + +CHAPTER 5. UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES +95 +4. In the case where the generator 1‘0 appears together with a bivector field on W, one can define +the 2-ary bracket on elements of degree ´1 which makes the anchor map a morphism as follows, +t1 ‘ 0, 0 ‘ Bxi ^ Bxju2 :“ p|ϕ| ´ 2q0 ‘ Bxi ^ Bxj. +5. The k-ary brackets can be chosen to be the same as in the structure given by Proposition 4.4.17 +on i˚ +W Xi`1pV q for i ě 1. +6. the 3-ary bracket can be chosen to be zero when evaluated at 1 ‘ 0 and with two other elements +of i˚ +W X2pV q. +Corollary 5.3.5. XpWq does not come from a Lie algebroid of rank 1 ` dpd`1q +2 +. +Remark 5.3.6. In general, it is hard to compute the generators of XpWq. +But, there is a case +where we know all generators: it is when W is a complete intersection with isolated singularity at +zero such that IW is generated by weight homogeneous polynomials ϕ1, . . . , ϕr. In that case, XW pCdq +is generated by IW XpCdq, the Euler vector field, and the Hamiltonian vector fields (see [HM93, Sie96]). +The computation of the Lie 8-algebroid remains complicated. +Conclusion: +As a particular case of Section 4, we notice that a Lie 8-algebroid can be associated to any +affine variety. Explicit computations are difficult. Some applications to Mohsen’s resolution of +singularities will be given in Section 7.2 +There are still many open questions on the geometric meaning of this Lie 8-algebroid struc- +ture. Although, we have not answered many questions , we state concepts, lemmas, and counter- +examples that we hope to be able to use in the future. + +CHAPTER 6 +Universal Q-manifolds of a singular foliation +The aim of this chapter is to lay the ground for the subsequent results. More precisely, to explain how +Theorem 4.2.1 extends the results on singular foliations of Sylvain Lavau’s PhD [Lav17] followed by +a referred version by C. Laurent-Gengoux, S. Lavau and T. Strobl in [LLS20]. The results of Section +Chapter 4 extend the latter for arbitrary Lie-Rinehart algebras and also to the infinite case, and it +still holds even when we do not have a geometric resolution. We introduce the notion of longitudinal +vector fields on a NQ-manifold and prove a new result on their cohomology. +This chapter is taken from the textbook [LGLR22] in which I am co-author. We refer the reader +to [Vor10, BP13, LMP20, LLS20] for more details on this topic. +Throughout of this chapter M is a smooth, real analytic or complex manifold and K P tR, Cu. We +denote by O the sheaf of functions on M. +6.1 +Q-manifolds +Let us first define N-graded manifolds. +6.1.1 +Graded manifolds +In words, as the name suggests graded manifolds are for graded vector spaces what manifolds are for +vector spaces, in the sense that roughly speaking manifolds look locally like Rn and graded manifolds +are locally like Rn ˆ V for some graded vector space V . In this chapter we introduce the notion of +graded manifolds, their vector fields, their morphisms, etc. +Definition 6.1.1. A (positively) graded manifold over the base manifold M is a sheaf +E : U ÞÑ EpUq +96 + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +97 +of graded commutative algebras over K such that every m P M admits an open neighborhood U Ă M +on which the sheaf structure takes the form +EpUq “ OU bK SpE˚ +´‚q +for some graded vector space E “ ‘8 +i“1E´i. Sections of the sheaf E are called functions on E. It is +convenient to denote a graded manifold as a pair pM, Eq where E is implicit. +Remark 6.1.2. A function ξ P Ej is a formal sum +ξ “ +`8 +ÿ +i“0 +ξpiq +(6.1) +with ξpiq P E an element of polynomial-degree i and degree j. For degree reasons, the sum must be +finite. +Local coordinates of a graded manifold +Recall that for U Ă M an open set, one has pE˚ +i qU +„ +ÝÑ U ˆ KrkpE˚ +i q for every i ě 1. Hence, the graded +coordinates on the graded manifold pM, Eq is the data made of: +In degree 0: a system of coordinates px1, . . . , xnq of M on U +In degree i ě 1: a local trivialization pξ1 +i , . . . , ξ +rkpE˚ +i q +i +q of E˚ +i on U. +That is, a system of graded coordinates of pM, Eq on U is +px1, . . . , xn, ξ1 +1, . . . , ξ +rkpE˚ +1 q +1 +, . . . , ξ1 +i , . . . , ξ +rkpE˚ +i q +i +, . . . q. +Elements of EpUq are "polynomials" in tpξj +i qj“1,...,rkpE˚ +´iq, i ě 1u with coefficients in OpUq. +Example 6.1.3. The sheaf of differential forms pM, E “ ΩpMqq on a manifold M is a graded manifold +since for every point m P M, it takes the form OU bK ^‚T ˚ +mM where U is an open neighborhood of +m. Exterior forms can be seen as sections on the graded vector bundle E´1 “ TM. +Example 6.1.4. Let k be a positive integer. A finite dimensional vector space E and its dual E˚ can +be seen as graded vector bundles of respective degree ´k and k over a point. E is a graded manifold +over M “ tptu, with functions isomorphic to ^E˚ for k odd and SpE˚q for k even. +Definition 6.1.5. A morphism of graded manifolds between the two graded manifolds pM, Eq and +pM1, E1q with respective base manifolds M and M1 is a pair made of a smooth or real analytic or +holomorphic map φ: M ÝÑ M1 called the base map and a sheaf morphism over it, i.e. a family of +graded algebra morphisms: +E1pU1q Ñ Epφ´1pU1qq, +compatible with the restriction maps, such that +Φpfαq “ φ˚pfqΦpαq. +(6.2) + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +98 +for all f P O1 +U1 and α P E1pU1q. +A homotopy between two morphisms of graded manifolds Φ, Ψ: pM, Eq ÝÑ pN, E1q is a morphism +of graded manifold +pM, Eq ˆ pr0, 1s, Ωpr0, 1sqq ÝÑ pM1, E1q +whose restrictions to t “ 0 resp. t “ 1 coincide with Φ and Ψ respectively. +Vector fields on graded manifolds +Vector fields on manifolds are derivations of its algebra of functions. For a graded manifold, the analog +of functions are the sheaf of sections ΓpS‚pE˚qq. Since it is not commutative but graded commutative, +one has to consider graded derivations. A graded derivation of degree k of E is the data, for every +U Ă M of a linear map +Q: E‚pUq ÝÑ E‚`kpUq, +compatible with all restriction maps, that increases the degree by `k and satisfies: +QrFGs “ QrFsG ` p´1qkiFQrGs +for every F P EipUq, G P EpUq. Since we think geometrically, we say "vector fields of degree k" instead +of graded derivation. +Definition 6.1.6. Let pM, Eq be a graded manifold. For U Ă M and k P Z let +XkpEqpUq :“ DerkpEpUqq +be the EpUq-module of derivation of degree k on EpUq. The correspondence U ÞÝÑ X‚pEqpUq is a sheaf +of E-module. Its sections are called vector fields on E. +Let us list some important facts on vector fields on E: +1. the E-module X‚pEq :“ ‘kPZXkpEq of vector fields on E is naturally graded. The E-module +X‚pEq of vector fields on E is a graded Lie subalgebra of the graded Lie algebra HomKpE, Eq +whose graded Lie bracket is the graded commutator. Precisely, the graded Lie bracket +rP, Qs “ P ˝ Q ´ p´1qklQ ˝ P +(6.3) +of two vector fields P, Q of degree k, l respectively is a vector field of degree k ` l. It is easily +checked that the bracket (6.3) fulfills +(a) rP, Qs “ ´p´1qjkrQ, Ps, +(graded skew-symmetry) +(b) p´1qjlrP, rQ, Rss ` p´1qjkrQ, rR, Pss ` p´1qklrR, rP, Qss “ 0, +(graded Jacobi identity) +(c) rP, fQs “ PrfsQ ` frP, Qs, +(Leibniz identity) +for f P O and P, Q, R are vector fields of degree j, k and l respectively. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +99 +2. Their description in local coordinates: notice that any homogeneous element e P ΓpE´kq cor- +responds to a vertical1 vector field ιe P X´kpEq (i.e. it is O-linear) of degree ´k defined by +contraction with e +ιepξq :“ xξ, ey, +ξ P ΓpE˚q +(6.4) +and we extend by O-linear derivation, where x¨ , ¨y is the dual pairing between E˚ and E. Notice +that ιe is by construction of polynomial-degree ´1. +Let pU, x1, . . . , xnq be a coordinate chart of M and pξj +i qj“1,...,rkpE˚ +i q with i ě 1 be a homogeneous +local trivialization of E˚ +´i, it should be understood that ξj +i is the j-th elements of the local +frame in ΓpE˚ +´iq. Let pej +iqj“1,...,rkpE´iq, i ě 1 be the dual basis of pξj +i qj“1,...,rkpE˚ +´iq, i ě 1, then for +every pair i, j, ιej +i “ +B +Bξj +i is the partial derivative with respect to ξj +i P ΓpE˚ +´iq. By choosing a +TM-connection on E, it is easy to check that for any k P Z the family +ˆ +ξj1 +i1 d ¨ ¨ ¨ d ξjl +il +B +Bxj +˙ +l ě 0 +i1 ¨ ¨ ¨ il “ k +j1, . . . , jl +j “ 1, . . . , n +Y +˜ +ξj1 +i1 d ¨ ¨ ¨ d ξjl +il +B +Bξj +i +¸ +l ě 0 +i1 ¨ ¨ ¨ il ´ i “ k +j1, . . . , jl +i ě 1, j “ 1, . . . , rkpE´iq +form a basis for XkpEqpUq up to permutations of the ξj1 +i1 d ¨ ¨ ¨ d ξjl +il ’s. +Here we adopt the +convention i0 “ j0 “ 0 and ξ0 “ 1 P ΓpS0pE˚qq » O. Whence, any vector field Q P XkpEqpUq +admits coordinates decomposition as follows +Q “ +ÿ +l ě 0 +i1 ¨ ¨ ¨ il “ k +j1, . . . , jl +j “ 1, . . . , n +1 +l! +jQj1¨¨¨jl +i1¨¨¨il ξj1 +i1 d ¨ ¨ ¨ d ξjl +il +B +Bxj +` +ÿ +i ě 1, l ě 0 +i1 ¨ ¨ ¨ il ´ i “ k +j1, . . . , jl +j “ 1, . . . , rkpE´iq +1 +l! +ijQj1¨¨¨jl +i1¨¨¨il ξj1 +i1 d ¨ ¨ ¨ d ξjl +il +B +Bξj +i +. +for some functions Qj1¨¨¨jl +i1¨¨¨il P O. These functions can be chosen in a unique manner to satisfy, e.g. +ijQ +jσp1q¨¨¨jσplq +iσp1q¨¨¨iσplq “ ϵpσqQj1¨¨¨jl +i1¨¨¨il for any permutation σ of t1, . . . , lu. +For example, if Q is of degree `1, then it can be written in these notations as +Q “ +ÿ +1 ď u ď rkpE´1q +j “ 1, . . . , n +jQu +1 ξu +1 +B +Bxj +` +ÿ +i ě 1, “ l ě 0 +i1 ¨ ¨ ¨ il ´ i “ 1 +j1, . . . , jl +j “ 1, . . . , n +1 +l! +ijQj1¨¨¨jl +i1¨¨¨il ξj1 +i1 d ¨ ¨ ¨ d ξjl +il +B +Bξj +i +. +This following lemma says that vector fields of polynomial-degree ´1 are all the types (6.4). +Lemma 6.1.7. Let pM, Eq be a graded manifold. For i ě 1, a vector field P P X´ipEq of polynomial- +degree ´1 and of degree ´i, is of the form ιe for some section e P ΓpE´iq. +Proof. Note that a vector field P P X´ipEq of polynomial-degree ´1 is vertical, i.e. Ppfq “ 0, since +functions of M are of polynomial-degree zero. +Hence, in local coordinates pU, x1, . . . , xnq M and +pξj +i qj“1,...,rkpE˚ +´iq with be a homogeneous local trivialization of E˚ +´i and pej +iqj“1,...,rkpE´iq be the dual +1A vector field P P XpEq is said to be vertical if it is linear with respect to functions on M, in other words if Prfs “ 0 +for all f P O. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +100 +basis of pξj +i qj“1,...,rkpE˚ +´iq: a polynomial-degree ´1 vector field P P X´ipEq of degree ´i is forced to be +of the form +P|U “ +rkpE´iq +ÿ +j“1 +fjpxq B +Bξj +i +(6.5) +with fj P C8pUq. Now, choose a local section eU P ΓUpE´iq of the form, +eU :“ +ÿ +jě1 +fjej +i. +(6.6) +It is then clear that ιeU “ P|U. +Since local sections of E´i given on different coordinates chart +domains Ua and Ub coincide on Ua X Ub and then lift to a global section of E´i. Thus, ιe “ P for some +e P ΓpE´iq. +6.1.2 +NQ-manifolds +Definition 6.1.8. A dg-manifold or NQ-manifold is a positively graded manifold pM, Eq endowed +with a degree `1 homological vector field Q on E, i.e., Q P X1pEq is such that Q2 “ 0. +They shall be denoted as a triple pM, E, Qq. +Example 6.1.9. Given a finite dimension Lie algebra pg, r¨ , ¨ sq of dimension d. We assume that g +is concentrated in degree ´1. It is clear that pM “ tptu, E “ ^‚g˚q is a graded manifold over M “ +tptu. This graded manifold carries a dg-manifold structure. Precisely, we define the corresponding +homological vector field as follows: fix a basis peiqi“1,...,n of g and let these global coordinate functions +pξiqi“1,...,n on g be its dual. We have +rei, ejs “ +nÿ +l“1 +λl +ijel +for some coefficients λl +ij P K. One can check that the degree `1 vector field +Q “ 1 +2 +nÿ +i,j,l“1 +λl +ijξi ^ ξj B +Bξl +corresponds to the Chevalley-Eilenberg differential pdCE, ^‚g˚q. +Therefore, Q2 “ 0. +Notice that +Q2 “ 0 is equivalent to the Jacobi identity for r¨ , ¨s. +Example 6.1.10. Given a differential graded vector bundle ppE´iqiě1, dq over M. There is a natural +dg manifold given by its sheaf of sections pM, E “ ΓpSpE˚qq. +In particular, the deferential map +d: E ÝÑ E is dualized as a degree `1 map S1pE˚q ÝÑ S1pE˚q that we extend to a C8pMq-linear +derivation on E squared to zero. +Example 6.1.11. Let E “ Tr1sM be the shifted bundle of M. It induces a graded manifold structure +pM, E “ ΩpMqq over M. This graded manifold carries a dg-manifold structure Q that corresponds to +the de Rham differential on ΩpMq. In terms of coordinates, the homological vector field Q reads +nÿ +i“1 +dxi +B +Bxi +. +Let us introduce some vocabularies that will need to use. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +101 +Definition 6.1.12. Let pM, E1, Q1q and pM, E, Qq be two NQ-manifolds. +1. A linear map Φ: E ÝÑ E1 is said to be of polynomial-degree/degree j P Z provided that, for all +function α P E of polynomial-degree/degree i, Φpαq is of polynomial-degree/degree i ` j. Any +map Φ: E ÝÑ E1 of degree i decomposes w.r.t the polynomial-degree as follows: +Φ “ +ÿ +rPZ +Φprq +with Φprq : E ÝÑ E1 a map of polynomial-degree r. +Remark 6.1.13. When Φ: E ÝÑ E1 is a graded morphism of algebras, necessarily one has Φprq “ 0 +for all r ă 0. Furthermore, for all n, r P N and all ξ1, . . . , ξk P ΓpV q one has: +Φprqpξ1 d ¨ ¨ ¨ d ξnq “ +ÿ +i1`¨¨¨`in“r +Φpi1qpξ1q d ¨ ¨ ¨ d Φpinqpξnq. +(6.7) +Obviously, in this case Φ is determined uniquely by the image of ΓpV q. +Definition 6.1.14. Let pM, E, Qq and pM, E1, Q1q be two NQ-manifolds over M with sheaves of +functions E and E1 respectively. A morphism of NQ-manifold over M from pM, E1, Q1q to pM, E, Qq is +a morphism of graded manifolds Φ: E ÝÑ E1 (of degree 0) over the identity map which intertwines Q +and Q1, i.e., +Φ ˝ Q “ Q1 ˝ Φ. +(6.8) +Remark 6.1.15. It is important to notice that +• morphisms of NQ-manifolds over M are by definition O-linear, since they are defined over the +identity map +• the component Φprq of polynomial-degree r ě 0 of any O-linear map Φ: E ÝÑ E1 maps ΓpE˚q to +ΓpSr`1pE1˚qq. By O-linearity, it gives rise to a section φr P ΓpSr`1pE1˚q b Eq. Therefore, one +has +Φprqpξq “ xφr, ξy +(6.9) +for all ξ P ΓpE˚q. It follows that Φ is entirely determined by the collection +` +φr P ΓpSk`1pE1˚q b Eq +˘ +rě0 +when Φ is an algebra morphism or a Ξ-derivation for some map Ξ: E ÝÑ E1. In such case, for +r ě 0, φr P ΓpSr`1pE1˚q b Eq is then called the r-th Taylor coefficient of Φ. We also call the +0-th Taylor coefficient φ0 : E1 Ñ E the linear part of Φ. The latter is a chain map +¨ ¨ ¨ +� E1 +´3 +φ0 +� +d1p3q � E1 +´2 +φ0 +� +d1p2q � E1 +´1 +φ0 +� +ρ1 +� TM +id +� +¨ ¨ ¨ +� E´3 +dp3q � E´2 +dp2q � E´1 +ρ +� TM +(6.10) + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +102 +6.1.3 +NQ-manifolds - Lie 8-algebroids +We have seen in Section 2.2 that Lie 8-algebroids (possibly infinite dimension) over O are one-to-one +with co-differentials of the graded symmetric algebra, which are compatible with the action of O. This +correspondence provides a simple characterization of Lie 8-algebroids over an arbitrary commutative +unital algebra O. In the finite dimensional case, we can work without co-differentials by using T. +Voronov’s higher derived brackets construction [Vor10]. Roughly speaking, it is shown in [Vor10] that +(finite dimensional) Lie 8-algebroids as in Remark 2.1.13 are the same as NQ-manifolds. However, it +is important to note the latter correspondence fails in infinite dimension case, since the identification +Γ pS‚pE˚qq » Γ pS‚pEq˚q fails. We will not explain entirely the construction here, but we refer the +reader to [Vor04, Vor05, Vor10] also to [BP13, LMP20] for more details on the construction. +The following statement is similar to our Proposition 2.2.4 the difference lies in the fact that ours +remains valid in infinite dimension. +Proposition 6.1.16 (T. Voronov). Let pM, Eq be a graded manifold of finite dimension in each degree. +There is a one-to- one correspondence between: +1. (finitely generated) negatively graded Lie 8-algebroids pE, pℓkqkě1, ρq over M, +and +2. homological vector fields Q P X`1pEq. +The relation stated in Theorem 6.1.16 can be enlightened in terms of the k-ary brackets pℓkq as +follows: +r¨ ¨ ¨ rrQ, ιe1s , ιe2s , . . . , ιeksp´1q “ ιℓkpe1,...,ekq, +for all e1, . . . , ek P ΓpEq +(6.11) +ρpeqrfs “ rQ, ιesp0qpfq, +for all f P O, e P ΓpE´1q +(6.12) +In particular2, +1. for all f P O, e P ΓpE´1q +xQp1qrfs, ey “ ρpeqrfs, +2. for all α P ΓpE˚q and e P ΓpEq: +A +Qp0qrαs, e +E +“ xα, ℓ1peqy , +3. for all homogeneous elements e1, e2 P ΓpEq and α P ΓpE˚q +A +Qp1qrαs, e1 d e2 +E +“ ρpe1qrxα, e2ys ´ ρpe2qrxα, e1ys ´ xα, ℓ2pe1, e2qy, +with the understanding that the anchor ρ vanishes on E´i when i ě 2. +2Our sign’s convention are those of [LLS20]. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +103 +4. for every k ě 3, the k-ary brackets ℓk : ΓpSk +KpEqq ÝÑ ΓpEq and the polynomial-degree k ´ 1 +component Qpk´1q : ΓpE˚q Ñ ΓpSk +KpE˚qq of Q are dual to each other, i.e. +A +Qpk´1qrαs, e1 d ¨ ¨ ¨ d ek +E +“ xα, ℓkpe1, . . . , ekqy , +for α P ΓpE˚q and e1, . . . , ek P ΓpEq. +Convention 6.1.17. From now on, unless otherwise mentioned, we shall simply say "Lie 8-algebroids +over M" for "finitely generated 8-algebroids over M". Also, Lie 8-algebroids pE, pℓkqkě1, ρq over M +shall be denoted as pE, Qq. +Remark 6.1.18. Whether we see Lie 8-algebroids as NQ-manifolds or as co-differentials, these two +approches have in common to give the same notion of Lie 8-morphism of 8-algebroids. This can be +seen directly by writing the conditions (2.3.1) and (6.8) in terms of the k-ary brackets. +6.2 +Universal Q-manifolds +This section can be understood as a consequence of the main results of the first part of the thesis. +We recover the result on universal Q-manifolds of a singular foliation [LLS20, Lav17], whose existence +was proved under the condition that geometric resolutions exist. We recall that the existence of such +a resolution is not guaranteed, unlike the case of resolution by free modules. We refer the reader to +Appendix B and B.3 for more details on resolutions of modules and geometric resolutions of singular +foliations. +Theorem 6.2.1. Let F be a singular foliation on a manifold M. +1. Any resolution of F by free O-modules (which may not be a geometric resolution) +¨ ¨ ¨ +d +ÝÑ P´3 +d +ÝÑ P´2 +d +ÝÑ P´1 +ρ +ÝÑ F ÝÑ 0 +(6.13) +carries a Lie 8-algebroid structure over F whose unary bracket is ℓ1 :“ d. +2. In particular, when F admits a geometric resolution pE, d, ρq, there exists a Lie 8-algebroid +pE, Qq over F whose linear part is pE, d, ρq. +Proof. Apply Theorem 4.2.1 to F seen as a Lie-Rinehart algebra over O. +Proposition 6.2.2. Let F be a singular foliation over M. Given, +a) a Lie 8-algebroid pM, E1, Q1q that terminates in F, i.e, ρ1pΓpE´1qq Ď F, +b) a universal Lie 8-algebroid pM, E, Qq of F, +then +1. there exists a Lie 8-morphism from pM, E1, Q1q to pM, E, Qq. +2. and any two such morphisms are homotopic. +Proof. Apply Theorem 4.2.4. +Corollary 6.2.3. Two universal Lie 8-algebroid of a singular foliation are homotopy equivalent. +Moreover, the homotopy equivalence between them is unique up to homotopy. +Proof. Apply Corollary 4.2.5. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +104 +6.2.1 +The complex defined by adQp0q +Let F be a singular foliation on M that admits a universal algebroid pE, Qq. For a fixed k P N0 Yt´1u +consider the bicomplex defined on Hom‚ +O +´ +ΓpE˚q, Γ +´ +Sk`1 +K +pE˚q +¯¯ +where the horizontal differential +Bh (resp. vertical differential Bv) are given by left composition (resp. right composition with Qp0q), +namely, for all Ψpkq P Hom‚ +O +´ +ΓpE˚q, Γ +´ +Sk`1 +K +pE˚q +¯¯ +one has, +Bh ´ +Ψpkq¯ +:“ +$ +& +% +Ψpkq ˝ Qp0q +when Ψpkq restricts to ΓpE˚ +´iq, with i ‰ 1, +Ψpkq ˝ ρ˚ +when Ψpkq restricts to ΓpE˚ +´1q, +(6.14) +and +Bv ´ +Ψpkq¯ +:“ +$ +& +% +ρ˚ ˝ Ψpkq +when Ψpkq is of degree i and restricts to ΓpE˚ +´i´1q, +Qp0q ˝ Ψpkq +otherwise. +(6.15) +With the understanding that Γ +` +S0 +KpE˚q +˘ +» O. The total differential is given by the formula, +B +´ +Ψpkq¯ +“ Bh ´ +Ψpkq¯ +´ p´1q|Ψpkq|Bv ´ +Ψpkq¯ +for all +Ψpkq P Hom‚ +O +´ +ΓpE˚q, Γ +´ +Sk`1 +K +pE˚q +¯¯ +. +The following diagram pictures the idea of the bicomplex. The total degree is given by the anti- +diagonals lines. +... +... +... +Ò +Ò +Ò +Ñ +HomO +ˆ +ΓpE˚ +´2q, Γ +´ +Sk`1 +K +pE˚q +¯ +k`3 +˙ +Bh +Ñ +HomO +ˆ +ΓpE˚ +´1q, Γ +´ +Sk`1 +K +pE˚q +¯ +k`3 +˙ +¨ ˝ρ˚ +Ñ +HomO +´ +F˚, Γ +´ +Sk`1 +K +pE˚qk`3 +¯¯ +Ñ +0 +Bv Ò +Bv Ò +Dv Ò +Ñ +HomO +ˆ +ΓpE˚ +´2q, Γ +´ +Sk`1 +K +pE˚q +¯ +k`2 +˙ +Bh +Ñ +HomO +ˆ +ΓpE˚ +´1q, Γ +´ +Sk`1 +K +pE˚q +¯ +k`2 +˙ +¨˝ ρ˚ +Ñ +HomO +´ +F˚, Γ +´ +Sk`1 +K +pE˚qk`2 +¯¯ +Ñ +0 +Bv Ò +Bv Ò +Bv Ò +Ñ +HomO +ˆ +ΓpE˚ +´2q, Γ +´ +Sk`1 +K +pE˚q +¯ +k`1 +˙ +Bh +Ñ +HomO +ˆ +ΓpE˚ +´1q, Γ +´ +Sk`1 +K +pE˚q +¯ +k`1 +˙ +¨ ˝ρ˚ +Ñ +HomO +ˆ +F˚, Γ +´ +Sk`1 +K +pE˚q +¯ +k`1 +˙ +Ñ +0 +Ò +Ò +Ò +0 +0 +0 +(6.16) +There is no cohomology for the total complex which is governed by B ” r ¨ , Qp0qs since the lines +are exact (see Proposition B.1.16). When k “ ´1 the bicomplex (6.16) is just the line complex: +¨ ¨ ¨ +Bh +ÝÑ HomO +` +ΓpE˚ +´2q, O +˘ +Bh +ÝÑ HomO +` +ΓpE˚ +´1q, O +˘ +Bh +ÝÑ HomO pF˚, Oq ÝÑ 0 +(6.17) +which is also exact. +6.2.2 +A result on longitudinal vector fields and examples +In this section, we study the cohomology of longitudinal vector fields, which will help in proving the +main results stated in the beginning of Chapter 8.2. +Let F be a singular foliation over M. +Definition 6.2.4. Let E be a splitted graded manifold over M with sheaf of function E. A vector +field L P XpEq is said to be a longitudinal vector field for F if there exists vector fields X1, . . . , Xk P F +and functions Θ1, . . . , Θk P E such that +Lpfq “ +kÿ +i“1 +XirfsΘi, +@f P O. +(6.18) + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +105 +We also need to define the following +Definition 6.2.5. A vector field X P XpMq is said to be an infinitesimal symmetry of F, if rX, Fs Ă F. +The Lie algebra of infinitesimal symmetries of F is denoted by, spFq +Example 6.2.6. Here are some examples. +1. Vertical vector fields on a graded manifold are longitudinal for any singular foliation. +2. For any Q-manifold pE, Qq over a manifold M. The homological vector field Q P XpEq is a +longitudinal vector field for F :“ ρpΓpE´1qq: in local coordinates pU, x1, . . . , xnq on M and a +local trivialization ξ1, ξ2, . . . of graded sections in ΓpE˚q. The vector fields Q are of the form: +Q “ +ÿ +j +ÿ +k, |ξk|“1 +Qj +kpxqξk B +Bxj +` +ÿ +j +ÿ +k,ι1,...,ιk +1 +k!Qj +ι1,...,ιkpxqξ1 d ¨ ¨ ¨ d ξk B +Bξj +(6.19) +“ +ÿ +k, |ξk|“1 +ξk +˜ÿ +j +Qj +kpxq B +Bxj +¸ +` ¨ ¨ ¨ +Take Xk :“ ř +j Qj +kpxq B +Bxj P FpUq for every k. One has for all f P C8pUq, Qpfq “ ř +k ξkXkrfs. +3. For pE, Qq a Q-manifold and F :“ ρpΓpE´1qq its basic singular foliation. For any extension of +a symmetry X P spFq of F to a degree zero vector field p +X P XpEq, the degree `1 vector field +rQ, p +Xs is longitudinal for F. +Let us show this last point using local coordinates px1, . . . , xnq on M and a local trivialization +ξ1, ξ2, . . . of graded sections in ΓpE˚q. The vector fields Q and p +X take the form: +Q +“ +ÿ +j +ÿ +k, |ξk|“1 +Qj +kpxqξk B +Bxj +` +ÿ +j +ÿ +k,ι1,...,ιk +1 +k!Qj +ι1,...,ιkpxqξ1 d ¨ ¨ ¨ d ξk B +Bξj +p +X +“ +X ` +ÿ +j +ÿ +k,ι1,...,ιk +1 +k!Xj +ι1,...,ιkpxqξ1 d ¨ ¨ ¨ d ξk B +Bξj +(6.20) +where X “ +nÿ +i“1 +Xipxq B +Bxi +. By using Equation (6.20) we note that all the terms of rQ, p +Xs are +vertical except maybe for the ones where the vector field X appears. For k ě 1, the vector field +rQj +ι1,...,ιkξ1 d ¨ ¨ ¨ d ξk B +Bξj , Xs is vertical; and for every fix k, one has +« nÿ +j“1 +Qj +kξk B +Bxj +, X +ff +“ ξk +« nÿ +j“1 +Qj +k +B +Bxj +, X +ff +. +Thus, +« nÿ +j“1 +Qj +k +B +Bxj +, X +ff +P F, since X is a symmetry for F and +nÿ +j“1 +Qj +k +B +Bxj +P F. +Remark 6.2.7. Longitudinal vector fields are stable under the graded Lie bracket. We denote by +LpEq the graded Lie algebra of longitudinal vector fields for F. +Remark 6.2.8. Let us study vector fields on E. +1. Sections of E are identified with derivations under the isomorphism mapping e P ΓpEq ÞÝÑ ιe P +XpEq. This allows us to identify a vertical vector field with (maybe infinite) sums of tensor +products of the form Θ b e with Θ P E, e P ΓpEq. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +106 +2. Any connection on ΓpE˚q induces a vector field of degree zero r∇X P XpEq by setting for f P O, +r∇Xpfq :“ Xrfs. Once a connection is chosen, we have for all k P Z +XkpEq » +à +jě1 +Ek`jbOΓpE´jq‘EkbOXpMq » ‘jě1ΓpSpE˚qk`jbE´jq‘ΓpSpE˚qkbTMq. (6.21) +Thus, one can realize a vector field P P XkpEq as a sequence P “ pp0, p1, . . .q, where p0 P +ΓpSpE˚qk b TMq and pi P ΓpSpE˚qk`i b E´iq for i ě 1 are called components of P. In the +diagram (6.23), P “ pp0, p1, . . .q is represented as an element of the anti-diagonal and pi is on +column i. We say that P is of depth n P N if pi “ 0 for all i ă n. In particular, vector fields of +depth greater or equal to 1 are vertical. Under the decomposition (6.21), the differential map +adQ takes the form +D “ Dh ` +ÿ +sě0 +Dvs +(6.22) +with D2 “ 0. Here Dh “ idbd +or +idbρ, and Dvs : ΓpSpE˚qkbE´iq Ñ ΓpSpE˚qk`s`1b E´i´sq +for i ě 0, s ě 0, we shall denote E0 :“ TM. We denote the latter complex by pX, Dq. They +can be represented as anti-diagonal lines in the following commutative diagram, whose lines are +complexes of O-modules +... +... +... +¨ ¨ ¨ +� ΓpSpE˚qk`2 b E´2q +� +idbd +� ΓpSpE˚qk`2 b E´1q +� +idbρ +� ΓpSpE˚qk`2 b TMq +� +¨ ¨ ¨ +� ΓpSpE˚qk`1 b E´2q +� +... +Qbid ` ¨¨¨ +� +idbd +� ΓpSpE˚qk`1 b E´1q +� +... +Qbid ` ¨¨¨ +� +idbρ +� ΓpSpE˚qk`1 b TMq +� +... +Qbid ` ¨¨¨ +� +¨ ¨ ¨ +� ΓpSpE˚qk b E´2q +� +... +Qbid ` ¨¨¨ +� +idbd +� ΓpSpE˚qk b E´1q +� +� +... +Qbid ` ¨¨¨ +� +idbρ +� ΓpSpE˚qk b TMq +� +� +... +Qbid ` ¨¨¨ +� +... +� +... +� +... +� +column 2 +column 1 +column 0 +(6.23) +Under this correspondence, we understand longitudinal vector fields as the following. +Lemma 6.2.9. A graded vector field P “ pp0, p1, . . .q P X is longitudinal if p0 P E bO F. +The following theorem is crucial for Chapter 8. +Theorem 6.2.10. Let pE, Qq be a universal Q-manifold of F. +1. Longitudinal vector fields form an acyclic complex. +More precisely, any longitudinal vector field on E which is a adQ-cocycle is the image through +adQ of some vertical vector field on E. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +107 +2. More generally, if a vector field on E of depth n is a adQ-cocycle, then it is the image through +adQ of some vector field on E of depth n ` 1. +Proof. Since pE, Qq is an universal Q-manifold of F, lines in (6.23) are exact. It is now a diagram +chasing phenomena. +Let P “ pp0, p1, . . . , q P X be a longitudinal element which is a D-cocycle. +By longitudinality there exists an element b1 P ΓpSpE˚q b E´1q such that pid b ρqpb1q “ p0. Set +P1 “ p0, b1, 0, . . .q, that is we extend b1 by zero on ΓpSpE˚q b Eď´2q and ΓpSpE˚q b TMq. It is clear +that P ´ DpP1q “ p0, p1 +1, p1 +2, . . .q is also a D-cocycle. In particular, we have Dhpp1 +1q “ 0 by exactness +there exists b2 P ΓpSpE˚qbE´2q such that Dhpb2q “ p1 +1. As before put P2 “ p0, 0, b2, 0, . . .q. Similarly, +P ´ DpP1q ´ DpP2q “ p0, 0, p2 +2, p2 +3, . . .q is a D-cocycle. By recursion, we end up to construct P1, P2, . . . +that satisfy P ´ DpP1q ´ DpP2q ` ¨ ¨ ¨ “ 0, that is, there exists an element B “ p0, b1, b2, . . .q P X such +that DpBq “ P. This proves item 1. +To prove item 2 it suffices to cross out in the diagram (6.23) the columns number 0, . . . , n ´ 1, +which does not break exactness. The proof now follows as for item 1. +In particular, we deduce from Theorem 6.2.10 the following exact subcomplex. +Corollary 6.2.11. Let pE, Qq be a universal Q-manifold of F. The subcomplex VQ of pXpEq, adQq +made of vertical vector fields P P XpEq that satisfy P ˝ Qpfq “ 0 for all f P O is exact. +Proof. Let P P XpEq be a vertical vector field which is an adQ-cocycle (note that we have automatically +P ˝ Qpfq “ 0 for all f P O). By Theorem 6.2.10 there exists a vertical vector field rP P XpEq such that +rQ, rPs “ P. Moreover, rP P VQ, since for all f P O, +0 “ rQ, rPspfq “ p´1q| rP| rP ˝ Qpfq. +The following remark will be used especially for the proof of Theorem 8.3.1 of Chapter 8. +Remark 6.2.12. For a cocycle P P VQ of degree 0 one has P p´1q “ 0 (for degree reason). +By +Corollary 6.2.11, P is the image by adQ of an element rP P VQ i.e. such that rQ, rPs “ P. Also, one +can choose rP p´1q “ 0: we have +rQ, rPsp´1q “ rQp0q, rP p´1qs “ P p´1q “ 0. +By exactness of adQp0q (see Section 6.2.1), we have P p´1q “ rQp0q, ϑs for some O-linear map ϑ P +HompΓpE˚q, ΓpS0pE˚qqq of degree ´2. Put sP :“ rP ´ rQ, ϑs, where ϑ is extended to a derivation of +polynomial-degree ´1. Clearly, +rQ, sPs “ P +and +sP p´1q “ rP p´1q ´ rQ, ϑsp´1q “ rP p´1q ´ rQp0q, ϑs “ 0. +Therefore, P “ adQp sPq with sP p´1q “ 0. +Here is direct consequence of Theorem 6.2.10 and Proposition B.1.7. +Corollary 6.2.13. Let F a singular foliation on M. Let pE, Qq be a universal Q-manifold of F. Then, +H‚ pXpEq, adQq » H‚ +ˆXpEq +LpEq, adQ +˙ +. +(6.24) +where adQ is induced by adQ on the quotient space. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +108 +Since all vertical vector fields are longitudinal, there is an isomorphism of O-modules +XpEq +LpEq » ΓpS‚pE˚qq bO +ˆXpMq +F +˙ +. +(6.25) +Let us describe the differential map adQ using this isomorphism: let pU, x1, . . . , xnq be local coordi- +nates on M and take local trivialisations ξ1, ξ2, . . . of graded sections in ΓpE˚q, also let ξ1, ξ2, . . . the +corresponding dual sections in ΓpEq. Notice that locally, any P P XpEq +LpEq is represented by a vector field +of the form +ÿ +j, i1,...,ik +fj +i1,...,ikpxqξi1 d ¨ ¨ ¨ d ξik b +B +Bxj +, +fj +i1,...,ik P C8pUq +since the vector fields in +B +Bθi are vertical vector fields. That is to say, P can be represented by an +element of the form P “ Θ b X with Θ P ΓUpS‚pE˚qq and X P XpUq, using the decomposition (6.25). +Since +rQ, Ps “ QrΘsX ` p´1q|Θ|Θ ¨ rQ, Xs +“ QrΘsX ` p´1q|Θ|Θ d +ÿ +k, |ξk|“1 +ξk +«ÿ +j +Qj +kpxq B +Bxj +, X +ff +` vertical vector fields +“ QrΘsX ` p´1q|Θ| +ÿ +k, |ξk|“1 +Θ d ξk rρpξkq, Xs ` vertical vector fields +we have +adQpPq “ QrΘs b X ` p´1q|Θ| +ÿ +k, |ξk|“1 +Θ d ξk b rρpξkq, Xs, +(6.26) +where X stands for the class of a vector field X P XpMq in XpMq +F +. Therefore, locally +adQ “ Q b id ` +ÿ +k, |ξk|“1 +id d ξk b ∇Bott +ξk +where ∇Bott : ΓpE´1qˆ XpMq +F +ÝÑ XpMq +F +, is the Bott connection associated to F, i.e. ∇Bott +e +X :“ rρpeq, Xs. +Remark 6.2.14. Assume that M “ Rd and F is a singular foliation that admits two leaves: t0u and +Rdzt0u. Then +XpEq +LpEq » ΓpS‚pE˚qq|0 bR +ˆ +Rd ‘ I0XpMq +F +˙ +. +The differential may be complicated to describe. +Example 6.2.15. Let F “ I0XpRdq and pE, Qq one of its universal Lie algebroid such that E´1 “» +glpRdq is the transformation Lie algebroid for the action of glpRdq on Rd. For this singular foliation, +XpRdq +F +» Rd. Thus, we obtain +XpEq +LpEq » ΓpS‚pE˚qq|0 bC8pRdq Rd. +and the Bott connection is simply the Chevalley-Eilenberg differential for the action of glpRdq on Rd. +In particular, it is easy to see that +H1pXpEqq “ H1 +ˆXpEq +LpEq +˙ +“ H1 +CEpglpRdq, Rdq “ 0 +In other words, the universal Lie 8-algebroid is, in that case, rigid, i.e. +any formal deformation +Q ` ϵQ1 ` ϵ2Q2 ` ¨ ¨ ¨ of its derivation Q is formally trivial [LGL22a]. + +CHAPTER 6. UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION +109 +Here is a second example where the universal Lie 8-algebroid is rigid. +Example 6.2.16. Let F be the singular foliation given by the action of Lie algebra g “ slp2, Rq on +R2 through the vector fields: +e “ x B +By, +f “ y B +Bx, +h “ x B +Bx ´ y B +By, +where e, f, h are the standard basis elements of slp2, Rq. The singular foliation F admits a universal +Lie 8-algebroid pE, Qq built on a geometric resolution ([LLS20], Example 3.31) of length 2 +0ÝÑE´2 +d +ÝÑ E´1 +ρ +ÝÑ TM, +where E´1 » slp2, Rq ˆ R2 is a trivial vector bundle of rank 3, and E´2 a trivial vector bundle of rank +1. Here, the bracket between two constant sections of E´1 is defined as being their bracket in slp2, Rq +and all other brackets between generators is trivial. Also, Q takes the form +Q “ xe˚ B +By `yf˚ B +Bx`h˚px B +Bx´y B +Byq`e˚df˚ B +Bh˚ ´2e˚dh˚ B +Be˚ `2f˚dh˚ B +Bf˚ `px2f˚`y2e˚´xyh˚q B +Bµ˚ +where µ is the generator of E´2, and pe˚, f˚, h˚, µ˚q is the dual basis of ppe, f, h, µqq. +In this case, one has +XpR2q +F +» R2 ‘ R +is generated by the classes of +B B +Bx, B +By, x B +Bx ` y B +By +F +. +Hence, as a graded vector space: +XpEq +LpEq » p^‚sl˚ +2 ‘ Rr2sq bR pR2 ‘ Rq. +The differential induced by Q is easily checked to be the Chevalley-Eilenberg differential for the natural +slp2, Rq-action on these modules. In particular, +H1pXpEqq “ H1 +ˆXpEq +LpEq +˙ +“ H1 +ˆ +slp2, Rq, XpR2q +F +˙ +“ 0, +since the Lie algebra slp2, Rq is semisimple. This implies that the universal Lie 8-algebroid pE, Qq is +rigid in the sense of [LGL22a]. +Conclusion: +We recalled the duality "Lie 8-algebroid ÐÑ Q-manifolds of finite rank in all degree" so that +the universal Lie 8-algebroid becomes a universal Q-manifold. We recover [LLS20] the notion +of universal Q-manifold pE, Qq of a singular foliation. We prove that adQ hasa no longitudinal +cohomology. +aWhich squares to zero on vector fields on E. + +CHAPTER 7 +Isotropy Lie algebras of a singular foliation +In this chapter, we look at an extension of the Androulidakis and Skandalis isotropy Lie algebra [AZ13] +of a singular foliation at a point. +Let us recall some definitions. +Definition 7.0.1. Let F be a singular foliation on a manifold M. Let Im “ tf P C8pMq | fpmq “ 0u. +1. The tangent space of F at a point m P M is the vector space TmF :“ tX|m | X P Fu whose +elements are evaluations of the vector fields of F at m. In other words, TmF is the image of the +evaluation map evm : F Ñ TmM, X ÞÑ X|m. +2. The set +! +m P M +ˇˇˇ +F +ImF ÝÑ TmF is bijective +) +is open and dense in M [AS09]. It is the set of +regular points of pM, Fq and is denoted by Mreg. +For any point m P M, the O-module Fpmq :“ tX P F | Xpmq “ 0u “ kerpevmq is a Lie subalgebra +of F. The evaluation map goes to quotient and induces the following exact sequence, +0 +� Fpmq +ImF +� +F +ImF +� TmF +� 0. +Since ImF Ď Fpmq is a Lie ideal, +Definition 7.0.2. the quotient space gm “ Fpmq +ImF is a Lie algebra, which is called the isotropy Lie +algebra of F at m. +Remark 7.0.3. The isotropy Lie algebras detect singular and regular points of pM, Fq. It measures +how far F is of being regular in neighborhood of a point. This can be stated as follows (see Lemma +1.1 of [AZ13]). +Proposition 7.0.4. For any point m P L of a leaf L Ă M, gm “ t0u if and only if, L is a regular +leaf. Also, the gm’s are all isomorphic as Lie algebras while m P L. +110 + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +111 +For A a Lie-Rinehart algebra over O, the same construction can be done for any maximal ideal m. +Let +A|m :“ ta P A | ρpaqrOs Ď mu. +The quotient A|m +mA is a Lie algebra over O{m. For O “ C8pMq and m “ Im for a point m P M, the +following sequence +0 +� A|Im +ImA +� � +� +A +ImA +� � TmρpAq +� 0 +(7.1) +is exact, where TmρpAq Ă Hompm{m2, O{mq is the image of A Ñ Hompm{m2, O{mq, a ÞÑ ρpaq. +Proposition 7.0.4 is however subtle to extend to this setting. +Proposition 7.0.5. For a finitely generated Lie-Rinehart A over C8pMq, there exists a Lie algebroid +pA, ρA, r¨ , ¨sAq such that A » ΓpAq for some vector bundle A Ñ M if and only if +A +ImA, as a vector +space, has constant rank at all points. In that case Apmq +ImA is kerpρA|mq for all m P M. In particular, it +is the case in a neighborhood of every point where the dimension of +A +ImA is minimal. +Proof. This is a consequence of Nakayama Lemma B.2.11. The germ Om of functions on a neighbor- +hood of U Ď M of m is a local ring [RS94] with maximal ideal still denoted by Im. The tensor product +OmbO A is a module over Om. Let r “ dimp +A +ImAq. By Nakayama Lemma B.2.11 any basis pe1, . . . , erq +of +A +ImA lifts to minimal generators on Om bO A in a neighborhood of m. Since r “ dimp +A +Im1Aq for all +points m1 P U, again Nakayama Lemma B.2.11 implies pe1, . . . , erq are minimal generators on a neigh- +borhood of m1. Any representative pe1, . . . , erq of pe1, . . . , erq are local sections of A in a neighborhood +U of m that span A on U so that A|U is a free module. This applies that A is a C8pUq-module, which +is projective. By Serre-Swan theorem, A “ ΓpAq for some vector bundle A. +7.1 +Specialization of a Lie 8-algebroid at a point +• Let pM, E, Qq “ pE‚, ℓ‚, ρq be a Lie 8-algebroid over a manifold M with anchor ρ. For every +point m P M, the k-ary brackets restrict to the graded vector space +evpE, mq :“ +˜ +à +iě2 +E´i|m +¸ +‘ kerpρmq +and equipped the latter with a Lie 8-algebra structure that we denote by IstropympE, Qq. For +every k ě 1, the restriction goes as follows: +tx1, . . . , xkuk :“ ℓkps1, . . . , skq|m +(7.2) +for all x1, . . . , xk P evpE, mq and s1, . . . , sk P ΓpEq sections of E such that sipmq “ xi with +i “ 1, . . . , k. These brackets are well-defined. It is clear that for k ‰ 2, since ℓk is linear over +functions. But it is not immediate that the 2-ary bracket is well-defined as well. Let us check +that. +On one hand, the new brackets t¨ ¨ ¨ uk have values in evpE, mq for degree reason, except may be +for the 2-ary bracket when applied to elements of degree ´1 (i.e. elements of the kernel of ρm) + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +112 +but in that case it is in the kernel of ρm since +ρmptx1, x2u2q “ ρmpℓ2ps1, s2q|mq +“ ρpℓ2ps1, s2qq|m +“ rρps1q, ρps2qs|m “ 0 +In the last line we have used the fact that the Lie bracket of two vector fields that vanish at m +is a vector field that vanishes again at m. +On the other hand, the 2-ary bracket t¨ , ¨ u2 is also well-defined when applied to elements of +degree less or equal to ´2, we need to verify when we take the bracket with at least an element +of degree ´1. Let pei +1, . . . ei +rkpE´iqq be a local trivialization of E´i on a neighborhood U of the +point m P M. For x1 P kerpρmq and x2 P E´i|m write +x1 “ +rkpE´1q +ÿ +k“1 +λke1 +kpmq, +x2 “ +rkpE´iq +ÿ +k“1 +µkei +kpmq +for some scalars pλiq in K. The scalars pλkq, pµkq extend to functions pfkq, pgkq on U. Therefore, +we have +tx1, x2u2 “ ℓ2ps1, s2q|m +with +s1 “ +rkpE´1q +ÿ +k“1 +fke1 +k, +s2 “ +rkpE´iq +ÿ +k“1 +gkei +k. +If rs2 is another extension of x2, then ps2 ´ rs2qpmq “ 0 and this is equivalent to pgk ´ rgkqpmq “ 0 +for k “ 1, . . . , rkpE´iq. It follows that +ℓ2ps1, s2 ´ rs2q|m “ +rkpE´iq +ÿ +k“1 +ℓ2 +` +s1, pfk ´ rgkqei +k +˘ +|m +“ +rkpE´iq +ÿ +k“1 +((((((( +pfk ´ rgkqpmqℓ2 +` +s1, ei +k +˘ +|m ` (((((((( +ρps1q|mrfk ´ rgksei +k +“ 0. +Hence, +evpE, mq : ¨ ¨ ¨ +t¨u1“ℓ1|m +ÝÑ +E´3|m +t¨u1“ℓ1|m +ÝÑ +E´2|m +t¨u1“ℓ1|m +ÝÑ +kerpρmq +comes equipped with a Lie 8-algebra whose brackets are pt¨ ¨ ¨ ukqkě1. +• Any Lie 8-morphism of algebroids Φ: pM, E1, Q1q Ñ pM, E, Qq induces a Lie 8-algebra morphism +Φ|m : S‚pV 1 +|mq Ñ S‚pV|mq since it is O-linear. +• We define the graded vector space +à +iě1 +H´ipE‚, mq +(7.3) +as the cohomology group of the complex +¨ ¨ ¨ +ℓ1|m� E´3|m +ℓ1|m � E´2|m +ℓ1|m � kerpρmq +� 0. + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +113 +One can check that when pM, E, Qq is universal for a singular foliation F, the graded space (7.3) +does not depend on the underlying geometric resolution of F and is denoted HpF, mq. +This construction can be extended to any Lie 8-algebroid over O for any maximal ideal I. Let +pE, ℓ‚, ρq be a Lie 8-alegrboid over O. Define +E´i|I “ +$ +’ +& +’ +% +te P E´1 | ρpeqrOs Ď Iu +for i “ 1 +E´i +IE´i +for i ě 1 +(7.4) +The construction is purely formal. Also, the cohomology of the complex +¨ ¨ ¨ +ℓ1|I � E´3|I +ℓ1|I � E´2|I +ℓ1|I � E´1|I +denoted by H‚pE, Iq does not depend on the choice of a free resolution of the Lie-Rinehart +algebra A, we denote it by H‚pE, Iq. +The isotropy Lie 8-algebra of a singular foliation +We assume that pM, E, Qq is universal for F. Note that the Lie 8-algberoid obtained by specialising +at some point m P M does not induce directly a Lie 8-algberoid on the graded space HpF, mq but the +2-ary bracket t¨ , ¨ u2 goes to quotient directly on elements of degree ´1 i.e. to H´1pF, mq, because +tdp2q +m px1q, x2u2 “ dp2q +m ptx1, x2u2q +for all x1 P E´2|m and x2 P kerpρmq. That endows H´1pF, mq with Lie algebra structure. +Proposition 7.1.1. The Androulidakis and Skandalis isotropy Lie algebra gm “ Fpmq +ImF of the singular +foliation F at a point m P M, is isomorphic to H´1pF, mq equipped with the induced Lie algebra +structure. +Proof. For m P M, we construct a Lie algebra isomorphism ζ : kerpρmq +impdp2q +m q Ñ gm as follows: For an +element u P kerpρmq, let ru be an extension of u to a local section on E´1. By construction, one has +ρpruq P Fpmq. Let rρm be the surjective linear map defined by +rρm : kerpρmq ÝÑ gm, u ÞÝÑ rρpruqs. +Since any other extension ru for u differs from the first one by a section in ImΓpE´1q, the map rρm is +well-defined. Surjectivity is due to the fact that every vector field of F vanishing at m P M is of the +form ρpeq with e a (local) section of E´1 whose value at m belongs to kerpρmq. In addition, it is not +hard to see that rρm is a morphism of brackets. +It remains to show that kerprρmq “ impdp2q +m q: let u P kerprρmq Ă kerpρmq and ru be a local section of +E´1 that extends u. By definition of u, the class of ρpruq is zero in gm, therefore, there exists some +functions fi P Im and Xi P F, i “ 1, . . . , k, local generators such that ρpruq “ +kÿ +i“1 +fiXi. This implies +that, +ρpru ´ +kÿ +i“1 +fieiq “ 0. + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +114 +where for i “ 1, . . . , k, ei is a (local) section of E´1 whose image through ρ is Xi. Since pE‚, d‚, ρq is +a geometric resolution, there exists a (local) section q P ΓpE´2q such that +ru “ +kÿ +i“1 +fiei ` dp2qq +(7.5) +By evaluating Equation (7.5) at m, we find out that u P impdp2q +m q. Conversely, for v P E´2|m, choose a +(local) section q of E´2 through v. Therefore, dp2qq P ker ρ, is a (local) extension of dp2q +m v P impdp2q +m q. +The image of dp2q +m v through rρm is obviously zero. This proves that kerprρmq “ impdp2q +m q. +However, if the underlying complex of pM, E, Qq is minimal at m then, for every i ě 2, the vector +space H´ipF, mq is canonically isomorphic to E´i|m. Also, H´1pF, mq is canonically isomorphic to +kerpρmq. +Definition 7.1.2. Let pM, E, Qq be a universal Lie 8-algebroid of a singular foliation F whose underly- +ing complex is minimal at m. Then, HpF, mq carries a Lie 8-algebra structure given by IstropympE, Qq +called the isotropy Lie 8-algebra of the singular foliation F at m. +One can show that this definition is independent of any choices made in the construction. +Remark 7.1.3. By Proposition 7.1.1, the isotropy Lie algebra of the singular foliation F at a point +m P M in the sense of Androulidakis and Skandalis, is isomorphic to the degree minus one component +H´1pF, mq » kerpρmq of the isotropy Lie 8-algebra of F at m. +Remark 7.1.4. For an arbitrary Lie-Rinehart algebra A and a maximal ideal I, it is still true that +pH‚pA, Iq, d “ 0q is quasi-isomorphic to pE|I, ℓ1|Iq and it is still true that the bracket of the universal +Lie 8-algrbroid over O constructed in Section 4.2 restricts to a 8-algebra structure on pE|I, ℓ1|Iq. By +Homotopy transfer, this Lie 8-algebra can be transferred to pH‚pA, Iq, d “ 0q. It is not clear whether +Proposition 7.1.1 holds true or not. +Lemma 7.1.5. Let pM, Fq be a singular foliation. Let pE, Qq be a universal Lie 8-algebroid of F and +let +pE, d, ρq : +¨ ¨ ¨ dp4q +ÝÑ E´3 +dp3q +ÝÑ E´2 +dp2q +ÝÑ E´1 +ρ +ÝÑ TM +(7.6) +be its linear part. +1. For all m P M, we have kerpρmq +impdp2q +m q » gm as Lie algebras. +2. The subset of regular points of F in M satisfies +Mreg “ tm P M | rkpdp2q +m q “ dimpker ρmqu. +It is open and dense in M. +3. The restriction of the foliation F to Mreg is the set of sections of a subbundle of TM, i.e., is a +regular foliation. +4. If pE, d, ρq is of finite length, then the regular leaves have the same dimension. +Proof. Item 1. is proved in Proposition 7.1.1 (see also Proposition 4.14 [LLS20]). For items 2. and 3. +we refer the reader to Proposition 1.5 of [AS09]. + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +115 +7.2 +A blow-up procedure for a singular foliation +We present in this section an interpretation of a blow-up procedure invented by Mohsen in [Moh21]. +7.2.1 +Grassmann bundle +For E a finite dimensional vector space, we denote by Gr´ℓpEq the set of all codimension ℓ P N vector +subspaces in E. Let us recall a few facts on Gr´ℓpEq see [KES00, Joe92, LB15], our main reference is +[vIR94]: +1. It is a metric space for the distance dpV, V 1q “ ∥PV ´ PV 1∥, where PV stands for the orthogonal +projection of E onto V Ă E with respect to an arbitrary metric on E. +2. The topology does not depend on the metric and makes Gr´ℓpEq a manifold. +3. It is moreover a compact manifold and an projective variety1. +Affine coordinates charts: One of the ways of defining the standard affine coordinates on the +Grassmanian Gr´ℓpEq is as follows (Example 1.24 of [vIR94]): Fix a basis e1, . . . , ed“dim E for E. +Any element V P Gr´ℓpEq, i.e. +a vector subspace V Ă E of codimension ℓ, can be viewed as a +d ˆ pd ´ ℓq matrix MV “ pC1, . . . , Cd´ℓq whose columns are formed by linearly independent column +vectors obtained by choosing a basis for V . The homogeneous coordinates of V in Gr´ℓpEq are the +components of the n ˆ pd ´ ℓq matrix MV . Any other choice of basis for V gives another maximal +rank matrix M1 +V and an invertible pd ´ ℓq ˆ pd ´ ℓq-matrix P P GLpd ´ ℓ, Kq such that MV “ M1 +V ˝ P. +In particular, since MV has full rank, there exists a family of integers 1 ď i1 ă ¨ ¨ ¨ ă id´ℓ ď d, such +that MV is equivalent to a matrix M1 +V whose submatrix made of the rows i1, ¨ ¨ ¨ , id´ℓ is the identity +matrix. +For example, if the first d ´ ℓ rows of MV are linearly independent, then the matrix is equivalent +to the matrix +˜ +Id´ℓ +A +¸ +where A “ paijq is a ℓ ˆ pd ´ ℓq-matrix. In that case, V admits a basis of the form +vj :“ ej ` +ℓÿ +k“1 +akjek, +j “ 1, . . . , d ´ ℓ. +(7.7) +V is completely determined by A. +One define an atlas on Gr´lpEq as follows: Consider the map +ψ1,...,d : Ml,d´lpKq ÝÑ Md,d´lpKq +A ÞÝÑ +˜ +Id´ℓ +A +¸ +. +1For the notion of projective variety and notations, we refer the reader e.g. to the book [vIR94]. + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +116 +For a permutation σ P Sd, we define the map ψσp1q,...,σpdq that associates every A P Ml,d´lpKq the +matrix in Md,d´lpKq that permutes the lines of +˜ +Id´ℓ +A +¸ +w.r.t σ, that is to say +ψσp1q,...,σpdqpAq :“ Ppσq ˝ ψ1,...,dpAq +where Ppσq is the permutation matrix associated to σ. +Statement: For every ordered integers 1 ď i1 ă ¨ ¨ ¨ ă id´ℓ ď d, the coordinates chart on Gr´ℓpEq is +the open set Ui1,¨¨¨ ,id´ℓ Ă Gr´ℓpEq consisting of all sub-vector space of E such that for every basis the +submatrix which is made of the rows i1, ¨ ¨ ¨ , id´ℓ is invertible, and the coordinate map is ψσp1q,...,σpdq +with σ is a permutation sending 1, . . . , d ´ ℓ on i1, ¨ ¨ ¨ , id´ℓ. +Grassmann bundle: For E´1 Ñ M a vector bundle of rank d, the disjoint union: +Gr´ℓpE´1q :“ +ž +mPM +Gr´ℓpE´1|mq +comes equipped with a natural manifold structure and +Π: Gr´ℓpE´1q ÝÑ M +is a fibration. It is called the Grassmann pd ´ ℓq-plane bundle. To fix some notations, the fiber at +m P M is +Π´1pmq “ tpV, mq | V P Gr´ℓpE´1|mqu . +For every open subset U Ă M on which E´1 is trivial, Π´1pUq » Gr´ℓpRdq ˆ U. +7.2.2 +A blow-up procedure +Settings: In what follows, we are given a foliated manifold pM, Fq with M connected. We assume +that a geometric resolution of finite length exists. Under these assumptions, all the regular leaves +have the same dimension. Let Mreg the set of the regular points of pM, Fq. Denote by ℓ the common +dimension of the regular leaves. +Remark 7.2.1. For most of the present discussion, the whole geometric resolution is not needed: is +it sufficient to assume that there exists vector bundles E´1, E´2 and that all regular leaves have the +same dimension. +Let pE‚, ℓ‚, ρq be a universal Lie 8-algebroid of F. Consider the underlying geometric resolution +pE, d, ρq : +¨ ¨ ¨ ℓ1“dp4q +ÝÑ E´3 +ℓ1“dp3q +ÝÑ E´2 +ℓ1“dp2q +ÝÑ E´1 +ρ +ÝÑ TM. +(7.8) +Notice that for every point m P Mreg, +dimpdp2q +m q “ dim kerpρmq “ rkpE´1q ´ ℓ. +Therefore, there exists a natural section of Π on Mreg defined by: +σ: Mreg ÝÑ Gr´ℓpE´1q, m ÞÝÑ impdp2q +m q +(7.9) + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +117 +We consider Ă +M :“ σpMregq the closure of the graph of σ in Gr´ℓpE´1q, and denote by π the restriction +of Π to Ă +M. +Recall from Section 7.1 that +evpE, mq : ¨ ¨ ¨ ÝÑ E´3|m +dp3q +m +ÝÑ E´2|m +dp2q +m +ÝÑ kerpρmq +comes equipped with a Lie 8-algebra pt¨ ¨ ¨ ukqkě1 whose unary bracket is dp‚q +m . +Theorem 7.2.2. The projection map +π: Ă +M Ă Gr´ℓpE´1q ÝÑ M, pV, mq ÝÑ m +(7.10) +fulfills the following properties +1. For each point m P M, the set +π´1pmq “ +" +V Ă E´1|m +ˇˇˇˇ D pmnq P MN +reg, such that, impdp2q +mnq ÝÑ +nÑ`8 V as mn ÝÑ +nÑ`8 m +* +is non-empty. +2. For all m P M, and V P π´1pmq one has, +(a) impdp2q +m q Ď V Ď kerpρmq. +(b) The 2-ary bracket t¨ , ¨ u2 on ker ρm restricts to V and the image of V in kerpρmq +impdp2q +m q » gm, is a +Lie subalgebra of codimension ℓ ´ dimpLmq, where Lm is the leaf through m. +3. For all m P Mreg, π´1pmq “ kerpρmq “ impdp2q +m q is reduced to a point in Gr´ℓpRrkpE´1qq. +4. Ă +M does not depend on the choice of a geometric resolution. +5. The projection π: Ă +M Ñ M is proper and onto. +Proof. Let us prove item 1 for m P M. By compactness of the Grassmanian manifold Gr´ℓpE´1|mq » +Gr´ℓpRrkpE´1qq, out of any sequence in MN +reg that converges to m, we can extract a sequence n ÞÑ +κ +´ +impdp2q +mnq +¯ +P Gr´ℓpE´1|mq that has a limit, where κ is a local trivialization of the vector bundle E´1 +which identifies the fibers E´1|m and E´1|mn. This proves item 1. +Let us show item 2.paq: +Let V +P π´1pmq and pmnq P MN +reg such that mn +ÝÑ +nÑ`8 m +and +impdp2q +mnq ÝÑ +nÑ`8 V . Let v P impdp2q +m q. We have v “ dp2q +m u for some u P E´2|m. Choose a (local) section +ru of E´2 through u. It implies that dp2q +mnrupmnq ÝÑ +nÑ`8 dp2q +m u, hence dp2q +m u P V . Thus, impdp2q +m q Ď V . For +any element v P V , there exists a sequence vn P kerpρmnq “ impdp2q +mnq, n P N that converges to v. In +particular, ρmnpvnq “ 0 for all n. Hence, by continuity, one has v P kerpρmq. +To prove item pbq, choose a local frame e1, . . . , er, . . . eℓ`r of E´1 such that e1pmq, . . . , erpmq is an +orthogonal basis of V for an arbitrary Hermitian structure on E´1. +For i, j P t1, . . . , l ` ru, let +pck +ijq P OpUq be a family of functions over U such that +ℓ2pei, ejq “ +l`r +ÿ +k“1 +ck +ijek. + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +118 +In particular, for every ti, ju Ă t1, . . . , l ` ru, +teipmq, ejpmqu2 “ +l`r +ÿ +k“1 +ck +ijpmqekpmq. +Let v1, v2 P V with v1 “ +rÿ +i“1 +αieipmq and v2 “ +rÿ +i“j +βieipmq. There exists sequences +vn +1 “ +r`l +ÿ +i“1 +αi +neipmq ÝÑ +nÑ`8 v1 +and +vn +2 “ +r`l +ÿ +i“1 +βi +neipmq ÝÑ +nÑ`8 v2 +with vn +1 , vn +2 P ker ρxn, for all n P N. In particular, the sequences pαi +nq, pβi +nq P KN with i P t1, . . . , r ` lu +satisfy αi +n ÝÑ +nÑ`8 αi, +βi +n ÝÑ +nÑ`8 βi +(αi “ βi “ 0 for i ě r ` 1). For every n P N we have, +r`l +ÿ +i,j,k“1 +αi +nβj +nck +ijpmnqekpmnq “ tvn +1 , vn +2 u2 P ker ρxn “ impdp2q +mnq. +(7.11) +Since +r`l +ÿ +i,j,k“1 +αi +nβj +nck +ijpmnqekpmnq ÝÑ +nÑ`8 +r`l +ÿ +i,j,k“1 +αiβjck +ijpmqekpmq “ tv1, v2u2, +one has, tv1, v2u2 P V . +Item 3. is a consequence of 2.paq, since m P Mreg if and only if kerpρmq “ impdp2q +m q (by item 1. of +Lemma 7.1.5). Item 4. follows from the existence of homotopy equivalence between any two geometric +resolutions. Item 5. follows from the fact that the projection Π is with compact fibers. This concludes +the proof. +The corollary below is a direct consequence of item 2.(b) of Theorem 7.2.2. +Corollary 7.2.3. There is a natural inclusion +Ă +M ãÑ +ž +mPM +Grℓ´dimpLmqpgmq. +(7.12) +Proof. Let m P M. Since elements V P π´1pmq satisfy impdp2q +m q Ď V Ď kerpρmq, they correspond +injectively to a (unique) sub-vector space of codimension ℓ ´ dim Lm in gm. In particular, this implies +π´1pmq ãÑ Grℓ´dimpLmqpgmq. +Lemma 7.2.4. For every m P M, the set Km “ tV P Gr´ℓpE´1|mq | V Ď ker ρmu Ă Gr´ℓpE´1|mq is +locally an affine variety. +Proof. Let e1, . . . , ed be a local frame of E´1 on an open subset U Ă M. One has, +ρpeiq “ +dim M +ÿ +k“1 +fk +i +B +Bxk +P F, +(7.13) + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +119 +for some local functions fk +i +P +C8pUq. Without any lost of generality, consider for example the +standard coordinates chart U1,...,d´ℓ for the grassmanian Gr´ℓpE´1|mq. Let V P U1,...,d´ℓ and let paijq +be the homogeneous coordinates of V . Define the sections +rvj :“ ej ` +ℓÿ +k“1 +akjek, +j “ 1, . . . , d ´ ℓ +(7.14) +By construction, the rvj’s, evaluated at m, form a basis for V . We have, +ρprvjq “ +dim M +ÿ +k“1 +˜ +fk +j ` +ℓÿ +s“1 +asjfk +s +¸ +B +Bxk +. +V Ď ker ρm if and only if +fk +j pmq ` +ℓÿ +s“1 +asjfk +s pmq “ 0, +j “ 1, . . . , d ´ ℓ +k “ 1, . . . , dim M. +(7.15) +Therefore, Km is defined by the polynomial Equations (7.15). +Theorem 7.2.5. If F is a polynomial singular foliation on M P +␣ +CN, RN( +, then Ă +M is a locally affine +variety. +Proof. We can choose a polynomial geometric resolution. +Denote by x “ px1, . . . , xNq the local +coordinates on W Ď M. We are using notations of Lemma 7.2.4. In Equation (7.13), the functions +fk +i are polynomial in px1, . . . , xNq. By item 1, Theorem 7.2.2, every element V of Ă +M “ Ť +xPM π´1pxq +is obtained as a limit +impdp2q +xn q ÝÑ +nÑ`8 V P π´1pxq +with pxnq P MN +reg, such that, xn +ÝÑ +nÑ`8 x. Fix n P N0. In the notations of Lemma 7.2.4, take V “ +impdp2q +xn q. Thus, in the coordinate chart U1,...,d´ℓ (see Section 7.2.1), the coordinates pan +ijq of impdp2q +xn q +satisfy the polynomial equations +fk +j pxnq ` +ℓÿ +s“1 +an +sjfk +s pxnq “ 0, +j “ 1, . . . , d ´ ℓ +k “ 1, . . . , N. +(7.16) +One has, +fk +j pxnq ` +ℓÿ +s“1 +an +sjfk +s pxnq “ 0 ÝÑ +nÑ`8 fk +j pxq ` +ℓÿ +s“1 +asjfk +s pxq “ 0, +where for all s, j, an +sj +ÝÑ +nÑ`8 asj. Therefore, Ă +M is given in local coordinates WˆU1,...,d´ℓ by elements +that satisfy +1. Equation (7.15) +2. and that are limit of elements of (7.15) in nearby regular points, elements which are unique. +Hence, it is, on this affine variety, the irreducible components of (7.15) that projects onto M. + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +120 +The pull-back of F to Ă +M: Let X P F. There exists a section υ of the vector bundle p: E´1 Ñ M +such that ρpυq “ X. Consider the linear vector field r +X P XpE´1q defined as follows +r +Xrp˚fs :“ p˚pρpυqrfsq, @ f P C8pMq, +x r +Xrαs, ey :“ ρpυqrxα, eys ´ xα, ℓ2pυ, eqy, @ α P ΓpE˚q, e P ΓpE´1q. +Notice that r +X depends on the choice of the almost Lie algebroid bracket ℓ2. The following items hold +(see [LGLR22], Lemma 2.1.19. p. 61). +1. The flow φ r +X +t : E´1 Ñ E´1 of r +X when it is defined, is a vector bundle isomorphism over φX +t . +2. The diagram below commutes, +E´1 +φĂ +X � +ρ +� +E´1 +ρ +� +TM +dφX +t +� TM +(7.17) +where φX +t is the flow of X. +In particular, φ r +X +t preserve the grassmanian Gr´ℓpE´1q. +Proposition 7.2.6. For every X P F, choose υ P ΓpE´1q such that ρpυq “ X. The vector field r +X +induces a vector field Ă +Ă +X on Gr´ℓpE´1q such that +1. Ă +Ă +X is tangent to Ă +M. +2. Ă +Ă +X projects onto X. +Proof. Since for every x P M, φ r +X +t |x is an isomorphism of E´1|x, therefore φ r +X +t |x preserves Π´1pxq “ +Gr´ℓpE´1|xq. We define Ă +Ă +X for pV, xq P Gr´ℓpE´1q by +Ă +Ă +XpV q :“ d +dt|t“0 +´ +φ +r +X +t |x +¯ +|V P TV Gr´ℓpE´1|xq. +(7.18) +This shows item 1. +Let us show item 1, φ r +X +t +preserves Ă +M: to see this take pV, xq P Ă +M, let xn +ÝÑ +nÑ`8 x be such that +imdp2q +xn +ÝÑ +nÑ`8 V with pxnq Ă Mreg. For every n P N0, one has +φ +r +X +t |xn +´ +imdp2q +xn +¯ +“ imdp2q +φX +t pxnq, +since ρ ˝ φ r +X +t “ dφX +t ˝ ρ. Thus, +φ +r +X +t |xpV q “ +lim +nÑ`8 φ +r +X +t |xn +´ +imdp2q +xn +¯ +“ +lim +nÑ`8 +´ +imdp2q +φX +t pxnq +¯ +P π´1 ` +φX +t pxq +˘ +. +By consequence, for V P Ă +M, we have Ă +Ă +XpV q P TV Ă +M, by Equation (7.18). + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +121 +Corollary 7.2.7. Ă +Ă +X does not depend on the choice of the almost bracket ℓ2. +Proof. This is an obvious consequence of item 3. in Proposition 7.2.6, since π: Ă +M ÝÑ M is invertible +on an open (dense) subset. +Denote by rF the module generated by the Ă +Ă +X’s, with X P F. For F polynomial, rF is a singular +foliation on the locally affine variety Ă +M, because r +X is still polynomial. +Corollary 7.2.8. If F is a polynomial singular foliation, then it is lifted to a projective singular +foliation rF on the locally affine variety Ă +M. +Generalization. The same construction would apply if, instead of considering impdp2qq Ď E´1, we +consider impdpi`1qq Ď E´i or impρq Ď TM for i ě 1. Denote by Ă +Mi the transformation that we +would obtain by such a construction. A lift rFi of F can be defined exactly in the same manner by +considering a lift r +X of X P F on E´i (or TM) associated to ℓ2 : ΓpE´1qˆΓpE´iq Ñ ΓpE´iq or to rX, ¨ s. +More precisely, consider for i ě 0, the Grassmann bundle Gr´ℓipE´iq, with E0 :“ TM, where ℓi :“ +rkpdi`1 +m q (with the understanding that dp1q “ ρ for i “ 0) at regular points. +These bundles are +manifolds that project to M through a proper map. Over Mreg, there exists a unique V P Gr´ℓipE´iq +such that dpiq +m pV q “ 0. Theses define maps +σi : Mreg ãÝÑ Gr´ℓipE´iq. +We define Ă +Mi :“ σipMregq. This is not a manifold in general. However, it is a locally affine variety if +F is a polynomial singular foliation on RN or, CN as in Theorem 7.2.5. Moreover, +1. independent of the choice of a geometric resolution, as in Theorem 7.2.2, +2. the projection Ă +Mi Ñ M is proper and onto, as in Theorem 7.2.2, +3. F lifts to a singular foliation on Ă +Mi, and it is one-to-one to Mreg, as in Proposition 7.2.6. +For i “ 0 the same conclusion holds for +σ0 : M ÝÑ Grℓ0pTMq +m ÞÝÑ TmF. +Let us give some examples. +Example 7.2.9. Notice that Gr´ℓpCℓq “ tptu, so that if dpiq is into on an open dense subset, the +construction degenerates, in this case Gr´ℓpCℓq ˆ M » M +1. If F is a projective singular foliation, then Ă +M1 » M: because ΓpE´1q » F and E´1 +ρÑ TM is +injective on the open dense subset Mreg, i.e. ℓ0 “ rkpE´1q. Hence Gr´ℓ0pE´1q » M. +2. If F admits open dense orbits, Ă +M0 » M, since Gr´ dim MpTMq » M. +3. Let W Ď M “ Cd be an affine variety generated by (independent functions) ϕ, ψ P Crx1, . . . , xds, +i.e. IW “ xϕ, ψy. Let FW be the singular foliation of (polynomial) vector fields vanishing on W. +At regular points x P Mreg, TxFW “ d, hence + +CHAPTER 7. ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION +122 +(a) Ă +M0 » M. +(b) Also, +• E´1 “ pCd ‘ Cdq ˆ M, ρ: ei ‘ 0 ÞÑ ϕ B +Bxi and 0 ‘ ei ÞÑ ´ψ B +Bxi , for all i “ 1, . . . , d. +• E´2 “ Cd ˆ M, and dp2q : ei ÞÑ ψ +K ei ‘ ϕ +K ei with, K “ gcdpϕ, ψq +impdp2q +x q “ ker ρx “ +B ψpxq +Kpxqu ‘ ϕpxq +Kpxqu, with u P Cd +F +For a convergent sequence pynq in MzW, pker ρynq converges if and only if r ψpynq +Kpynq : ϕpynq +Kpynqs +converges in P1pCq. In that case, Ă +M1 is the closure of the graph tpy, rψpyq: ϕpyqsq, y P +MzWu, which is the blow up of Cd along W. +• Ă +M2 » M since dp2q +x +is injective at regular points, so that Gr´dpE´2q » M. +4. Let W be the affine variety defined by ϕ P Crx1, . . . , xds. Consider the singular foliation Fϕ “ +tX P XpCdq | Xrϕs “ 0u. +For every y P Cd, pTyFϕqK “ x∇yϕy. +For convergent sequence +yn ÝÑ +nÑ`8 y P W. The sequence impρynq converges if and only if ∇ynϕ converges in Gr´pd´1qpCdq, +that is +” +Bϕ +Bx1 pynq: ¨ ¨ ¨ : +Bϕ +Bxd pynq +ı +converges in the projective space Pd´1pCq. Therefore, Ă +M0 is the +closure of the graph of the map, y ÞÑ py, +” +Bϕ +Bx1 pyq: ¨ ¨ ¨ : +Bϕ +Bxd pyq +ı +q which is the blow up of Cd along +the singular locus of W. +Let us conclude this chapter by an open question. Can we desingularize a singular affine variety +W Ď Cd by applying the constructions above to the singular foliation F “ XpWq of vector fields +tangent to W? We should then understand Ă +W0 as the Nash modification of W [LU81]. The meaning +of Ă +W1 is unclear. We would like to relate the Ă +Wi’s and the rFi together and go further in the universal +Lie 8-algebroid, e.g. to understand the role of the 3-ary bracket in this procedure. +Conclusion: +In this chapter, we recall the notion of isotropy Lie (8)-algebra of a singular foliation (and of +a Lie-Rinehart algebra). +Then, we use the geometric resolution (i.e. the resolution on which the universal Lie 8- +algebroid is built) to recover several notions of resolution of singularities: one being due to +Nash and a second one to Mohsen. + +CHAPTER 8 +Symmetries of singular foliations through Lie 8-algebroids +This chapter is one of the main application of results in Chapter 8.3.1 and Section 6.2. These results +are taken from my article [Lou22]. +We introduce the notion of weak symmetry actions of a lie algebra g on a singular foliation F and +study the interaction of those on the universal Lie 8-algebroids of F. Also, in the subsequent chapter, +we apply these results to the problem of extending a strict Lie algebra action on a sub-affine variety +on the ambient space. +Convention 8.0.1. Throughout this chapter, M stands for a smooth or complex manifold, or an +affine variety over C. We denote the sheaf of smooth or complex, or regular functions on M by O +and the sheaf of vector fields on M by XpMq, and Xrfs stands for a vector field X P XpMq applied to +f P O. Also, K stands for R or C. +8.1 +Definitions and examples +Definition 8.1.1. Let F Ă XpMq be a singular foliation over M. +• A diffeomorphism φ: M ÝÑ M is said to be a symmetry of F, if φ˚pFq “ F. +• A vector field X P XpMq is said to be an infinitesimal symmetry of F, if rX, Fs Ă F. The Lie +algebra of infinitesimal symmetries of F is denoted by spFq. +In particular, F Ă spFq, since rF, Fs Ă F. +Proposition 8.1.2. [AS09, GY18] Let M be a smooth or complex manifold. The flow of an infinites- +imal symmetry of F, if it exists, is a symmetry of F. +As we will see later, one of the consequences of our future results is that any symmetry X P spFq +of a singular foliation F admits a lift to a degree zero vector field on any universal NQ-manifold over +F that commutes with the homological vector field Q. This allows us to have an alternative proof and +123 + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS124 +interpretation of Proposition 8.1.2 (see Section 8.2). +Let pg, r¨ , ¨sgq be a Lie algebra over K “ R or C, depending on the context. From now on and in +the sequel g is concentrated in degree ´1. +Definition 8.1.3. A weak symmetry action of the Lie algebra g on a singular foliation F on M is a +K-linear map ϱ: g ÝÑ XpMq that satisfies: +• @ x P g, rϱpxq, Fs Ď F, +• @ x, y P g, ϱprx, ysgq ´ rϱpxq, ϱpyqs P F. +When x ÞÝÑ ϱpxq is a Lie algebra morphism, we speak of strict symmetry action of g on F. There +is an equivalence relation on the set of weak symmetry actions which is defined as follows: two weak +symmetry actions, ϱ, ϱ1 : g ÝÑ XpMq are said to be equivalent if there exists a linear map ϕ: g ÝÑ F +such that ϱ ´ ϱ1 “ ϕ. +Remark 8.1.4. It is important to notice that when F is a regular foliation and M{F is a manifold, +any weak symmetry action of a Lie algebra g on F induces a strict action of g over M{F. Definition +8.1.3 is a way of extending this idea to all singular foliations. +Here is a list of some examples. +Example 8.1.5. Let π: M ÝÑ N be a submersion. Since any vector field on N comes from a π- +projectable vector field on M, therefore any Lie algebra morphism g ÝÑ XpNq can be lifted to a weak +symmetry action g ÝÑ XpMq on the regular foliation Γpker dπq, and any two such lifts are equivalent. +Furthermore, any weak action of a Lie algebra g on a singular foliation F on N can be lifted to a +class of weak symmetry actions on the pull-back foliation π´1pFq, (see Definition 1.9 in [AS09]). +Example 8.1.6. Let F be a singular foliation on M. For any point m P M, consider gm “ Fpmq +ImF the +isotropy Lie algebra of F at m (see Definition 7.0.2). Let us denote its Lie bracket by r¨ , ¨sgm. +1. Consider ϱ: gm Ñ Fpmq Ă XpMq a section of the projection map, +ImF� � +� Fpmq +� � gm +ϱ +� +(8.1) +Then, rϱpxq, ImFs Ă ImF and ϱprx, ysgmq ´ rϱpxq, ϱpyqs P ImF. Hence, the map ϱ: gm Ñ XpMq +is a weak symmetry action of the singular foliation ImF. A different section ϱ1 of the projection +map yields an equivalent weak symmetry action of gm on ImF. An obstruction class for having +a strict symmetry action equivalent to ϱ will be given later in Section 9. +2. In particular, for k ě 1, let us denote by gk +m the isotropy Lie algebra of the singular foliation +Ik +mF at m. Any section ϱk : gk +m ÝÑ XpMq of the projection map +Ik`1 +m +F� � +� IkF +� � gk +m +ϱk +� +(8.2) +is a weak symmetry action of the Lie algebra gk +m on the singular foliation Ik`1 +m +F. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS125 +Example 8.1.7. Let pA, r¨ , ¨sA , ρq be an almost Lie algebroid on a smooth manifold M, and let +F “ ρpΓpAqq. Assume there exists p P M such that A|p is a Lie algebra and ρ|p “ 0. The question of +finding a map A|p ÝÑ ΓpAq such that the composition +A|p ÝÑ ΓpAq +ρ +ÝÑ XpMq +(8.3) +be a Lie algebra morphism, i.e., to ask whether the singular foliation F comes from the transformation +Lie algebroid of A|p, can be formulated as a weak symmetry action of A|p on the singular foliation +IpF as follows: Consider a linear map +A|p ÝÑ ΓpAq +(8.4) +a ÞÝÑ ra +such that for all a P A|p and ra a section of A such that ra|p “ a. It is easily checked that the map in +(8.4) is not a Lie algebra morphism, but it satisfies +Ć +ra, bsA|p ´ +” +ra,rb +ı +A P IpΓpAq +and +” +Ą +A|p, IpΓpAq +ı +Ă IpΓpAq. +Therefore, the map a ÞÝÑ ρpraq is a weak symmetry action of A|p on IpF. Notice that the isotropy Lie +algebra g1 +p “ IpF +I2pF of the singular foliation IpF, is Abelian. In Chapter 9, we show that in this case, +the obstruction of having a Lie algebra morphism in (8.3) is a cocycle of Chevalley-Eilenberg of A|p +valued in g1 +p. +The next example comes from [LGR22], and follows the same patterns as in Examples 8.1.5 and +8.1.6. It is based on the notion of Ehresmann connection. Let us first recall quickly this concept +for the sake of completeness and clarity. There are several equivalent manners of viewing Ehresmann +connections (see [Iva93] Section 9, Page 76), the most relevant approach in this context is the following: +An Ehresmann connection on a smooth fiber bundle1 π: E Ñ M is a vector subbundle H of TE, such +that TE “ H ‘V , where V :“ tξ P TE | π˚pξq “ 0u is called the "vertical bundle" whose fiber at e P E +is Ve “ TepEπpeqq. The subbundle H is called the "horizontal bundle". Given a connection as defined +above, for every e P E, the linear map π˚ : TeE Ñ TπpeqM restricts to an isomorphism, He Ñ TπpeqM, +whose inverse TπpeqM Ñ He is called the horizontal lift. +Example 8.1.8. Let pM, Fq be a singular foliation on a smooth manifold M and L Ă M a leaf. Let +rL, Ms be a neighborhood of L in M equipped with some projection2 π: rL, Ms Ñ L. According to +[LGR22], upon replacing rL, Ms be a smaller neighborhood of L if necessary, there exists an Ehresmann +connection (that is a vector sub-bundle H Ă TrL, Ms with H ‘ kerpπ˚q “ TrL, Ms) which satisfies +that ΓpHq Ă F. Such an Ehresmann connection is called an Ehresmann F-connection and induces a +C8pLq-linear section ϱH : XpLq Ñ Fproj of the surjection Fproj Ñ XpLq, where Fproj stands for vector +fields of F π-projectable on elements of XpLq. The section ϱH is a weak symmetry action of XpLq +on the transverse foliation T :“ Γpker π˚q X F. When the Ehresmann connection H is flat, ϱH is +bracket-preserving, and defines a strict symmetry of XpLq on the transverse foliation T . +1this generalizes connections to arbitrary fiber bundle π : E Ñ M, here the total space E is itself a smooth manifold, +and has its own tangent bundle TE Ñ E. +2such projection comes with the Tubular neighborhood theorem [dS01]. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS126 +Example 8.1.9. Consider, for a fixed k, the singular foliation Fk :“ Ik +0 XpRdq generated by all vector +fields vanishing to order k at the origin. The action of the Lie algebra glpRdq on Rd which is given by, +glpRq ÝÑ XpRdq, paijq1ďi,jďd ÞÝÑ +ÿ +1ďi,jďd +aijxi +B +Bxj +is a strict symmetry of Fk. +Example 8.1.10. Let ϕ :“ pϕ1, . . . , ϕrq be a r-tuple of homogeneous polynomial functions in d +variables over K. Consider the singular foliation Fϕ (see [LGL22b] Section 3.2.1) which is generated +by all polynomial vector fields X P XpKdq that satisfy Xrϕis “ 0 for all i P t1, . . . , ru. The action +K Ñ XpKdq, λ ÞÑ λÝÑ +E , is a strict symmetry of Fϕ. Here ÝÑ +E stands for the Euler vector field. +Example 8.1.11. Let W Ă Cd be an affine variety and IW Ă Crx1, . . . , xds its corresponding ideal. +Let us denote by XpWq :“ DerpCrx1, . . . , xds{IW q the Lie algebra of vector field of W. Any vector +field on W can be extended (not unique) to a vector field on Cd (see Section 5.1.1). +Let FW :“ IW XpCdq the singular foliation made of vector fields vanishing on W. Since every +vector field on W can be extended to a vector field on Cd tangent to W. Any Lie algebra morphism +ϱ: g ÝÑ XpWq extends to a linear map rϱ: g ÝÑ XpCdq that makes this diagram commutes +XpCdq +�� +g +rϱ +� +ϱ � XpWq +This extension rϱ is a weak symmetry action of g on FW over the ambient space Cd. Two different +extensions yield equivalent symmetry actions. +8.2 +A Lie 8-morphism lifting a weak symmetry of a foliation +We recall that O is the sheaf of smooth/complex functions on a smooth/complex manifold M, or the +algebra of regular functions on an affine variety over C. We refer the reader to the Chapters 6 and 4 +for the notion of (universal) Lie 8-algebroid of a singular foliation. We denote them by pE, Qq and +their functions by E. +For the sake of clarity, let put this chapter in context, and fix some notations. +Let pg, r¨ , ¨sgq a Lie algebra and pE, Qq a Lie 8-algebroid over M. In the sequel, the Lie algebra +g is concentrated in degree ´1. The differential graded Lie algebra pXpEq, r¨ , ¨s , adQq of vector fields +on E is shifted by 1, i.e. a derivation of degree k in XkpEq is of degree k ´ 1 as an element of the +shifted space XkpEqr1s. The graded symmetric Lie bracket on XpEqr1s is of degree `1 and given on +homogeneous elements u, v P XpEqr1s as +tu, vu :“ p´1q|v|ru, vs. +In the sequel, we write pXpEqr1s, r¨ , ¨s , adQq instead of pXpEqr1s, t¨ , ¨u, adQq. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS127 +Let pS‚ +Kg, Qgq respectively pS‚ +KpXpEqr1sq, ¯Qq be the corresponding formulations in terms of co- +derivations of the differential graded Lie algebras pg, r¨ , ¨sgq and pXpEqr1s, r¨ , ¨s , adQq. Precisely, Qg is +the co-derivation defined by putting for every homogeneous monomial x1 ^ ¨ ¨ ¨ ^ xk P Sk +Kg, +Qgpx1 ^ ¨ ¨ ¨ ^ xkq :“ +ÿ +1ďiăjďk +p´1qi`j´1rxi, xjsg ^ x1 ^ ¨ ¨ ¨ pxi ¨ ¨ ¨ pxj ¨ ¨ ¨ ^ xk, +(8.5) +and ¯Q “ ¯Qp0q ` ¯Qp1q is the co-derivation of degree `1 where the only non-zero Taylor coefficients are, +¯Qp0q : S1 +KpXpEqr1sq +adQ +ÝÑ XpEqr1s and ¯Qp1q : S2 +KpXpEqr1sq +t¨ ,¨u +ÝÑ XpEqr1s. +The following is a particular case of Example 2.3.5, see also [LM94]. +Definition 8.2.1. A Lie 8-morphism3 Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq is a graded coalgebra +morphism ¯Φ: pS‚ +Kg, Qgq ÝÑ pS‚ +K pXpEqr1sq , ¯Qq of degree zero which satisfies, +¯Φ ˝ Qg “ ¯Q ˝ ¯Φ. +(8.6) +In order words, it is the datum of degree zero linear maps +´ +¯Φk : Sk`1 +K +g ÝÑ X´kpEqr1s +¯ +kě0 that satisfies +ÿ +1ďiăjďn`2 +p´1qi`j´1 ¯Φnprxi, xjsg, x1, . . . , pxij, . . . , xn`2q “ rQ, ¯Φn`1px1, . . . , xn`2qs ` +ÿ +i ` j “ n +i ď j +σ P Si`1,j`1 +ϵpσqr¯Φipxσp1q, . . . , xσpi`1qq, ¯Φjpxσpi`2q, . . . , xσpn`2qqs +(8.7) +where pxij means that we take xi, xj out of the list. When there is no risk of confusion, we write Φ for +¯Φ. +Remark 8.2.2. It is important to notice that: +1. Definition 8.2.1 and Definition 6.1.14 are compatible when M “ tptu. Therefore, morphisms in +both sense match. +2. In [MZ12], Definition 8.2.1 corresponds to the definition of actions of a Lie 8-algebras of finite +dimension on Lie 8-algebroids of finite rank. Here we only have a Lie algebra. In contrast to +theirs, we do not assume that g is finite dimensional. +Remark 8.2.3. It follows from the axioms (8.7) that for all x, y P g, rQ, Φ0pxqs “ 0 and +Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqs “ rQ, Φ1px, yqs. +(8.8) +In particular, if the homological vector field Q vanishes at some point m P M, then the map x ÞÝÑ +pP P XpEq, P|m ÞÑ rΦ0pxq, Ps|mq endows the vector space XpEq|m » pSpE˚q b Eq|m with a g-module +structure. +Moreover, the restriction of the map Φ1 : ^2 g ÝÑ X´1pEq|m at m is a 2-cocycle of +Chevalley-Eilenberg. +Remark 8.2.4. Let pE, Qq be a Lie 8-algebroid and F its basic singular foliation. +Any Lie 8- +morphism Φ: pg, r¨ , ¨sgq ÝÑ pX‚pEqr1s, r¨ , ¨s , adQq gives a weak symmetry action of g on F. If Q|m “ 0 +for some point m P M, the g-action defined in Remark 8.2.3, is independent of the equivalence class +of the weak symmetry action. +3Here, we use "ù" to emphasize that Φ is not a DGLA morphism. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS128 +The following lemma explains what the 0-Taylor coefficient of a Lie 8-morphism as in Definition +8.2.1 induces on the linear part of pE, Qq. More details will be given in Proposition 8.2.10 and Remark +9.1.1. +Lemma 8.2.5. The 0-th Taylor coefficient Φ0 : g ÝÑ X0pEq induces +1. a linear map ϱ: g ÝÑ XpMq, x ÞÝÑ pϱpxqrfs :“ Φ0pxqrfs, f P Oq and +2. a linear map x P g ÞÝÑ ∇x P Der0pEq, i.e. for each x P g, ∇x : E ÝÑ E is a degree zero map +that satisfies +∇xpfeq “ f∇xpeq ` ϱpxqrfse, for f P O, e P ΓpEq. +such that +xΦ0pxqp0qpαq, ey “ ϱpxqrxα, eys ´ xα, ∇xpeqy, for all α P ΓpE˚q, e P ΓpEq. +(8.9) +Φ0pxqp0q stands for the polynomial-degree zero of Φ0pxq. +Proof. Using Lemma 6.1.7, we have for every x P g, and e P ΓpEq, +rΦ0pxq, ιesp´1q “ ι∇xe, +for some K-bilinear map ∇x : ΓpE´‚q ÝÑ ΓpE´‚q that depends linearly on x P g and that satisfies +∇xpfeq “ f∇xpeq ` ϱpxqrfse, for f P O, e P ΓpEq. +(8.10) +To see (8.10), compute rΦ0pxq, ιfesp´1q: +ι∇xpfeq “ rΦ0pxq, ιpfeqsp´1q +“ Φ0pxqrfsιe ` frΦ0pxq, ιesp´1q +“ ιϱpxqrfse`∇xe. +In particular, one has for all α P ΓpE˚q, e P ΓpEq, +xΦ0pxqp0qpαq, ey “ Φ0pxqp0qrxα, eys ´ rΦ0pxqp0q, ιesp´1qpαq +“ ϱpxqrxα, eys ´ xα, ∇xpeqy. +8.2.1 +Homotopies +The following is a particular case of Definition 2.3.14 of Section 2.3.2. We rewrite it in this special +context for the sake of clarity. +Definition 8.2.6. Let ¯Φ, ¯Ψ: pS‚ +Kg, Qgq ù +` +S‚ +KpXpEqr1sq, ¯Q +˘ +be Lie 8-morphisms. We say ¯Φ, ¯Ψ are +homotopic over the identity of M if the following conditions hold: +1. there a piecewise rational continuous path t P ra, bs ÞÑ Ξt : pS‚ +Kg, Qgq ù +` +S‚ +KpXpEqr1sq, ¯Q +˘ +made +of Lie 8-morphisms that coincide with ¯Φ and ¯Ψ at t “ a and b, respectively, + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS129 +2. and a piecewise rational path t P ra, bs ÞÑ Ht of Ξt-co-derivations of degree ´1 such that +dΞt +dt “ ¯Q ˝ Ht ` Ht ˝ Qg. +(8.11) +Remark 8.2.7. Homotopy equivalence in the sense of the Definition 8.2.6 is an equivalence rela- +tion, and it is compatible with composition of Lie 8-morphisms. +Also, we "glue" infinitely many +equivalences, as in Lemma 2.3.22. +Convention 8.2.8. In the sequel, Qg and ¯Q will be in implicit. +Remark 8.2.9. Definition 8.2.6 is slightly more general than the equivalence relation [MZ12]. In +[MZ12], it is explained that Lie 8-oid morphisms are Maurer-Cartan elements in some Lie 8-algebroid +g‘E of certain form, and they define equivalence as gauge-equivalence of the Maurer-Cartan elements. +This gauge equivalence corresponds to homotopies as above for which all functions are smooth. Also, +we do not require nilpotence unlike in Definition 5.1 of [MZ12]. Last, we do not assume g to be of +finite dimension. +Proposition 8.2.10. Let g be a Lie algebra and pE, Qq a Lie 8-algebroid over M. +1. Any Lie 8-morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq induces a weak symmetry action +of g on the basic singular foliation F “ ρpΓpE´1qq of pE, Qq. +2. Homotopic Lie 8-morphisms Φ, Ψ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq induce equivalent weak +symmetry actions ϱa, ϱb of g on the basic singular foliation F. +Proof. Item 1. is a consequence of Remark 8.2.3. Indeed, take ϱ: g ÝÑ XpMq as in Lemma 8.2.5 (1.) +We claim that ϱ is a weak symmetry action of g on F: Let x, y P g, and e P ΓpE´1q and f P O. +• rΦ0pxq, Qs “ 0 entails, +A +Φ0pxqp0q ” +Qp1qpfq +ı +, e +E +“ +A +Qp1q ´ +Φ0pxqp0qrfs +¯ +, e +E +ϱpxqrxQrfs, eys ´ xQrfs, ∇xpeqy “ ρpeqrϱpxqs, +(by Lemma 8.2.5 (2.)) +ϱpxqrρpeqsrfs ´ ρp∇xpeqqrfs “ ρpeqrϱpxqs +By consequence, rϱpxq, ρpeqs “ ρp∇xpeqq P F. Therefore, rϱpxq, Fs Ď F. +• By Lemma 6.1.7, there exists a skew-symmetric linear map η: ^2 g ÝÑ ΓpE´1q such that +Φ1px, yqp´1q “ ιηpx,yq. Therefore, the polynomial-degree zero of Equation (8.8) evaluated at an +arbitrary function f P O yields: +Φ0prx, ysgqp0qpfq ´ rΦ0pxq, Φ0pyqsp0qpfq “ rQ, Φ1px, yqsp0qpfq +ùñ Φ0prx, ysgqpfq ´ +” +Φ0pxqp0q, Φ0pyqp0qı +pfq “ +” +Qp1q, Φ1px, yqp´1qı +pfq +ùñ ϱprx, ysgqrfs ´ rϱpxq, ϱpyqsrfs “ +” +Qp1q, ιηpx,yq +ı +pfq +“ ρpηpx, yqqrfs. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS130 +Since f is arbitrary, this proves item 1. Using Proposition 2.3.17, Φ „ Ψ implies for x P g that +Ψpxq ´ Φpxq “ ¯Q ˝ Hpxq `  + +H ˝ Qgpxq +“ rQ, Hpxqs +(8.12) +with H : g ÝÑ X´1pEq a linear map. Let β : g ÝÑ ΓpE´1q be a linear map such that Hpxqp´1q “ +ιβpxq. Taking the polynomial-degree zero of both sides in Equation (8.12) and evaluating at f P O +we obtain that +pϱapxq ´ ϱbpxqq rfs “ +” +Qp1q, Hpxqp´1qı +“ +” +Qp1q, ιβpxq +ı +rfs “ ρpβpxqqrfs. +Since f is arbitrary, this proves item 2. +Proposition 8.2.10 tells us that Lie 8-morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq induces +weak symmetry action on the base manifold M. The aim of the next section is to look at the opposite +direction. It responds to the following question: Do any weak symmetry action of a Lie algebra on a +singular foliation comes from a Lie 8-morphism? If so, can we extend uniquely? +Now we define what we call "lift" of a weak symmetry action of a Lie algebra g on a singular +foliation F to a Lie 8-algebroid pE, Qq over F. +Definition 8.2.11. Let F be a singular foliation over M and pE, Qq a Lie 8-algebroid over F. Consider +a weak symmetry action ϱ: g ÝÑ XpMq of g on F. +• We say that a Lie 8-morphism of differential graded Lie algebras +Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq +lifts the weak symmetry action ϱ to pE, Qq if for all x P g, f P O, Φ0pxqpfq “ ϱpxqrfs. +• When Φ exists, we say then Φ is a lift of ϱ on pE, Qq. +8.3 +Main statements +We now state the main theorem of this chapter. Proposition 8.2.10 showed that a Lie 8-morphism +between a Lie algebra g and a Lie 8-algebroid pE, Qq induces a weak action of g on the basic foliation. +In this section, we show that any weak symmetry action of a Lie algebra g on a singular foliation F +arises this way. +More precisely, +Theorem 8.3.1. Let F a be a singular foliation over a smooth manifold (or an affine variety) M and +g a Lie algebra. Let ϱ: g ÝÑ XpMq be a weak symmetry action of g on F. The following assertions +hold: +1. for any universal Lie 8-algebroid pE, Qq of the singular foliation F, there exists a Lie 8- +morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq that lifts ϱ to pE, Qq, + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS131 +2. any two such Lie 8-morphisms are homotopy equivalent over the identity of M, +3. any two such lifts of any two equivalent weak symmetry actions of g on F are homotopy equivalent. +Remark 8.3.2. Again, Lie 8-morphisms in item 1 of Theorem 8.3.1 are g-actions on pE, Qq in [MZ12]. +Remark 8.3.3. The item 1 in Theorem 8.3.1 means that there exists a linear map Φ0 : g ÝÑ X0pEq +such that +Φ0pxqrfs “ ϱpxqrfs, and rQ, Φ0pxqs “ 0, +@x P g, f P O. +(8.13) +This morphism is not a graded Lie algebra morphism, but there exist a linear map Φ1 : ^2g ÝÑ X´1pEq +such that for all x, y, z P g, +Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqs “ rQ, Φ1px, yqs. +(8.14) +Also, +Φ1 prx, ysg, zq ´ rΦ0pxq, Φ1py, zqs` ö px, y, zq “ rQ, Φ2px, y, zqs +(8.15) +for some linear map ^3g ÝÑ X´2pEq. +These sets of compatibility conditions continue to higher +multilinear maps. +Corollary 8.3.4. Any symmetry X P XpMq of the singular foliation F can be lifted to a degree zero +vector field Z P X0pEq that commutes with Q, i.e. such that rZ, Qs “ 0. +Proof. To construct Z, it suffices to apply Theorem 8.3.1 for g “ R and take Z to be the image of 1 +through Φ0 : R ÝÑ X0pEq. +Remark 8.3.5. In particular, Corollary 8.3.4 has the following consequences: +1. for any admissible t, the flow ΦZ +t : E ÝÑ E of Z induces an isomorphism of vector bundles +E´1 ÝÑ E´1. Since rQ, Zs “ 0, the following diagram commutes, +ΓpE´1q +ρ +� +pΦZ +t qp0q +� ΓpE´1q +ρ +� +XpMq +pϕX +t q˚ +� XpMq +where φX +t is the flow of X at t. +2. Consequently, for any open set U Ă M which is stable under ϕX +t , there exists an invertible +matrix Mt +X with coefficients in OpUq that satisfies +` +φX +t +˘ +˚ +¨ +˚ +˚ +˝ +X1 +... +Xn +˛ +‹‹‚“ Mt +X +¨ +˚ +˚ +˝ +X1 +... +Xn +˛ +‹‹‚, +for some generators X1, . . . , Xn of F over U. As announced earlier, we recover Proposition 8.1.2, +that is, +` +φX +t +˘ +˚ pFq “ F. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS132 +Let pE, Qq and pE1, Q1q be two universal Lie 8-algebroids of F. A direct consequence of Ricardo +Campos’s Theorem 4.1 in [Cam19] is that the differential graded Lie algebras pX‚pEqr1s, r¨ , ¨s , adQq +and +` +X‚pEqr1s, r¨ , ¨s , adQ1˘ +are homotopy equivalent over the identity of M. This leads to the following +statement. +Corollary 8.3.6. Let ϱ: g ÝÑ XpMq be a weak symmetry action of a Lie algebra g on F. Then, there +exist Lie 8-morphisms, Φ: g ù pX‚pEqr1s, r¨ , ¨s , adQq and Ψ: g ù +` +X‚pE1qr1s, r¨ , ¨s , adQ1˘ +that lift +ϱ, and Φ, Ψ make the following diagram commute up to homotopy +g +Φ +� +Ψ +� +pX‚pEqr1s, r¨ , ¨s , adQq � +„ +� ` +X‚pE1qr1s, r¨ , ¨s , adQ1˘ +. +(8.16) +Proof. The composition of Φ with the horizontal map in the diagram (8.16) is a lift of the action ϱ. +It is necessarily homotopy equivalent to Ψ by item p2q in Theorem 8.3.1. +8.3.1 +Proof of 8.3.1 +This section is devoted to the proof of the main results, i.e. Theorem 8.3.1. +Let F be a singular foliation, and pE, Qq a universal Lie 8-algebroid of F. We start with the +following lemma. +Lemma 8.3.7. For every weak symmetry Lie algebra action of g on F there exists a linear map, +Φ0 : g Ñ X0pEq, such that rQ, Φ0pxqs “ 0 and Φ0pxqrfs “ ϱpxqrfs for all x P g, f P O. +Proof. For x P g, let y +ϱpxq P X0pEq be any arbitrary extension of ϱpxq P spFq to a degree zero vector +field on E. Since ϱpxq is a symmetry of F, the degree `1 vector field r y +ϱpxq, Qs is also a longitudinal +vector field on E, see Example 6.2.6 item 3. In addition, r y +ϱpxq, Qs is a adQ-cocycle. By item 1 of +Theorem 6.2.10, there exists a vertical vector field Y pxq P apEq of degree zero such that +rQ, Y pxq ` y +ϱpxqqs “ 0. +(8.17) +Let us set for x P g, Φ0pxq :“ Y pxq ` y +ϱpxq. +By construction, we have, rQ, Φ0pxqs “ 0 and +Φ0pxqrfs “ ϱpxqrfs for all x P g, f P O. +We will need the following lemma. +Lemma 8.3.8. Assume pE, Qq is a universal Lie 8-algebroid over M. Let ¯Φ: pS‚ +Kg, Qgq ÝÑ pS‚ +KXpEqr1s, ¯Qq +be a coalgebra morphism which is a Lie 8-morphism up to polynomial-degree n ě 0, i.e. +`¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ +˘piq “ 0 for all integer i P t0, . . . , nu. +Then, ¯Φ can be lengthened to a 8-morphism up to polynomial-degree n ` 1. +Proof. For convenience, we omit the variables x P S‚ +Kg. The identity, +¯Q ˝ +`¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ +˘ +` +`¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ +˘ +˝ Qg “ 0 + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS133 +taken in polynomial-degree n ` 1 yields, +0 “ +` ¯Q ˝ p¯Φ ˝ Qg ´ ¯Q ˝ ¯Φq +˘pn`1q “ rQ, p¯Φ ˝ Qg ´ ¯Q ˝ ¯Φqpn`1qs, +since Qp0q +g +“ 0 and +`¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ +˘piq “ 0 for i P t0, . . . , nu. It is clear that for all n ě 0 the map +`¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ +˘pn`1q : Sn`2 +K +g ÝÑ X´npEqr1s take value in vertical vector fields on E. By virtue of +Lemma 6.2.11 there exists a map ζ : Sn`2 +K +g ÝÑ X´n´1pEqr1s such that +rQ, ¯Φpn`1q ` ζs “ ¯Φpnq ˝ Qp1q +g +´ ¯Qp1q ˝ ¯Φpnq. +(8.18) +By redefining the polynomial-degree n`1 of ¯Φ as ¯Φpn`1q :“ ¯Φpn`1q`ζ. One obtains a Lie 8-morphism +up to polynomial-degree n ` 1. The proof continues by recursion. +Proof of Theorem 8.3.1. Let us show Item 1. Note that Lemma 8.3.7 gives the existence of a linear +map Φ0 : g ÝÑ X0pEq such that, rQ, Φ0pxqs “ 0 for all x P g. For x, y P g, consider +Λpx, yq “ Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqs. +(8.19) +Since ϱprx, ysgq ´ rϱpxq, ϱpyqs P F for all x, y P g, and since ρ: ΓpE´1q ÝÑ F surjective, we have +ϱprx, ysgq ´ rϱpxq, ϱpyqs “ ρ pηpx, yqq for some element ηpx, yq P ΓpE´1q depending linearly on x and y. +Now we consider the vertical vector field of degree ´1, ιηpx,yq P X´1pUFq which is defined on ΓpE˚q as: +ιηpx,yqpαq :“ xα, ηpx, yqy for all α P ΓpE˚q, +and extended it by derivation on the whole space. For every f P O, +` +Λpx, yq ´ rQ, ιηpx,yqs +˘ +pfq “ pϱprx, ysgq ´ rϱpxq, ϱpyqs ´ ρpηpx, yqq rfs +(by definition of Φ0) +“ 0 +(by definition of η) +It is clear that Λpx, yq ` rQ, ιηpx,yqs is a adQ-cocycle. +Also, +` +Λpx, yq ` rQ, ιηpx,yqs +˘p´1q: for every +α P ΓpE˚q, +rQ, ιηpx,yqsp´1qpαq “ rQp0q, ιηpx,yqspαq “ (((((((( +( +Qp0qrxα, ηpx, yqys ` (((((((( +( +xQp0qrαs, ηpx, yqy “ 0, +where the first term (resp. the second term) is cancelled by O-linearity of Qp0q (resp. for degree reason). +Hence, by Corollary 6.2.11 and Remark 6.2.12, the degree zero vector field Λpx, yq ` rQ, ιηpx,yqs is of +the form rQ, Υpx, yqs for some vertical vector field Υpx, yq P X´1pEq of degree ´1 with Υpx, yqp´1q “ 0. +For all x, y P g, we define the Taylor coefficient Φ1 : ^2 g ÝÑ XpEq as Φ1px, yq :“ Υpx, yq ` ιηpx,yq. By +construction, we have the following relation, +Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqs “ rQ, Φ1px, yqs, @x, y P g +(8.20) +Consider for x, y, z P g, +ϑpx, y, zq “ Φ1 prx, ysg, zq ´ rΦ0pxq, Φ1py, zqs` ö px, y, zq. +(8.21) +Here, ö px, y, zq stands for circular permutation of x, y and z with Koszul sign. For degree reason +ϑpx, y, zq is O-linear. Moreover, ϑpx, y, zq is a adQ-cocycle: +rQ, Φ1prrx, ysg, zsgqs ` ö px, y, zq “ ´ rΦ0 prx, ysgq , Φ0pzqs ` ö px, y, zq +“ rrΦ0pzq, Qs, Φ1px, yqs ´ rrΦ1px, yq, Φ0pzqs, Qs` ö px, y, zq +“ rQ, rΦ0pxq, Φ1py, zqss` ö px, y, zq. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS134 +Here, we have used the fact that rQ, Φ0pxqs “ 0 for all x P g, and the Jacobi identity for the Lie +brackets r¨ , ¨sg and r¨ , ¨s. +By Corollary 6.2.11, there exists a derivation of degree ´2 denoted by +Φ2px, y, zq P X´2pEqr1s that satisfies, +ϑpx, y, zq “ rQ, Φ2px, y, zqs. +(8.22) +So far, in the construction of the Lie 8-morphism, we have shown the existence of a Lie 8- +morphism ¯Φ: S‚ +Kg ÝÑ S‚ +K pXpEqr1sq up to polynomial-degree 2 that is p¯Φ ˝ Qgqpiq “ p ¯Q ˝ ¯Φqpiq with +i “ 0, 1, 2. The proof continues by recursion or by applying directly Lemma 8.3.8. This proves the +part 1. of the theorem. +Before proving item 3 of Theorem 8.3.1 we will need the following lemma. For convenience, we +sometimes omit the variables in g. +Lemma 8.3.9. For any two Lie 8-morphisms Γ, Ω: pS‚ +Kg, Qgq ù pS‚ +KpXpEqr1sq, ¯Qq which coincide +up to polynomial-degree n ě 1, i.e. Γpiq “ Ωpiq, for 0 ď i ď n, their difference in polynomial-degree +n ` 1, namely, +Γpn`1q ´ Ωpn`1q : Sn`2 +K +g ÝÑ X´n´1pEqr1s +is valued in adQ-coboundary. +Proof. Indeed, a direct computation yields +¯Q ˝ pΓ ´ Ωq “ pΓ ´ Ωq ˝ Qg ùñ ¯Qp0q ˝ pΓ ´ Ωqpn`1q ´ ppΓ ´ Ωq ˝ Qgqpn`1q +looooooooooomooooooooooon +“0 +“ 0 +ùñ rQ, Γpn`1q ´ Ωpn`1qs “ 0 +ùñ Γpn`1q ´ Ωpn`1q “ rQ, Hpn`1qs +(by item 1 of Theorem 6.2.10) +for some linear map Hpn`1q : Sn`2 +K +g ÝÑ X´n´2pEqr1s. +Let us show item 2 of Theorem 8.3.1. Let Φ, Ψ: g ÝÑ XpEqr1s be two different lifts of the action +g ÝÑ XpMq. We denote by ¯Φ, ¯Ψ: S‚ +Kg ÝÑ S‚ +KpXpEqr1sq the unique comorphisms given respectively +by the Taylor coefficients +$ +& +% +¯Φprq : Sr`1 +K +g Φr +ÝÑ X´rpEqr1s +¯Ψprq : Sr`1 +K +g Ψr +ÝÝÑ X´rpEqr1s +, for r ě 0. +(8.23) +For any x P g, the degree zero vector field Φ0pxq ´ Ψ0pxq P X0pEq is vertical. Moreover, we have, +rQ, Φ0pxq ´ Ψ0pxqs “ 0. By Corollary 6.2.11 there exists a vector field H0 P X´1pEq of degree ´1, +such that Ψ0pxq ´ Φ0pxq “ rQ, H0pxqs +g +Ψ0´Φ0 +� +H0 +� +X´1pEqr1s +adQ � X0pEqr1s +(8.24) +Consider the following differential equation +$ +& +% +dΞt +dt +“ ¯Q ˝ Ht ` Ht ˝ Qg, +t P r0, 1s +Ξ0 +“ ¯Φ +(8.25) + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS135 +where pΞtqtPr0,1s is as in Definition 8.2.6, and for t P r0, 1s, Ht is the unique Ξt-co-derivation where the +only non-zero polynomial-degree is Hp0q “ H0. Equation (8.25) gives a homotopy between ¯Φ and Ξ1. +When we consider the polynomial-degree zero component in Equation (8.25), one obtains +dΞp0q +t +dt +“ ¯Qp0q ˝ Hp0q +t +` Hp0q +t +˝ Qp0q +g +“ rQ, H0s +“ Ψ0 ´ Φ0 “ ¯Ψp0q ´ ¯Φp0q. +Therefore, Ξp0q +t +“ ¯Φp0q `tp¯Ψp0q ´ ¯Φp0qq, and ¯Φ „ Ξ1 with ¯Ψp0q “ Ξp0q +1 . Using Lemma 8.3.9, the image of +any element through the map ¯Ψp1q ´ Ξp1q +1 : S2 +Kg ÝÑ X´1pEqr1s is a adQ-coboundary. Thus, ¯Ψp1q ´ Ξp1q +1 +can be written as +¯Ψp1q ´ Ξp1q +1 +“ rQ, Hp1qs, +with Hp1q : S2 +Kg ÝÑ X´2pEqr1s. +(8.26) +Let us go one step further by considering the differential equation on r0, 1s given by +$ +& +% +dΘt +dt +“ ¯Q ˝ Ht ` Ht ˝ Qg +Ξ0 +“ ¯Ξ1 +(8.27) +Here Ht is the extension of Hp1q as the unique Θt-co-derivation where all its arities vanish except the +polynomial-degree 1 which is given by Hp1q. In polynomial-degree zero, pΘp0q +t qtPr0,1s is constant and +has value Θp0q +1 +“ ¯Ψp0q. In polynomial-degree one, we have, +dΘp1q +t +dt +“ ¯Qp0q ˝ Hp1q +t +“ rQ, Hp1qs “ ¯Ψp1q ´ Ξp1q +1 . +Hence, Θp1q +t +“ ¯Φp1q ` tp¯Ψp1q ´ Ξp1q +1 q with ¯Ψpiq “ Θpiq +1 +for i “ 0, 1. We then continue this procedure +by gluing all these homotopies as in the proof of item 2 of Theorem 4.2.4. We obtain at last a Lie +8-morphism Ω such that ¯Φ „ Ω and Ωpiq “ ¯Ψpiq for i ě 0. That means Ω “ ¯Ψ, therefore ¯Φ „ Ψ. This +proves item 2. of Theorem 8.3.1. +Let us prove item 3 of Theorem 8.3.1. Given two equivalent weak symmetry actions ϱ, ϱ1 of g on +a singular foliation F, i.e. ϱ, ϱ1 differ by a linear map g ÝÑ XpMq of the form x ÞÑ ρpβpxqq for some +linear map β : g ÝÑ ΓpE´1q. Let Φ, Φ1 : g ù pX‚pEqr1s, r¨ , ¨s , adQq be a lift into a Lie 8-morphism +of the action ϱ and ϱ1 respectively. One has for all x P g and f P O, +` +Φ0pxq ´ Ψ0pxq ´ rQ, ιϕpxqs +˘ +pfq “ ρpϕpxqqrfs ´ xQpfq, ϕpxqy +“ 0. +Since rQ, Φ0pxq ´ Ψ0pxq ´ rQ, ιϕpxqss “ 0, by Corollary 6.2.11 there exists a vertical derivation pHpxq P +X´1pEq of degree ´1 depending linearly on x P g such that +Φ0pxq ´ Ψ0pxq “ rQ, pHpxq ` ιϕpxqs. +Let Hpxq :“ pHpxq ` ιϕpxq, for x P g. The proof continues the same as for item 2 of Theorem 8.3.1 + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS136 +8.3.2 +Particular examples +We recall that for a regular foliation F on a manifold M, the Lie algebroid TF Ă TM, whose sections +form F, is a universal Lie 8-algebroid of F. Its corresponding Q-manifold is given by the leafwise De +Rham differential on Γp^‚T ˚Fq. +Example 8.3.10. Let F be a regular foliation on a manifold M. Any weak symmetry action g ÝÑ +XpMq, x ÞÝÑ ϱpxq, of F, can be lifted to Lie 8-morphism Φ: g ù pX‚pEqr1s, r¨ , ¨s , adQq given +explicitly as follows: +x P g ÞÝÑ Φ0pxq “ Lϱpxq P X0p^‚T ˚Fq +(8.28) +x ^ y P ^2g ÞÝÑ Φ1px, yq “ ιχpx,yq P X´1p^‚T ˚Fq +(8.29) +and +` +Φi : ^i`1 g ÝÑ X´ip^‚T ˚Fq +˘ +” 0, for all i ě 2, where χpx, yq :“ ϱprx, ysgq ´ rϱpxq, ϱpyqs for +x, y P g. Also, LX stands for the Lie derivative on multi-forms w.r.t X P XpMq, and ιX is the internal +product. +Example 8.3.11. Let F be a singular foliation on a manifold M together with a strict symmetry +action ϱ: g ÝÑ XpMq such that g Ă F. Hence, C8pMqg is a singular foliation which is the image of +the transformation Lie algebroid g ˆ M. The universality theorem (see [LLS20, LGL22b]) provides +the existence of a Lie 8-morphism ν : g ÝÑ UF. Let us call its Taylor coefficients νn : ^n`1 g ÝÑ +E´n´1, n ě 0. We may take for example the 0-th and 1-th Taylor coefficients of a Lie 8-morphism +that lifts ϱ as: +Φ0pxq :“ rQ, ιν0pxqs P X0pUFq, for x P g. +Φ1px, yq :“ rQ, ιν1px,yqsp´1q ´ +ÿ +kě0 +rrQ, ιν0pxqs, ιν0pyqspkq P X´1pUFq, for x, y P g. +Note that in this case the action ϱ is equivalent to zero, therefore by item 3 of Theorem 8.3.1 the Lie +8-morphism Φ is homotopic to zero. +8.4 +Lifts of weak symmetry actions and Lie 8-algebroids +In this section, g is a finite dimensional Lie algebra that we see as the trivial vector bundle over M +with fiber g. +The following theorem says that any lift of strict symmetry action of g on a singular foliation F +induces a Lie 8-algebroids with some special properties and vice versa. See [MZ12], Prop. 3.3, for a +proof of the following statement. +Proposition 8.4.1. Let pE, Qq be a Lie 8-algebroid over a singular foliation F. Any Lie 8-morphism +Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq with g of finite dimension induces a Lie 8-algebroid pE‘g, Q1q +with +Q1 :“ dCE ` Q ` +ÿ +kě1,i1,...,ik“1,...,dimpgq +1 +k!ξi1 d ¨ ¨ ¨ d ξikΦk´1pξi1, . . . , ξikq, +(8.30) +where dCE is the Chevalley-Eilenberg complex of g, and ξ1, . . . , ξdimpgq P g˚ is the dual basis of some +basis ξ1, . . . , ξdimpgq P g and for all k ě 0, Φk : Sk`1g ÝÑ X´kpEqr1s is the k-th Taylor coefficients of Φ. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS137 +In the dual point of view, (8.30) corresponds to a Lie 8-algebroid over the complex +¨ ¨ ¨ +ℓ1 +ÝÑ E´3 +ℓ1 +ÝÑ E´2 +ℓ1 +ÝÑ g ‘ E´1 +ρ1 +ÝÑ TM +(8.31) +whose brackets satisfy +1. the anchor map ρ1 sends an element x ‘ e P g ‘ E´1 to ϱpxq ` ρpeq P ϱpgq ` TF, +2. the binary bracket satisfies +ℓ2 pΓpE´1q, ΓpE´1qq Ă ΓpE´1q +and +ℓ2pΓpE´1q, xq Ă ΓpE´1q, @ x P g +3. the g-component of the binary bracket on constant sections of g ˆ M is the Lie bracket of g. +Conversely, if there exists a Lie 8-algebroid pE1, Q1q whose underlying complex of vector bundles is of +the form (8.31) and that satisfies item 1, 2 and 3, then there is a Lie 8-morphism +Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq +which is defined on a given basis ξ1, . . . , ξd of g by: +Φk´1pξi1, . . . , ξikq “ pr ˝ r¨ ¨ ¨ rrQ1, ιξi1s, ιξi2s, . . . , ιξiks Ă XpEqr1s, k P N, +(8.32) +where pr stands for the projection map XpE1qr1s ÝÑ XpEqr1s. +Proof. We explain the idea of the proof. A direct computation gives the first implication. Conversely, +let us denote by Q1 the homological vector fields of Lie 8-algebroid whose underlying complex of +vector bundles is of the form (8.31). The map defined in Equation (8.32) is indeed a lift into a Lie +8-morphism of the weak symmetry action ϱ: +• It is not difficult to check that, for any ξ P g, one has rQ, Φ0pξqs “ 0. +• The fact that Φ defines a Lie 8-morphism can be found using Voronov trick [Vor04, Vor05], i.e, +doing Jacobi’s identity inside the null derivation +0 “ pr ˝ r¨ ¨ ¨ rrrQ1, Q1s, ιξi1s, ιξi2s, . . . , ιξiks. +(8.33) +A direct computation of Equation (8.33) falls exactly on the requirements of Definition 8.2.1. +Remark 8.4.2. Let us compute Equation (8.33) for a small number of generators (e.g k “ 2, 3) in +order to show how it works: from the identity +””“ +Q1, Q1‰ +, ιξi1 +ı +, ιξi2 +ı +“ 0, +one obtains by using twice the Jacobi identity the following relation, +“ +Q1, +““ +Q1, ξi1 +‰ +, ξi2 +‰‰ +´ +““ +Q1, ξi1 +‰ +, +“ +Q1, ξi2 +‰‰ +“ 0. +(8.34) +One should notice that rrQ1, ξi1s , ξi2s splits into two parts. One part where the Chevalley-Eilenberg +acts to give +““ +dCE, ξi1 +‰ +, ξi2 +‰ +“ ιrξi1,ξi2sg, while the other part is +““ +Q1 ´ dCE, ξi1 +‰ +, ξi2 +‰ +. Hence, by putting +them in Equation (8.34), afterwards projecting on X pS‚pE˚qq, we get +pr ˝ +” +Q1, ιrξi1,ξi2sg +ı +` pr ˝ rQ1, +”” +Q1 ´ dCE, ξi1 +ı +, ξi2 +ı +s ´ pr ˝ +““ +Q1, ξi1 +‰ +, +“ +Q1, ξi2 +‰‰ +“ 0. +From here, we deduce that +Φ0prξi1, ξi2sgq “ rQ, Φ1pξi1, ξi2qs ` rΦ0pξi1q, Φ0pξi2qs. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS138 +Here is an application of Theorem 8.4.1. +Corollary 8.4.3. +1. Let g be a Lie algebra and G its Lie group. +2. Let pM, Fq a singular foliation together with a weak symmetry action ϱ: g ÝÑ XpMq. +The following assertions hold: +1. On G ˆ M the C8pG ˆ Mq-module generated by +$ +& +% +pÐÝu , ϱpuqq +u P g +p0, Xq +X P F +(8.35) +is a singular foliation that we denote by pXpGq ˆϱ Fq. +2. If pE, d, ϱq is a geometric resolution of pM, Fq, then +¨ ¨ ¨ ÝÑ p˚E´3 +d +ÝÑ p˚E´2 +d +ÝÑ g ‘ p˚E´1 +ρ1 +ÝÑ TpG ˆ Mq +(8.36) +with ρ1pe, uq “ pÐÝu , ϱpuq ` ρpeqq, is a geometric resolution of pXpGqˆϱFq. Here, p: GˆM ÝÑ M +is the projection on M. +Let pE, Qq be a universal Lie 8-algebroid structure of pM, Fq. Let Φk : ^k`1 g ÝÑ X´kpEq be the +Taylor coefficients of a Lie 8-morphism g ù XpEq that lifts ϱ. Then +Q1 :“ dG +dR ` Q ` +ÿ +kě1,i1,...,ik“1,...,dimpgq +1 +k!ξi1 d ¨ ¨ ¨ d ξikΦk´1pξi1, . . . , ξikq, +(8.37) +with pξ1, . . . , ξdimpgqq, pξ1, . . . , ξdimpgqq be dual basis of g and g˚ +1. is a universal Lie 8-algebroid of pXpGq ˆϱ Fq +2. whose coefficients are left invariant for the action of G on G ˆ M given by g ¨ ph, mq :“ pgh, mq. +Conversely, a left invariant Lie 8-algebroid on (8.36) can be interpreted as Taylor coefficients of a lift +of ϱ. +8.4.1 +A more general statement of Proposition 8.4.1 +We end the chapter with a generalization of Proposition 8.4.1. +In the previous section, Proposition 8.4.1 is stated in the finite dimensional context, i.e. it needs g +to be finite dimensional and the existence of a geometric resolution for the singular foliation F. In this +section we prove that given a weak symmetry action of a Lie algebra g (may be of infinite dimensional) +on a Lie-Rinehart algebra F Ă XpMq (we do not require F being locally finitely generated), such Lie +8-algebroid described at the sections level of the complex (8.31) as Proposition 8.4.1 exists. +We state the following theorem in the context of singular foliations, but the same statement and +the same proof are valid word-by-word by replacing F by a Lie-Rinehart algebra. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS139 +Theorem 8.4.4. Let g be a (possibly infinite dimensional) Lie K-algebra and let ϱ: g ÝÑ XpMq be +a weak symmetry action of g on a singular foliation F. Let ppK´iqiě1, d, ρq be a free resolution of the +singular foliation F over O. The complex of trivial vector bundles over M +¨ ¨ ¨ +d +ÝÑ E´3 +d +ÝÑ E´2 +d +ÝÑ g ‘ E´1 +ρ1 +ÝÑ TM +(8.38) +where ΓpE´1q “ K´i, comes equipped with a Lie 8-algebroid structure +1. whose unary bracket is d and whose anchor map ρ1, sends an element x ‘ e P g ‘ E´1 to +ϱpxq ` ρpeq P ϱpgq ` TF, +2. the binary bracket satisfies +ℓ2 pΓpE´1q, ΓpE´1qq Ă ΓpE´1q +and +ℓ2pΓpE´1q, Γpgqq Ă ΓpE´1q, +3. the g-component of the binary bracket on constant sections of g ˆ M is the Lie bracket of g. +Remark 8.4.5. When we have ϱpgq X TmF “ t0u for all m in M, Equation (8.38) is a free resolution +of the singular foliation C8pMqϱpgq ` F and we can apply directly the Theorem 4.2.1. Otherwise, we +need to show there is no obstruction in degree ´1 while doing the construction of the brackets if the +result still needs to hold. +Proof. (of Theorem 8.4.4) The complex of Equation (8.38) being exact everywhere except in degree +´1 we cannot apply directly Theorem 2.1 in [LGL22b] but we can mimic the proof given for Theorem +2.1 in [LGL22b] to construct the higher brackets when there is no obstruction in degree ´1. For +convenience, let us denote R´1 :“ Γpgq ‘ ΓpE´1q and R´i :“ ΓpE´iq for i ě 2. Given a natural +number k ě 0, we consider the total complex +ˆ +z +Page +pkq +‚ pRq, D “ rd, ¨sRN +˙ +of the following bicomplex +... +... +... +Ò +Ò +Ò +¨ ¨ ¨ +Ñ +HomO +´Äk`1 R |´k´3, R´3 +¯ +dÑ +HomO +´Äk`1 R |´k´3, R´2 +¯ +dÑ +HomO +´Äk`1 R |´k´3, dR´2 +¯ +Ñ +0 +δ Ò +δ Ò +δ Ò +¨ ¨ ¨ +Ñ +HomO +´Äk`1 R |´k´2, R´3 +¯ +dÑ +HomO +´Äk`1 R |´k´2, R´2 +¯ +dÑ +HomO +´Äk`1 R |´k´2, dR´2 +¯ +Ñ +0 +δ Ò +δ Ò +δ Ò +¨ ¨ ¨ +Ñ +HomO +´Äk`1 R |´k´1, R´3 +¯ +dÑ +HomO +´Äk`1 R |´k´1, R´2 +¯ +dÑ +HomO +´Äk`1 R |´k´1, dR´2 +¯ +Ñ +0 +Ò +Ò +Ò +0 +0 +0 +"-3 column" +"-2 column" +"-1 column" +(8.39) +The map δ stands for the vertical differential which is defined for all Φ P HomO +´Äk`1 R, R +¯ +by +δpΦq pr1, . . . , rk`1q :“ Φ ˝ d pr1 d . . . d rk`1q, +@ r1, . . . , rk`1 P R, +where here d acts as an O-derivation on r1 d . . . d rk`1 P Äk R and the horizontal differential given +by +Φ ÞÑ d ˝ Φ. +Since the line the bicomplex is exact, the total complex +ˆ +z +Page +pkq +‚ pRq, D “ rd, ¨sRN +˙ +is also exact. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS140 +Construction of the 2-ary bracket: its construction is almost the same as in [LGL22b] we adapt +what has been done to our case. We first construct a 2-ary bracket on R´1 to extend on every degree. +For all k ě 1, let us denote by pep´kq +i +qiPIk a basis of ΓpE´kq. The set tXi “ ρpep´1q +i +q P F | i P I1u is a +set of generators of F. In particular, there exists elements ck +ij P O and satisfying the skew-symmetry +condition ck +ij “ ´ck +ji together with +rXi, Xjs “ +ÿ +kPI +ck +ijXk +@i, j P I1. +(8.40) +By definition of weak symmetry one has +rϱpξiq, ρpep´1q +j +qs P F +and +ϱprξi, ξjsqg ´ rϱpξiq, ϱpξjqs P F for all pi, jq P Ig ˆ I´1. +(8.41) +Here, pξiqiPIg is a basis for g. Since ppK´iqiě1, d, ρq is a free resolution of F, there exists two O-bilinear +maps χ: Γpgq ˆ ΓpE´1q Ñ ΓpE´1q, η: Γpgq ˆ Γpgq Ñ ΓpE´1q defined on generators ξi, ep´1q +j +by the +relations +rϱpξiq, ρpep´1q +j +qs “ ρpχpξi, ep´1q +j +qq +and +ϱprξi, ξjsgq ´ rϱpξiq, ϱpξjqs “ ρpηpξi, ξjqq. +We first define a naive 2-ary bracket on ΓpE´1q as follows: +1. an anchor map by ρ1pep´1q +i +q “ Xi, and ρ1pξiq “ ϱpξiq, for all i P I, Ig, +2. a degree `1 graded symmetric operation ˜ℓ2 on R‚ as follows: +(a) ˜ℓ2 +´ +ep´1q +i +, ep´1q +j +¯ +“ ř +kPI ck +ijep´1q +k +for all i, j P I´1, +(b) ˜ℓ2 +´ +ξi, ep´1q +j +¯ +“ χ +´ +ξi, ep´1q +j +¯ +, +(c) ˜ℓ2 pξi, ξjq “ rξi, ξjsg ` ηpξi, ξjq, +(d) rℓ2 is zero on the other generators, +(e) we extend ˜ℓ2 to R using O-bilinearity and Leibniz identity with respect to the anchor ρ1. +By paq, pbq, pcq, pdq, peq, ˜ℓ2 satisfies the Leibniz identity with respect to the anchor rρ and paq, pbq, pcq +makes the latter a bracket morphism. The map defined for all homogeneous r1, r2 P R‚ by +rd, ˜ℓ2sRNpr1, r2q “ d ˝ ˜ℓ2 pr1, r2q ` ˜ℓ2 pdr1, r2q ` p´1q|r1|˜ℓ2 pr1, dr2q , +(8.42) +is a graded symmetric degree `2 operation pRbRq‚ ÝÑ R‚`2, and rd, ˜ℓ2sRN|R´1 “ 0. It is O-bilinear, +i.e. for all f P O, r1, r2 P R +rd, ˜ℓ2sRNpr1, fr2q ´ frd, ˜ℓ2spr1, r2q “ 0. +We also have that ρprd, ˜ℓ2sRNpr1, fr2qq “ ρp˜ℓ2pdr1, r2qq “ 0, for all r1 P R´2, r2 P R´1, since ρ ˝ d “ 0. +Thus, rd, ˜ℓ2sRN|R´2ˆR´1 P dR´2, because ppE´iqiě1, d, ρq is a geometric resolution. +Therefore, rd, ˜ℓ2sRN is a degree `2 element in the total complex z +Page +p1qpRq. +The O-bilinear op- +erator rd, ˜ℓ2sRN is D-closed in z +Page +p1qpRq, since rd, rd, ˜ℓ2sRNsRN|Rď´2 “ 0. +So there exists τ2 P +‘jě2HomO +´Ä2 R´j´1, R´j +¯ +such as Dpτ2q “ ´rd, ˜ℓ2sRN. By replacing ˜ℓ2 by ˜ℓ2 ` τ2 we get a 2- +ary bracket ℓ2 of degree `1 which is compatible with the differential map d and the anchor map rρ. + +CHAPTER 8. SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS141 +Construction of higher brackets: notice that by construction of the 2-ary bracket ℓ2 one has, +Jacpr1, r2, r3q P dR´2 for all r1, r2, r3 P R´1. In other words, Jac P HomOpÄ3 R´1, dR´2q. A direct +computation shows +dJacpr1, r2, r3q “ Jacpdr1, r2, r3q ` p´1q|r1|Jacpr1, dr2, r3q ` p´1q|r1|`|r2|Jacpr1, r2, dr3q +for all r1, r2, r3 P R. Which means, rJac, dsRNpr1, r2, r3q “ 0 for all r1, r2, r3 P R. +Thus, DpJacq “ 0. It follows that, Jac is a D-coboundary, there exists an element ℓ3 “ ř +jě2 ℓj +3 P +z +Page +p2q +1 pRq with ℓj +3 P HompÄ3 R |´j´1, R´jq such that +Dpℓ3q “ ´Jac. +(8.43) +We choose the 3-ary bracket to be ℓ3. For degree reason, the remaining terms of the k-ary brackets +for k ě 3 have trivial components on the column ´1 of the bicomplex (8.39). From this point, the +proof continues exactly as in Section 4.3.2. +Example 8.4.6. We return to Example 8.1.8. In that case, the Lie algebra g “ XpLq that acts on +the singular foliation T . In that case, we have ϱpgq X TmT “ t0u for all m in rL, Ms. We can apply +directly Theorem 4.2.1, to obtain a Lie 8-algebroid structure on the complex +¨ ¨ ¨ +d +ÝÑ ΓpE´3q +d +ÝÑ ΓpE´2q +d +ÝÑ g ‘ ΓpE´1q +ρ1 +ÝÑ XprL, Msq. +(8.44) +Notice that here the Lie algebra g is infinite dimensional, therefore we are not allowed to use the duality +between Lie 8-algebroid and Q-manifold. Therefore, we cannot use the explicit Formula (8.32) to +define at lift Φ. However, we have to rely on the existence theorem 8.3.1 to assure the existence of a +lift Φ. +Conclusion: +We show that actions of a Lie algebra g on the leaf space M{F i.e. weak symmetry actions, +lift to Lie 8-algebra morphisms g ù XpEq on the DGLA of vector fields on an universal Lie +8-algebroid pE, Qq, provided it exists, in a unique up to homotopy manner. +We explain how to use chapter 4 to get rid of all finite ranks/dimensions assumptions, using +the universal Lie 8-algebroids of Lie-Rinehart algebras. + +CHAPTER 9 +On weak and strict symmetries: an obstruction theory +In this chapter, we apply the main theorems of 8.2 of Chapter 8 to define a class obstructing the +existence of strict symmetry action equivalent to a given weak symmetry action. +9.1 +Introduction +Recall that Theorem 8.3.1 assures that any weak symmetry action ϱ: g Ñ XpMq of a Lie algebra g on +a singular foliation F admits a lift to a Lie 8-morphism +Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq. +(9.1) +To understand the compatibility conditions between the low terms of Φ see Remark 8.3.3. By playing +with the definition of Φ we make the following observation. +Remark 9.1.1. What does Φ induces on the linear part of pE, Qq? We have seen in Lemma 8.2.5 +and Proposition 8.2.10 that the 0-th Taylor coefficient of the lift Φ induces a linear map x P g ÞÝÑ +p∇x : E´i ÝÑ E´iq for every i ě 1 that satisfies +∇xpfeq “ f∇xpeq ` ϱpxqrfse, for f P O, e P ΓpEq. +1. Also, for every x P g and e P ΓpE´1q, +ρp∇xpeqq “ rϱpxq, ρpeqs. +142 + +CHAPTER 9. ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY +143 +2. A direct computation gives +0 “ rrQ, Φ0pxqs , ιesp´1q “ rQ, rΦ0pxq, ιessp´1q ´ rΦ0pxq, rQ, ιessp´1q +“ +” +Qp0q, rΦ0pxq, ιesp´1qı +´ +” +Φ0pxqp0q, rQ, ιesp´1qı +“ +” +Qp0q, ι∇xpeq +ı +´ +” +Φ0pxqp0q, ιℓ1peq +ı +“ ιℓ1˝∇xpeq ´ ι∇x˝ℓ1peq +We have used graded Jacobi identity and the dual correspondence between Lie 8-algebroids and +NQ-manifolds (see Proposition 6.1.16). We recapitulate in the following commutative diagram: +¨ ¨ ¨ +d � ΓpE´2q +d +� +∇x +� +ΓpE´1q +ρ +� +∇x +� +F +adϱpxq +� +¨ ¨ ¨ +d � ΓpE´2q +d +� ΓpE´1q +ρ +� F +(9.2) +which means, +ℓ1 ˝ ∇x “ ∇x ˝ ℓ1 +and +ρ ˝ ∇x “ adϱpxq ˝ ρ. +Here, ℓ1 stands for the corresponding unary bracket of pE, Qq. Also, for X P XpMq, adX :“ +rX, ¨ s. +3. For x, y P g, and e P ΓpEq, the relation (8.14) yields +rΦ0prx, ysgq ´ rΦ0pxq, Φ0pyqs , ιesp´1q “ rrQ, Φ1px, yqs , ιesp´1q +ι∇rx,ysgpeq ´ rΦ0pxq, rΦ0pyq, ιessp´1q ` rΦ0pyq, rΦ0pxq, ιessp´1q “ +” +ι∇rx,ysgpeq ´ +” +Φ0pxqp0q, ι∇ypeq +ı +` +” +Φ0pyqp0q, ι∇xpeq +ı +“ rrQ, Φ1px, yqs , ιesp´1q . +Thus +rrQ, Φ1px, yqs , ιesp´1q “ ι∇rx,ysgpeq ´ ι∇x˝∇ypeq ` ι∇y˝∇xpeq. +(9.3) +On the other hand, Φ1px, yq admits a polynomial decomposition +Φ1px, yqp´1q ` +ÿ +iě0 +Φ1px, yqpiq “ ιηpx,yq ` +ÿ +iě0 +Φ1px, yqpiq +and that +«ÿ +iě0 +Φ1px, yqpiq, ιe +ffp´1q +“ +” +Φ1px, yqp0q, ιe +ı +“ ιγpx,yqpeq, + +CHAPTER 9. ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY +144 +for some linear map γpx, yq: ΓpE´‚q ÝÑ ΓpE|e|´1q depending linearly on x, y. Therefore, +rrQ, Φ1px, yqs , ιesp´1q “ +““ +Q, ιηpx,yq +‰ +, ιe +‰p´1q ` +«« +Q, +ÿ +iě0 +Φ1px, yqpiq +ff +, ιe +ffp´1q +“ ιℓ2pηpx,yq,eq ` +« +Q, +«ÿ +iě0 +Φ1px, yqpiq, ιe +ffffp´1q +´ +«ÿ +iě0 +Φ1px, yqpiq, rQ, ιes +ffp´1q +“ ιℓ2pηpx,yq,eq ` +» +–Qp0q, +«ÿ +iě0 +Φ1px, yqpiq, ιe +ffp´1qfi +fl ´ +” +Φ1px, yqp0q, rQ, ιesp´1qı +“ ιℓ2pηpx,yq,eq ` +” +Qp0q, ιγpx,yqpeq +ı +´ +” +Φ1px, yqp0q, ιℓ1peq +ı +“ ιℓ2pηpx,yq,eq ` ιℓ1pγpx,yqpeqq ´ ιγpx,yqpℓ1peqq +By equating the latter with (9.3) we can recapitulate as follows: +Conclusion: In general, the map g ÝÑ DerpEq, x ÞÑ ∇x is not a Lie algebra morphism even +when the action ϱ is strict. In fact, there exists a bilinear map γ : ^2 g ÝÑ EndpEqr1s of degree +0 that satisfies +∇rx,ysg ´ r∇x, ∇ys “ γpx, yq ˝ ℓ1 ´ ℓ1 ˝ γpx, yq ` ℓ2pηpx, yq, ¨ q, +here ℓ2 is the corresponding 2-ary bracket of pE, Qq, and η: ^2 g ÝÑ ΓpE´1q is such that +ϱprx, ysgq ´ rϱpxq, ϱpyqs “ ρpηpx, yqq. +4. Also, Equation (8.15) taken in polynomial-degree ´1 implies, that, for all α P ΓpE˚ +´1q +Φ1prx, ysg, zqp´1q ´ rΦ0pxqp0q, Φ1py, zqp´1qs` ö px, y, zq “ rQp0q, Φ2px, y, zqp´1qs. +ùñ ιηprx,ysg,zq ´ rΦ0pxqp0q, ιηpy,zqs` ö px, y, zq “ rQp0q, ιζpx,y,zqs, +with +ζ : ^3 g Ñ ΓpE´2q +ùñ xα, ηprx, ysg, zqy ´ +´ +Φ0pxqp0qrxα, ηpy, zqys ´ xΦ0pxqp0qpαq, ηpy, zqy +¯ +` ö“ xQp0qrαs, ζpx, y, zqy, +ùñ +xα, ηprx, ysg, zqy ´ +` +(((((((( +( +ϱpxqrxα, ηpy, zqys ´ (((((((( +( +ϱpxqrxα, ηpy, zqys ` xα, ∇xηpy, zqy +˘ +` ö px, y, zq “ +xα, ℓ1pζpx, y, zqqy. +We have used Equations (8.13) and (8.9) in the last line. Hence, +xα, ηprx, ysg, zq ´ ∇xηpy, zqy` ö px, y, zq “ xα, ℓ1pζpx, y, zqqy. +Since α is arbitrary, one obtains +∇xηpy, zq ´ ηprx, ysg, zq` ö px, y, zq “ ℓ1pζpx, y, zqq. +(9.4) + +CHAPTER 9. ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY +145 +In particular, if m P M is such that ℓ1|m “ 0 and ℓ2|m “ 0, then the map x ÞÑ ∇x defines an +action on the isotropy Lie algebra gm of F, since ∇x preserves the kernel of ρ. If in addition +ηpx, yq P ker ρm, then η|m : ^2 g ÝÑ gm is a cocycle of Chevalley-Eilenberg. +Remark 9.1.2. Notice that by using the duality which is given in Proposition 8.4.1, the linear map +x ÞÑ ∇x is simply +x ÞÑ ℓ1 +2px, ¨q, +where ℓ1 +2 is the 2-ary bracket between sections of g and E of the Lie 8-algebroid pQ1, E ‘ gq over the +complex +¨ ¨ ¨ +ℓ1 +ÝÑ E´3 +ℓ1 +ÝÑ E´2 +ℓ1 +ÝÑ g ‘ E´1 +ρ1 +ÝÑ TM +(9.5) +like in Proposition 8.4.1. Indeed, for α P ΓpE˚ +´1q and e P ΓpE´1q, +Φ0pxqp0q “ pr ˝ rQ1, ιxsp0q +“ pr ˝ rQ1p1q, ιxs. +This implies that: +xΦ0pxqp0qα, ey “ xQ1p1qα, x d ey +“ ρ1pxqrxα, eys ´  +ρ1peqrxα, xys ´ xα, ℓ1 +2px, eqy +“ ϱpxqrxα, eys ´ xα, ℓ1 +2px, eqy. +9.2 +An obstruction theory +Let us start with some generalities. Assume we are given +• a Lie algebra g, +• a weak symmetry action ϱ: g ÝÑ XpMq of g on a singular foliation F, together with η: ^2 g ÝÑ +ΓpE´1q such that x, y P g +ϱprx, ysgq ´ rϱpxq, ϱpyqs “ ρpηpx, yqq. +(9.6) +• an universal Lie 8-algebroid pE, QEq of F, +Theorem 8.3.1 assures ϱ: g Ñ XpMq admits a lift to a Lie 8-morphism +Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq. +(9.7) +Equivalently, if g is of finite dimension, (9.7) corresponds (by Proposition 8.4.1) to a Lie 8-algebroid +pE1, Q1q over M such that +• pE, QEq is included as a sub-Lie 8-algebroid in a Lie algebroid pE1, Qq over M, +• its underlying complex is, E1 +´1 :“ g ‘ E´1, and for any i ě 2, E1 +´i “ E´i, namely +¨ ¨ ¨ +d +ÝÑ E´3 +d +ÝÑ E´2 +d +ÝÑ g ‘ E´1 +ρ1 +ÝÑ TM, +(9.8) + +CHAPTER 9. ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY +146 +• we have, +ℓ1 +2px ‘ 0, y ‘ 0q “ rx, ysg ‘ ηpx, yq +and +ℓ1 +2px, ΓpE´1qq Ă ΓpE´1q +for all x P g. +Remark 9.2.1. It is important to note that the Lie 8-algebroid pE1, Q1q can be constructed directly +out of the weak symmetry action ϱ: g ÝÑ XpMq, even if g is of infinite dimension (see Theorem 8.4.4). +In what follows, we will use the 8-algebroid which is described on the complex (9.8). +Lemma 9.2.2. Let m P M. Assume that the underlying complex pE, ℓ1q is minimal at a point m, i.e. +ℓ1|m “ 0. The map +ν : g ÝÑ End +´ +E´1|m +¯ +, x ÞÝÑ ℓ1 +2px , ¨q|m +satisfies +(a) νprx, ysgq ´ rνpxq, νpyqs ` ℓ2p ¨, ηpx, yqq|m “ 0, +(b) νpzq +` +ηpx, yq|m +˘ +´ ηprx, ysg, zq|m` ö px, y, zq “ 0. +Proof. Since ℓ1|m “ 0, E1 +´1|m is a Lie algebra. The Jacobi identity on elements x, y P g, e P ΓpE´1q, +evaluated at the point m, implies that +νprx, ysgqpe|mq ´ rνpxq, νpyqspe|mq ` ℓ2pηpx, yq, eq|m “ 0. +This proves item (a). Likewise, Jacobi identity on elements x, y, z P g and since ℓ1|m “ 0 give: +ℓ1 +2pℓ1 +2px, yq, zq|m` ö px, y, zq “ 0 ùñ ℓ1 +2prx, ysg, zq|m ` ℓ1 +2pηpx, yq, zq|m` ö px, y, zq “ 0, +ùñ νpzq +` +ηpx, yq|m +˘ +´ ηprx, ysg, zq|m` ö px, y, zq “ 0. +Here we have used the definition of ℓ1 +2 on degree ´1 elements and Jacobi identity for the bracket r¨ , ¨sg. +This proves item (b). +By Lemma 9.2.2, E´1 is equipped with a g-module structure when ηpx, yq|m is for all x, y P g +valued in the center the Lie algebra E´1|m. The following proposition generalizes this remark. +Proposition 9.2.3. Let m P M and assume that +• the underlying complex pE, ℓ1q of pE, Qq is minimal at m, +• for all x, y P g, ηpx, yq|m is valued in the center1 ZpE´1|mq of E´1|m. +Then, +1. the restriction of the 2-ary bracket +ℓ1 +2 : g b ZpE´1|mq ÝÑ ZpE´1|mq +endows ZpE´1|mq with a g-module structure which does not depend neither on the choices of weak +symmetry action ϱ, a universal Lie 8-algebroid of F, nor of the Lie 8-morphism Φ: g ÝÑ XpEq. +1In particular, when the 2-ary bracket ℓ2 is zero at m, on elements of degree ´1 we have, ZpE´1|mq “ E´1|m. + +CHAPTER 9. ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY +147 +2. the restriction of the map η: ^2 g ÝÑ ΓpE´1q at m +η|m : ^2 g ÝÑ ZpE´1|mq +is a 2-cocycle for the Chevalley-Eilenberg complex of g valued in ZpE´1|mq, +3. the cohomology class of this cocycle does not depend on the representatives of the equivalence +class of ϱ, +4. if ϱ is equivalent to a strict symmetry action, then η|m is exact. +Proof. We may assume that, ℓ2|m “ 0 on E´1|m, i.e., ZpE´1|mq “ E´1|m. The first clause of item p1q +follows from item paq of Lemma 9.2.2 when ℓ2|m “ 0. It is easy to see that if we change the action ϱ to +ϱ ` ρ ˝ β for some vector bundle morphism β : g ÝÑ E´1, the new 2-ary bracket between sections of g +and E´1 made in the proof of Theorem 8.4.4 is modified by px, eq ÞÑ ℓ1 +2px, eq ` ℓ2pβpxq, eq. Therefore, +under the assumption, ℓ2|m “ 0, we obtain the last clause of item p1q. Item p2q follows from Item (b) +of Lemma 9.2.2 that tells that η|m : ^2 g ÝÑ E´1|m is a 2-cocycle for the Chevalley-Eilenberg complex +of g valued in E´1|m. +Let ϱ1 be a weak action of g on F which is equivalent to ϱ, i.e. +there exists a vector bundle +morphim β : g ÝÑ E´1 such that ϱ1pxq “ ϱpxq ` ρpβpxqq for all x P g. Let η1 : ^2 g ÝÑ E´1 be such +that ϱ1prx, ysgq ´ rϱ1pxq, ϱ1pyqs “ ρpη1px, yqq for all x, y P g. Following the constructions in the proof of +Theorem 8.4.4, this implies that +η1px, yq “ ηpx, yq ` βprx, ysgq ´ ℓ1 +2px, βpyqq ` ℓ1 +2py, βpxqq ´ ℓ2pβpxq, βpyqqq, +for all x, y P g. +(9.9) +Hence, if η1 +|m P H2pg, E´1|mq is exact, i.e. +there exists a linear map λ: g ÝÑ E´1|m such that +dCEpλq “ η1 +|m. Using Equation (9.9) and ℓ2|m “ 0, one gets dCEpβ|m ` λq “ η|m. This proves items +p3q and p4q. +Remark 9.2.4. When ℓ2|m ‰ 0. +The weak symmetry action ϱ is equivalent to strict one if the +Maurer-Cartan-like equation (9.9) has no solution with η1 +|m “ 0. +Let F be a singular foliation. Let us choose a universal Lie 8-algebroid pE, Qq such that pE, ℓ1q is +minimal at a point m P M. Such a structure always exists (see Proposition B.2.14). By Proposition +4.14 in [LLS20] the isotropy Lie algebra gm of the singular foliation F at the point m P M is isomorphic +to kerpρmq. The following is a direct consequence of Proposition 9.2.3. +Corollary 9.2.5. Let m P M be a point of M Assume that the isotropy Lie algebra gm of F at +m is Abelian. +Then, for any weak symmetry action ϱ of a Lie algebra action g on F such that +ϱprx, ysgq ´ rϱpxq, ϱpyqs P Fpmq for all x, y P g +1. gm is a g-module. +2. The bilinear map, η|m : ^2 g Ñ gm, is a Chevalley-Eilenberg 2-cocycle of g valued in gm. +3. Its class clpηq P H2pg, gmq does not depend on the choices made in the construction. +4. Furthermore, clpηq is an obstruction of having a strict symmetry action equivalent to ϱ. +Example 9.2.6. We return to Example 8.1.6 with m P M a leaf of F. Since the isotropy Lie algebra +gk +m is Abelian for every k ě 2 the following assertions hold by Corollary 9.2.5: + +CHAPTER 9. ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY +148 +1. For each k ě 1, the vector space gk`1 +m +is a gk +m-module. +2. The obstruction of having a strict symmetry action equivalent to ϱk is a Chevalley-Eilenberg +cocycle valued in gk`1 +m . +Here is a particular case of this example. +Example 9.2.7. Let F :“ I3 +0XpRnq be the singular foliation generated by vector fields vanishing to +order 3 at the origin. The quotient g :“ I2 +0XpRnq +I3 +0XpRnq is a trivial Lie algebra. There is a weak symmetry +action of g on F which assigns to an element in g a representative in I2 +0XpRnq. In this case, the isotropy +Lie algebra of F at zero is Abelian and ℓ1 +2pg, g0q|0 “ 0. Thus, the action of g on g0 is trivial. One +can choose η: ^2 g ÝÑ g0 such that η +´ +x2 +i +B +Bxi , x2 +i +B +Bxj +¯ +“ 2eij, with eij a constant section in a set of +generators of degree ´1 whose image by the anchor is x3 +i +B +Bxj . Therefore, η|0 +´ +x2 +i +B +Bxi , x2 +i +B +Bxj +¯ +‰ 0. This +implies that the class of η is not zero at the origin. Therefore, by item 2 of Corollary 9.2.5 the weak +symmetry action of g on F is not equivalent to a strict one. +Example 9.2.8. Consider again the Example 8.4.6 and let pE, ℓ‚, ρq be a universal Lie 8-algebroid +of T . Recall that a flat Ehresmann connection is a horizontal distribution whose sections are closed +under the Lie bracket of vector fields. +Let m P M. +Assume in Example 8.1.8 that the section +ϱH : XpLq Ñ Fproj satisfies +ϱHpra, bsq ´ rϱHpaq, ϱHpbqs “ ρpηpa, bqq P T pmq +for some bilinear map η: ^2 XpLq Ñ E´1. By Proposition 9.2.5, the isotropy Lie algebra gT +m of T at +m is a XpLq-module, and η is a cocycle of Chevalley-Eilenberg. This provides an obstruction for the +F-connection to be flat. +Also, we have the following consequence of Corollary 9.2.5 for Lie algebra actions on affine vari- +eties, as in Example 8.1.11. Before going to Corollary 9.2.12 let us write definitions and some facts. +Settings: Let W be an affine variety realized as a subvariety of Cd, and defined by some ideal +IW Ă Crx1, . . . , xds. We denote by XpWq :“ DerpOW q the Lie algebra of vector fields on W, where +OW is coordinates ring of W. +Definition 9.2.9. A point p P W is said to be strongly singular if for all f P IW , dpf ” 0 or +equivalently if for all f P IW and X P XpCdq, one has Xrfsppq P Ip. +Example 9.2.10. Any singular point of a hypersurface W defined by a polynomial ϕ P Crx1, . . . , xds +is strongly singular. +The lemma below is immediate. +Lemma 9.2.11. In a strongly singular point, the isotropy Lie algebra of the singular foliation F “ +IW XpCdq is Abelian. +Corollary 9.2.12. Let ϱ: g ÝÑ XpWq be a Lie algebra morphism. +1. Any extension rϱ as in Example 8.1.11 is a weak symmetry action for the singular foliation +F “ IW XpCdq. + +CHAPTER 9. ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY +149 +2. For any strongly singular point p in W if the class clpηq does not vanish the strict action ϱ: g ÝÑ +DerpOW q can not be extended to the ambient space. +Let us give an example of a Lie algebra action on an affine variety that do not extend to the +ambient space. +We hope to construct an example as follows +Example 9.2.13. Let W Ă Cd be an affine variety generated by a regular homogeneous polynomial +ϕ P O “ Crx1, . . . , xds of degree ě 2. Assume there exists two vector fields X, Y P XpCdq that satisfy +Xrϕs “ fϕ, Y rϕs “ gϕ, with f, g P I0, and such that rX, Y s “ ϕZ, for some Z P XpCdq. Consider the +action of the trivial g “ R2 on W that sends its canonical basis to X, and Y respectively. It is a weak +symmetry action on the singular foliation Fϕ :“ xϕyXpCdq and induces a Lie algebra map, g ÝÑ XpWq. +Notice that the universal Lie algebroid of Fϕ is a Lie algebroid (see Example of [LGL22b]) because, +0 +� Oµ bO XpCdq +ϕ B +Bµ bOid +� Fϕ. +is a O-module isomorphism. Here µ is a degree ´1 variable, so that µ2 “ 0. The universal algebroid +structure over that resolution is given on the set of generators by: +ℓ2 +ˆ +µ bO +B +Bxa +, µ bO +B +Bxb +˙ +:“ Bϕ +Bxa +µ bO Bxb ´ Bϕ +Bxb +µ bO Bxa +and ℓk :“ 0 for every k ě 3. Write Z “ +dÿ +i“1 +fi +B +Bxi +, with pfiqi“1,...,d Ă O. We have, +ηpe1, e2q :“ +dÿ +i“1 +fiµ bO +B +Bxi +. +where e1, e2 is the canonical basis of R2. +This Lie 8-algebroid structure satisfies all the assuptions of Proposition 9.2.5. Assume that the vector +field Z does not vanish at zero. Since the action is trivial at zero and η|0 ‰ 0, therefore its class is +non-zero. By consequence, such action cannot be extended to ambient space. +Let us make it explicit +Example 9.2.14. Let W Ă C2 be the affine variety generated by the polynomial ϕ “ FG with +F, G P Crx, ys “: O. We consider the vector fields U “ FXG, V “ GXF P XpC2q, where XF and XG +are Hamiltonian vector fields w.r.t the Poisson structure tx, yu :“ 1. Note that U, V are tangent to +W, i.e. Urϕs, V rϕs P xϕy. It is easily checked that rU, V s “ ϕXtF,Gu. +The action of the trivial Lie algebra g “ R2 on W that sends its canonical basis pe1, e2q to U, and +V respectively, is a weak symmetry action on the singular foliation Fϕ :“ xϕyXpC2q, and induces a +Lie algebra map, +ϱ: g ÝÑ XpWq. +(9.10) +A universal Lie algebroid of Fϕ is a Lie algebroid (see Example 3.19 of [LGL22b]) because, +0 +� Oµ bO XpC2q +ϕ B +Bµ bOid +� Fϕ + +CHAPTER 9. ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY +150 +is a O-module isomorphism. Here µ is a degree ´1 variable, so that µ2 “ 0. The universal algebroid +structure over that resolution is given on the set of generators by: +ℓ2 +ˆ +µ bO +B +Bx, µ bO +B +By +˙ +:“ Bϕ +Bx µ bO +B +By ´ Bϕ +By µ bO +B +Bx +(9.11) +and ℓk :“ 0 for every k ě 3. Write XtF,Gu “ BtF, Gu +By +B +Bx ´ BtF, Gu +Bx +B +By. Therefore, we can put +ηpe1, e2q :“ BtF, Gu +By +µ bO +B +Bx ´ BtF, Gu +Bx +µ bO +B +By. +(9.12) +Take for example, Fpx, yq “ y ´ x2 and Gpx, yq “ y ` x2. The isotropy Lie algebra gp0,0q of Fϕ +is Abelian, since zero is a strong singular point of W. By Corollary 9.2.5 (1), gp0,0q is a R2-module. +A direct computation shows that the action on gp0,0q is not trivial, but takes value in O µ bO +B +Bx. +Besides, Equation (9.12) applied to tF, Gu “ 4x gives +ηpe1, e2q “ ´4 µ bO +B +By. +(9.13) +If η|p0,0q were a coboundary of Chevalley Eilenberg, we would have (in the notations of Proposition +9.2.3) that +ηpx, yq|p0,0q “ βprx, ysR2q ´ ℓ1 +2px, βpyqq ` ℓ1 +2py, βpxqq P O µ bO +B +Bx, +for all x, y P g +(9.14) +for some linear map β : g ÝÑ gp0,0q. Therefore, Equation (9.14) is impossible by Equation (9.13) and +since η|p0,0q ‰ 0. In orther words, its class clpηq does not vanish. By Corollary 9.2.12 (2), the action ϱ +given in Equation (9.10) cannot be extended to ambient space. +Conclusion: +We applied the existence of a Lie 8-morphism to the question of lifting to M an g-action on +M{F, i.e. of making strict a weak g-action. +We apply this question to several geometric issues, e.g. neighbourhood of leaves, Lie algebra +actions on an affine variety. + +CHAPTER 10 +Bi-submersion towers +10.1 +Symmetries of bi-submersions +In this chapter, we introduce the notion “bi-submersion towers”. The work contained in this chapter is +entirely original, except for the notion below that arose in a discussion between C. Laurent-Gengoux, +L. Ryvkin, and I, and will be the object of a separate study. +Let us firstly recall the definition of bi-submersion. The concept of bi-submersion over singular folia- +tions has been introduced in [AS09]. +Definition 10.1.1. Let M be a manifold endowed with a singular foliation F. +A bi-submersion +B +s +� +t +� M over F is a triple pB, s, tq where: +• B is a manifold, +• s, t: B Ñ M are submersions, respectively called source and target, +such that the pull-back singular foliations s´1F and t´1F are both equal to the space of vector fields +of the form ξ ` ζ with ξ P Γpkerpdsqq and ζ P Γpkerpdtqq. Namely, +s´1F “ t´1F “ Γpkerpdsqq ` Γpkerpdtqq. +(10.1) +In that case, we also say that pB, s, tq is a bi-submersion over pM, Fq. +Example 10.1.2. Let F be a singular foliation over a manifold M. For x P M and X1, . . . , Xn P F +inducing generators for Fx :“ F{IxF. We know from [AS09] that there is an open neighborhood W of +px, 0q P M ˆ Rn such that pW, t, sq is a bi-submersion over F, where +spx, yq “ x +and +tpx, yq “ expx +˜ nÿ +i“1 +yiXi +¸ +“ ϕ +řn +i“1 yiXi +1 +pxq, +(10.2) +where for X P XpMq, ϕX +1 denotes the time-1 flow of X. Such bi-submersions are called path holonomy +bi-submersions [AZ13]. +151 + +CHAPTER 10. BI-SUBMERSION TOWERS +152 +Now we can introduce the following definition. +Definition 10.1.3. A bi-submersion tower over a singular foliation F on M, is a (finite or infinite) +sequence of manifolds and maps as follows +TB : ¨ ¨ ¨ +si`1 � +ti`1 +� Bi`1 +si +� +ti +� Bi +si´1 � +ti´1 +� ¨ ¨ ¨ +s1 +� +t1 +� B1 +s0 � +t0 +� B0, +(10.3) +together with a sequence Fi of singular foliations on Bi, with the convention that B0 “ M and F0 “ F, +such that +• for all i ě 1, Fi Ă Γpker dsi´1q X Γpker dti´1q, +• for each i ě 1, Bi`1 +si +� +ti +� Bi is a bi-submersion over Fi. +A bi-submersion tower over pM, Fq shall be denoted as pBi`1, si, ti, Fiqiě0. The bi-submersion tower +over F in (10.3) is said to be of length n P N if Bj “ Bn, sj “ tj “ id and Fj “ t0u for all j ě n. +Remark 10.1.4. Let us spell out some consequences of the axioms. For i ě 1, two points b, b1 P Bi +of the same leaf of Fi satisfy si´1pbq “ si´1pb1q and ti´1pbq “ ti´1pb1q. Also, for all b P Bi, TbFi Ă +pker dsi´1q|b X pker dti´1q|b. +Let us explain how such towers can be constructed out of a singular foliation. Let F be a singular +foliation on M. Then, +1. By Proposition 2.10 in [AS09], there always exists a bi-submersion B1 +s0 +� +t0 +� M over F. +2. The C8pB1q-module Γpker ds0q X Γpker dt0q is closed under Lie bracket. +When it is locally +finitely generated, it is a singular foliation on B1. Then, it admits a bi-submersion B2 +s1 +� +t1 +� B1 . +Therefore, we have obtained the two first terms of a bi-submersion tower. +3. We can then continue this construction provided that Γpker ds1q X Γpker dt1q is locally finitely +generated as a C8pB2q-module, and that it is so at each step1. +Definition 10.1.5. A bi-submersion tower TB “ pBi`1, si, ti, Fiq over pM, Fq is called exact bi- +submersion tower over pM, Fq when Fi`1 “ Γpkerpdsiqq X Γpkerpdtiqq for all i ě 0. It is called a path +holonomy bi-submersion tower (resp. path holonomy atlas bi-submersion tower) if Bi`1 +si +� +ti +� Bi is +a path holonomy bi-submersion (resp. a path holonomy atlas) for Fi for each i ě 0. When a path +holonomy bi-submersion tower is exact, we speak of exact path holonomy bi-submersion tower. +The following theorem gives a condition which is equivalent to the existence of a bi-submersion +tower over a singular foliation. The proof uses Lemma 10.1.17 which is stated in the next section. +Theorem 10.1.6. Let F be a singular foliation on M. The following items are equivalent: +1In real analytic case, the module Γpker ds1q X Γpker dt1q is locally finitely generated because of the noetherianity of +the ring of germs of real analytic functions [Fri67, Siu69]. + +CHAPTER 10. BI-SUBMERSION TOWERS +153 +1. F admits a geometric resolution. +2. There exists an exact path holonomy bi-submersion tower over pM, Fq. +Proof. For smooth maps φ, ψ: M ÝÑ N, we denote by ψΓpker dφq the space of ψ-projectable vector +fields in Γpker dφq Ă XpMq. +1 ñ 2 : Assume that F admits a geometric resolution pE, d, ρq. In particular, ρpΓpE´1qq “ F. Let +pB1, s0, t0q be a path holonomy bi-submersion over pM, Fq. Let b P B1 and Ub an open neighborhood +of b. Let pe1, . . . , erq be a local trivialization of E´1 on the open subset U “ t0pUbq Ă M. We define a +map on generators by +ÐÝ‚ : ΓUpE´1q ÝÑ s0ΓUbpker dt0q Ă XpB1q +(10.4) +ei ÞÝÑ ÐÝ +ei :“ ÐÝÝ +ρpeiq +and extend by s0-linear map on U, i.e. ÐÝ‚ is additive and for every i P t1, . . . , ru and f P C8pUq one +has ÐÝ +fei “ ps0q˚pfqÐÝ +ei . By Lemma 10.1.17, the map (10.4) is surjective. +Since for every i ě 1, ÐÝ +ei is s0-related to ρpeiq, in particular the map (10.4) restricts to a surjective +map +ker ρ|U ÝÑ ker +´ +ds0|ΓUbpker dt0q +¯ +“ Γpker ds0q X Γpker dt0q|Ub Ă XpUbq +(10.5) +e ÞÝÑ ÐÝe +By exactness in degree ´1, ker ρ|U “ dpΓUpE´2qq. Therefore, ker ρ is locally finitely generated. By +subjectivity of the map (10.5), Γpker ds0q X Γpker dt0q “: F1 is also locally finitely generated, in par- +ticular F1 is a singular foliation on B1. Thus, one can take a path holonomy bi-submersion pB2, s1, t1q +over pB1, F1q. The proof continues the same as the previous case. +Let us make a step further for clarity. Let b P B2 and Ub an open neighbourhood of b Let pe1, e2, e3 . . .q +be a local trivialization of E´2 on the open subset U “ t1pUbq Ă B1. Just like in the first step, define +the surjective s1-linear map, +ΓUpE´2q ÝÑ s1ΓUbpker dt1q Ă XpB2q +(10.6) +e ÞÝÑ ÐÝ +ei :“ ÐÝÝ +dpeiq. +which restricts to a surjective map +ker pd: ΓUpE´2q Ñ ΓUpE´1qq ÝÑ ker +` +ds1|Γpker dt1q +˘ +“ Γpker ds1q X Γpker dt1q Ă XpB2q, +(10.7) +e ÞÝÑ ÐÝe +since 0 “ dpeq “ ds1pÐÝe q for any e P ker ρ|U. By exactness in degree ´2, the C8pUq-module +ker pd: ΓUpE´2q Ñ ΓUpE´1qq “ dpΓUpE´3qq +is (locally) finitely generated, hence F2 :“ Γpker ds1q X Γpker dt1q is a singular foliation on B2. The +proof continues by recursion. +2 ñ 1 is proven by Lemma 10.1.7 and Remark 10.1.8 below. + +CHAPTER 10. BI-SUBMERSION TOWERS +154 +Lemma 10.1.7. Let F be a singular foliation on M. Assume that there exists a bi-submersion tower +TB “ pBi, ti, si, Fiqiě0 over F. Then, +¨ ¨ ¨ +� ker ds2 +� +dt2 � ker ds1 +� +dt1 � ker ds0 +� +dt0 +� TM +� +¨ ¨ ¨ +� B3 +t2 +� B2 +t1 +� B1 +t0 +� M. +(10.8) +is a complex of vector bundles, which is exact on the sections level2 if TB is an exact bi-submersion +tower, i.e. if Fi “ Γpker dsi´1q X Γpker dti´1q for all i ě 1. +Proof. For any element b P Bi`1 and any vector v P ker dsi Ă TbBi`1 one has +dtipvq P TtipbqFi, +(since Γpker dsiq Ă t´1 +i pFiq). +ùñ +dtipvq P pker dsi´1 X ker dti´1q |tipbq +by Remark 10.1.4. +ùñ +dtipvq P ker dsi´1 +and +dti´1 ˝ dtipvq “ 0, for all i ě 1. +This shows the sequence (10.8) is a well-defined complex of vector bundles. +Let us prove that it is exact when Fi “ Γpker dsi´1qXΓpker dti´1q for all i ě 1. Let ξ P Γ pker dsi´1q +be a ti´1-projectable vector field that projects to zero, i.e. +dti´1pξq “ 0. +This implies that ξ P +Γpker dsi´1q X Γpker dti´1q “ Fi. Since ti is a submersion, there exists a ti-projectable vector field +ζ P t´1 +i pFiq that satisfies dtipζq “ ξ. The vector field ζ can be written as ζ “ ζ1`ζ2 with ζ1 P Γ pker dtiq +and ζ2 P Γ pker dsiq, because t´1 +i pFiq “ Γpker dsiq`Γpker dtiq. One has, dtipζ2q “ ξ. A similar argument +shows that the map, Γpker ds0q dt0 +ÝÑ t˚ +0F, is surjective. This proves exactness in all degree. +Remark 10.1.8. One of the consequence of Lemma 10.1.7 is that: +1. If there exists a sequence of maps +M � +ε0 +� B1 � +ε1 � B2 � +ε2 +� ¨ ¨ ¨ +(10.10) +where for all i ě 0, εi is a section for both si and ti then the pull-back of (10.8) on M through +the sections pεiqiě0 i.e. +¨ ¨ ¨ +dt3� ε˚ +2,0 ker ds2 +dt2 � ε˚ +1,0 ker ds1 +dt1 +� ε˚ +0 ker ds0 +dt0 +� TM +(10.11) +is a complex of vector bundles, with the convention εn,0 “ εn ˝ ¨ ¨ ¨ ˝ ε0. +If TB is an exact +bi-submersion tower then, (10.11) is a geometric resolution of F. +2. In case that TB is an exact path holonomy bi-submersion tower, such a sequence (1) always +exists, since the bi-submersions pBi`1, si, tiq are as in Example 10.1.2. For such bi-sumersions, +the zero section x ÞÑ px, 0q is a section for both si and ti. +2Let us explain the notion of exactness at the level of sections when the base manifolds are not the same: what we +mean is that for all n ě 0, Γpker dtnq X Γpker dsnq “ ptn`1q˚pΓpker dsn`1qq. +Equivalently, it means that the pull-back of the vector bundles in (10.12) to any one of the manifold Bm with m ě n +is exact at the level of sections, i.e +Γpt˚ +n`1,m ker dsn`1q +dtn`1 � Γpt˚ +n,m ker dsnq +dtn � Γpt˚ +n´1,m ker dsn´1 q +(10.9) +is a short exact sequence of C8pBmq-modules, with tn,m “ tn ˝ ¨ ¨ ¨ ˝ tm for all m ě n. + +CHAPTER 10. BI-SUBMERSION TOWERS +155 +Corollary 10.1.9. Under the assumptions of Lemma 10.1.7, assume the tower of bi-submersion TB +is of length n ` 1. Then, the pull-back of the sequence of vector bundles +ker dsn +� +dtn � t˚ +n ker dsn´1 +� +� +¨ ¨ ¨ +� +dt2 � t˚ +2,n ker ds1 +� +dt1 � TBn`1 ˆTM ker ds0 +� +pr1 � TBn`1 +� +Bn`15 +(10.12) +is a geometric resolution of the pull-back foliation t´1 +0,npFq Ă XpBn`1q, where pr1 is the projection on +TBn`1 and for i ě 1, ti,j is the composition ti ˝ ¨ ¨ ¨ ˝ tj : Bj`1 Ñ Bi. +Proof. By Lemma 10.1.7, the complex in Equation (10.12) is exact. By construction, the projection +of the fiber product TBn`1 ˆTM ker ds0 to TBn`1 induces the singular foliation t´1 +0,npFq. +10.1.1 +Lift of a symmetry to the bi-submersion tower +Let us investigate what an action ϱ: g Ñ XpMq of a Lie algebra g on pM, Fq would induce on a +bi-submersion tower TB over F. +We start with some vocabulary and preliminary results. +Definition 10.1.10. Let pB, s, tq be a bi-submersion of a singular foliation F on a manifold M. We +call lift of a vector field X P XpMq to the bi-submersion pB, s, tq a vector field r +X P XpBq which is +both s-projectable on X and t-projectable on X. +The coming proposition means that the notion of lift to a bi-submersion only makes sense for +symmetries of the singular foliation. +Proposition 10.1.11. If a vector field on M admits a lift to a bi-submersion pB, s, tq, then it is a +symmetry of F. +Proof. Let r +X P XpBq be a lift of X P XpMq. Since r +X is s-projectable, r r +X, Γpker dsqs Ă Γpker dsq. +Since r +X is t-projectable, r r +X, Γpker dtqs Ă Γpker dtq. Hence: +r r +X, s´1pFqs “ r r +X, Γpker dsq ` Γpker dtqs +“ r r +X, Γpkerpdsqs ` r r +X, Γpker dtqs +Ă Γpker dsq ` Γpker dtq “ s´1pFq. +In words, r +X is a symmetry of s´1F. Since r +X projects through s to X, X is a symmetry of F. +We investigate the existence of lifts of symmetries of F to bi-submersions over F. +Remark 10.1.12. For a given X P spFq, +1. the lift r +X to a given bi-submersion is not unique, even when it exists. However, two different lifts +of a X P spFq to a bi-submersion pB, s, tq differ by an element of the intersection Γpkerpdsqq X +Γpkerpdtqq. +2. r +X is a symmetry of Γpkerpdsqq X Γpkerpdtqq: +r r +X, Γpkerpdsqq X Γpkerpdtqqs Ă Γpkerpdsqq X +Γpkerpdtqq, since r +X is s-projectable and t-projectable. + +CHAPTER 10. BI-SUBMERSION TOWERS +156 +As the following example shows, the lift of a symmetry to a bi-submersion may not exist. +Example 10.1.13. Consider the trivial foliation F :“ t0u on M. For any diffeomorphism φ: M ÝÑ +M, pM, id, φq is a bi-submersion over F. Every vector field X P XpMq is a symmetry of F. If it exists, +its lift has to be given by, r +X “ X since the source map is the identity. But r +X “ X is t-projectable if +and only if X is φ-invariant. A non φ-invariant vector field X therefore admits no lift to pM, id, φq. +However, internal symmetries, i.e. elements in F admit lifts to any bi-submersion. +Proposition 10.1.14. Let pB, s, tq be any bi-submersion of a singular foliation F on a manifold M. +Every internal symmetry, i.e. every vector field in F, admits a lift to pB, s, tq. +Proof. Let X P F. Since s: B ÝÑ M is a submersion, there exists Xs P XpBq s-projectable on X. +Since t is a submersion, there exists Xt P XpBq t-projectable on X. By construction Xs P s´1pFq and +Xt P t´1pFq. Using the property (10.1) of the bi-submersion pB, s, tq, the vector fields Xs and Xt +decompose as +$ +& +% +Xs “ Xs +s ` Xs +t +with Xs +s P Γpkerpdsqq, Xs +t P Γpkerpdtqq, +Xt “ Xt +s ` Xt +t +with Xt +s P Γpkerpdsqq, Xt +t P Γpkerpdtqq. +By construction, Xs +t is s-projectable to X and t-projectable to 0 while Xt +s is s-projectable to 0 and +t-projectable to X. It follows that, r +X :“ Xt +s ` Xs +t , is a lift of X to the bi-submersion pB, s, tq. +Let us make the Proposition 10.1.14 more precise. +Corollary 10.1.15. Let pB, s, tq be any bi-submersion of a singular foliation F on a manifold M. +Every vector field X P F admits a lift r +X P XpBq on pB, s, tq. Moreover, r +X can be decomposed as +r +X “ ÝÑ +X ` ÐÝ +X +where +1. the vector field ÝÑ +X P XpBq (resp. ÐÝ +X P XpBq) is t-related (resp. s-related) with X P XpMq +2. the vector field ÝÑ +X (resp. ÐÝ +X) is tangent to the fibers of s (resp. t). +Proof. Take ÝÑ +X :“ Xt +s P Γpker dsq and ÐÝ +X :“ Xs +t P Γpker dtq in the Proof 10.1.14. +Remark 10.1.16. Upon choosing generators for F, +F ÝÑ Γpker dtq +X ÞÝÑ ÐÝ +X +can not be a O-linear map. +The following lemma is immediate. +Lemma 10.1.17. Let pB, s, tq be any bi-submersion of a singular foliation F on a manifold M. Any +t-projectable vector field of Γpker dsq (resp. s-projectable of Γpker dtq) is of the form ÝÑ +X (resp. ÐÝ +X) for +some X P F. + +CHAPTER 10. BI-SUBMERSION TOWERS +157 +We can now state one of the important results of this section. It uses several concepts introduced +in [AS09], which are recalled in the proof. +Proposition 10.1.18. Let F be a singular foliation on a manifold M. Any symmetry X P spFq admits +a lift +1. to any path holonomy bi-submersion pB, s, tq, +2. to Androulidakis-Skandalis’ path holonomy atlas, +3. to a neighborhood of any point in a bi-submersion through which there exists a local bisection +that induces the identity. +Remark 10.1.19. In cases (1) or (2) in Proposition 10.1.18, a linear lift +X Ñ r +X +can be defined on the whole space spFq of symmetries of F. As an immediate consequence of Remark +10.1.12, we obtain that for all X, Y P spFq, +Č +rX, Y s ´ r r +X, rY s P Γpker dsq X Γpker dtq. +(10.13) +Proof of Proposition 10.1.18. Let X P spFq. Assume that pB, s, tq “ pW, s0, t0q is a path holonomy +bi-submersion associated to some generators X1, . . . , Xn P F as in Example 10.1.2. +Fix pu, y “ +py1, . . . , ynqq P W Ă M ˆ Rn, set Y :“ řd +i“1 yiXi. Since dϕY +1 pXq “ pϕY +1 q˚pXq P X ` F, there exists +Zy P F depending in smoothly on y such that dt0pX, 0q “ X ` Zy. Take rZy P t´1 +0 pFq such that +dt0p rZyq “ Zy. One has, +dt0 +´ +pX, 0q ´ rZy +¯ +“ X “ ds0pX, 0q. +We can write rZy “ rZ1 +y ` rZ2 +y, with rZ1 +y P Γpker ds0q, rZ2 +y P Γpker dt0q. By construction, r +X :“ pX, ´ rZ1 +yq +is a lift of X to the bi-submersion pW, s0, t0q. This proves item 1. +If XB P XpBq and XB1 P XpB1q are two lifts of the symmetry X on the path holonomy bi- +submersions pB, s, tq and pB1, s1, t1q respectively, then pXB, XB1q is a lift of X on the composition +bi-submersion B ˝ B1. This proves item 2, since the path holonomy atlas is made of fibered prod- +ucts and inverse of holonomy path holonomy bi-submersions [AS09]. Also Xa is a symmetry of pB, t, sq. +Item 2 in Proposition 2.10 of [AS09] states that if the identity of M is carried by pB, s, tq at some +point v P B, then there exists an open neighborhood V Ă B of v that satisfies s|V “ s0 ˝ g and +t|V “ t0 ˝ g, for some submersion g: V ÝÑ W, for W of the form as in item 1. Thus, for all X P spFq +there exists a vector field r +X P XpV q fulfilling ds|V p r +Xq “ dt|V p r +Xq “ X. This proves item 3. +Definition 10.1.20. A symmetry of the tower of bi-submersion TB “ pBi`1, si, ti, Fiqiě0 is a family +X “ pXiqiě0, with the i-th component Xi in spFiq, such that dsi´1pXiq “ dti´1pXiq “ Xi´1 for all +i ě 1. We denote by spTBq the Lie algebra of symmetries of TB. +The next theorem gives a class of bi-submersion tower to which any symmetry of the base singular +foliation F lifts. + +CHAPTER 10. BI-SUBMERSION TOWERS +158 +Theorem 10.1.21. Let F be a foliation. Let TB be a path holonomy bi-submersion tower (or an +exact path holonomy atlas bi-submersion tower). A vector field X P XpMq is a symmetry of F, i.e. +rX, Fs Ă F, if and only if it is the component on M of a symmetry of TB. +Proof. It is a direct consequence of Proposition 10.1.11 and of item 1. resp. item 2. in Proposition +10.1.18. It is due to the fact that the tower TB is generated by path holonomy bi-submersions, and +then we can lift symmetries at every stage of the tower TB. +Remark 10.1.22. Let pXiqiě0 be a lift of X0 :“ X P spFq. +For i ě 1, ∇i +X :“ adXi preserves +Γpker dsi´1q, since Xi is si´1-projectable. Altogether, they define a chain map p∇i +Xqiě0 at the section +level of the complex (10.8), on projectable vector fields in (10.9), since for every i ě 0 and any +ti-projectable vector field ξ P ker dsi, +dtiprXi`1, ξsq “ rdtipXi`1q, dtipξqs +“ rXi, dtipξqs, +that is dti ˝ ∇i`1 +X +“ ∇i +X ˝ dti. +Remark 10.1.23. In [GZ21], under some assumptions, it is shown that if a Lie group G acts on a +foliated manifold pM, Fq, then it acts on its holonomy groupoid. It is likely that this result follows +from Theorem 10.1.21, this will be addressed in another study. +10.1.2 +Lifts of actions of a Lie algebra on a bi-submersion tower +We end the section with the following constructions and some natural questions. +Let TB “ pBi`1, si, ti, Fiqiě0 be an exact path holonomy bi-submersion tower over a singular +foliation pM, Fq. +By Theorem 10.1.21, any vector field X P spFq lifts to a symmetry pXiqiě0 of TB. Once a lift is +chosen, we can define a linear map, +X P spFq ÞÑ pXiqiě1 P spTBq. +Let ϱ: g Ñ XpMq be a strict symmetry action of a Lie algebra g on pM, Fq. For x P g, there exists +pϱpxqiqiě0, with ϱpxqi P spFiq Ă XpBiq a symmetry of TB such that ϱpxq0 “ ϱpxq P spFq, by Theorem +10.1.21. Consider the composition, +x P g ÞÝÑ ϱpxq P spFq ÞÝÑ pϱpxqiqiě0 P spTBq ÞÑ ϱpxq1 P XpB1q. +(10.14) +Lemma 10.1.24. For all x, y P g, +rϱpxq, ϱpyqs1 ´ rϱpxq1, ϱpyq1s “ dt1pC1px, yqq +with C1px, yq P Γpker ds1 Ñ B2q a t1-projectable vector field, for some bilinear map +C1 : ^2 g ÝÑ Γpker ds1 Ñ B2q. +Proof. This follows from Lemma 10.1.7, because rϱpxq, ϱpyqs1 ´rϱpxq1, ϱpyq1s P Γpker ds0qXΓpker dt0q. + +CHAPTER 10. BI-SUBMERSION TOWERS +159 +Theorem 10.1.25. The map C1 : ^2 g ÝÑ Γpker ds1 Ñ B2q of Lemma 10.1.24 satisfies for all +x, y, z P g, +C1prx, ysg, zq ` ∇2 +ϱpxqpC1py, zqq` ö px, y, zq “ dt2pC2px, y, zqq +(10.15) +for some tri-linear map C2 : ^3 g ÝÑ Γpker ds2 Ñ B3q. Here, ∇2 is as in Remark 10.1.22. +Proof. For x, y, z P g, +dt1 pC1prx, ysg, zqq ` ö px, y, zq “ rϱprx, ysgq, ϱpzqs1 ´ rϱprx, ysgq1, ϱpzq1s` ö px, y, zq +“ ((((((((( +( +rrϱpxq, ϱpyqs, ϱpzqs1 ´ rrϱpxq, ϱpyqs1, ϱpzq1s` ö px, y, zq +“ ´(((((((((( +( +rrϱpxq1, ϱpyq1s, ϱpzq1s ` rdt1pC1px, yqq, ϱpzq1s` ö px, y, zq +“ dt1prC1px, yq, ϱpzq2sq` ö px, y, zq. +We have used Jacobi identity and dt1pϱpzq2q “ ϱpzq1. This implies that +dt1 +` +C1prx, ysg, zq ´ rC1px, yq, ϱpzq2s` ö px, y, zq +˘ +“ 0. +(10.16) +Again Lemma 10.1.7 implies the result. +Here is a natural question: +Question: Can we continue the construction in Theorem 10.1.25 to a Lie 8-morphism? +There is another natural type of questions: +Question: Can a strict symmetry action ϱ: g Ñ XpMq of a Lie algebra g on a singular foliation +pM, Fq lift to a given geometric resolution pE‚, d, ρq of F? +Discussion +• Can we answer this question with what we have? +• We know by Theorem 8.3.1 that ϱ lifts on any universal Lie 8-algebroid pE, Qq of F to a Lie +8-morphism Φ: g ÝÑ X‚pEq. If we can choose at least the polynomial-degree zero of the Taylor +coefficient Φ1 : ^2 g Ñ X´1pEq to be zero, then the answer is yes. +• If yes, can we assume that the previous action to preserve the dg-almost Lie algebroid? Again, +if yes, then the polynomial-degree `1 can be chosen to be zero. +It seems that being able to suppress the higher Taylor coefficients has a strong geometric meaning. +We intend to study that problem in a subsequent paper. +Question: Can a bi-submersion tower be equipped with a (local) Lie 8-groupoid structure? + +Appendices +160 + +APPENDIX A +Tensor algebra +We have used almost everywhere in the thesis, modules or vector spaces that arise as the quotient +of the tensor algebra. It is worth it to dedicate a section to recall the construction and some basic +facts on tensor algebras. In this chapter we assume that the reader is familiar with the notion of +O-modules, graded O-modules, and vector spaces. +It is known that many algebras such as the exterior algebra, symmetric algebra, Clifford algebras +[Tod11], universal enveloping algebras [Bek] and many other algebras are the quotient of tensor al- +gebras. These make the tensor algebra a fundamental and a very useful notion. In this thesis we +have dealt a lot with O-multilinear maps, it simply means that these maps are O-linear w.r.t each +argument while we fix the other arguments. The tensor algebra is used to characterize multilinear +relations between algebraic objects related to modules or vector spaces. There are several ways to +construct the tensor algebra, we refer the reader e.g. to chapter 16 of [Lan05] or [And, Kei05] for more +details on the matter. +When O “ K the reader may replace "O-module" by "K-vector space" and the construction of the +tensor algebra that we give below works the same. +A.1 +Tensor product +Let V and W and Z be O-modules. The tensor product V bO W of V and W over O is the O-module +which is defined by the following universal property +Theorem A.1.1. There exists a O-bilinear map, p: V ˆ W Ñ V bO W, that satisfies the following +properties: +1. any O-bilinear map B : V ˆ W Ñ Z admits a unique O-linear map γ : V bO W Ñ Z, such that +161 + +APPENDIX A. TENSOR ALGEBRA +162 +γ ˝ p “ B. In other words, such that the following diagram commutes: +V ˆ W +p +� +B +� +V bO W +γ +� +Z +(A.1) +2. Furthermore, if there is another O-module Q and a O-bilinear map, p1 : V ˆ W Ñ Q with the +property (A.1), then there exists a unique isomorphism i: V bO W Ñ Q such that p1 “ i ˝ p. +Proof. We first show "uniqueness" when such O-module exits, then show existence. +Uniqueness: Suppose that there exist two pairs pVbW, p: VˆW Ñ VbOWq and pV ˜bW, ˜p: VˆW Ñ +V ˜bOWq satisfying the property A.1.1. By using two times A.1.1, we deduce the existence of a unique +O-linear map i: V bO W Ñ V ˜bW, and another one j : V ˜bOW Ñ VbW such that i ˝ p “ ˜p and +j ˝ ˜p “ p. This implies that j ˝ i ˝ p “ p and i ˝ j ˝ ˜p “ ˜p. By unicity of the factorization in A.1.1, we +must have i ˝ j “ id and j ˝ i “ id. This proves item 2. +Existence: Let us consider the free O-module P generated by elements of the Cartesian product +V ˆ W, i.e. elements of P are formal finite sums of elements of the form fpv, wq, with f P O and +v P V, w P W. Next, we divide by the submodule R Ă P of P generated by elements of the following +types: +pv1 ` v2, wq ´ pv1, wq ´ pv2, wq, +pv, w1 ` w2q ´ pv, w1q ´ pv, w2q, +pfv, wq ´ fpv, wq, and pv, fwq ´ fpv, wq +which are the relations that elements of the tensor product must satisfy with v, v1, v2 P V and +w, w1, w2 P W and f P O. In those notations, the tensor product of V and W over O is defined +as the quotient +V bO W :“ P{R. +For v P V, w P W, we denote the class of pv, wq P V ˆ W by v b w P V bO W. This quotient comes +with the natural map, p: V ˆ W Ñ V bO W, pv, wq ÞÑ v b w. For any O-linear map B : V ˆ W Ñ Z, +the O-linear map q: P Ñ Z which is given on the basis of P by px, yq ÞÑ qppx, yqq :“ Bpx, yq clearly +goes to quotient to define a O-linear map sq: V bO W Ñ Z, since e.g. +q ppv1 ` v2, wq ´ pv1, wq ´ pv2, wqq “ qppv1 ` v2, wqq ´ qppv1, wqq ´ qppv2, wqq +“ Bpv1 ` v2, wq ´ Bpv1, wq ´ Bpv2, wq +“ 0, +by bilinearity of B +also, +qppfv, wq ´ fpv, wqq “ qppfv, wqq ´ fqppv, wqq, +by O-lineraity of q +“ Bpfv, wq ´ fBpv, wq “ 0, +by O-bilineraity of B. +We did everything to get, sq ˝ p “ B. Therefore, we can take γ “ sq to satisfy (A.1). + +APPENDIX A. TENSOR ALGEBRA +163 +Proposition A.1.2. Let V, W, Z be O-modules. We have the following isomorphisms +1. V bO W » W bO V. +2. O bO V » V, and for any ideal I Ă O, Also, O +I bO V » +V +IV +3. pV bO Wq bO Z » V bO pW bO Zq. +Proof. For item 1, the map pv, wq P V ˆ W ÞÑ w b v P V bO W goes to quotient to the twist map +v b w ÞÑ w b v and give the isomorphism whose inverse is w b v P W bO V ÞÑ v b w P V bO W. +For item 2, the map pf, vq P O ˆ V Ñ fv also induces an isomorphism whose inverse is v P V ÞÑ +1 b v P O bO V. A similar map gives the second clause. +For item 3, the isomorphism is, obviously, pv1 bv2qbv3 ÞÑ v1 bpv2 bv3q. This is trivially extended +to more O-modules. +Remark A.1.3. A similar construction as in Theorem A.1.1 can be done for a finite family of O- +modules V1, . . . , Vr, as in the case of bilinear maps. Then V1 bO ¨ ¨ ¨ bO Vr is a universal object that +factorizes r-multilinear maps, defined on V1 ˆ ¨ ¨ ¨ ˆ Vr. We naturally have, +pV1 bO V2q bO V3 » V1 bO pV2 bO V3q » V1 bO V2 bO V3. +So, in this thesis, we make no difference in how we denote the elements pv1 b v2q b v3, v1 b pv2 b v3q, +or v1 b v2 b v3. +Remark A.1.4. It is important to notice that if C and C1 are graded O-algebras, then C bO C1 is a +graded O-algebra with product +pc1 b c1 +1qpc2 b c1 +2q :“ p´1q|c1 +1||c2|c1c2 b c1 +1c1 +2 +(A.2) +for homogeneous elements c1, c2 P C and c1 +1, c1 +2 P C1. Moreover, if C and C1 are unitary with units +1C and 1C1, respectively, then the algebra C bO C is also unitary with unit 1C b 1C1. Here, to avoid +explosion of notations, we denote the product of two elements a, b by ab. Also, we will not make +notational distinctions between the unit elements. +Convention A.1.5. For Φ: C Ñ C1 and Ψ: C2 Ñ C3 two homogeneous morphisms of Z-graded O- +modules, then Φ b Ψ : C bO C2 Ñ C1 bO C3 stands for the following morphism: +pΦ b Ψqpx b yq “ p´1q|Ψ||x|Φpxq b Ψpyq, for all homogeneous x P C, y P C2. +Remark A.1.6. In virtue of Remark A.1.4 and Convention A.1.5, if Φ: C Ñ C1 and Ψ: C2 Ñ C3 are +graded algebra morphisms, then ΦbΨ is also a graded algebra morphism w.r.t to the product defined +in (A.2): +Φ b Ψppc1 b c1 +1qpc2 b c1 +2qq “ p´1q|c1 +1||c2|Φ b Ψpc1c2 b c1 +1c1 +2q +“ p´1q|c1 +1||c2|Φpc1c2q b Ψpc1 +1c1 +2q +“ p´1q|c1 +1||c2|Φpc1qΦpc2q b Ψpc1 +1qΨpc1 +2q +“ +` +Φpc1q b Φpc1 +1q +˘ ` +Ψpc2q b Ψpc1 +2q +˘ +“ +` +Φ b Ψpc1 b c1 +1q +˘ ` +Φ b Ψpc2 b c1 +2q +˘ +, +since Φ, Ψ are of degree 0. + +APPENDIX A. TENSOR ALGEBRA +164 +A.2 +The tensor algebra of a linear space +Let V be an O-module. For k P N0, the k-th tensor power T k +OV over O of V (elements of polynomial- +degree k) is the tensor product of V with itself k times, namely +T k +OV :“ V bO ¨ ¨ ¨ bO V +loooooooomoooooooon +k times +we also adopt the convention T 0 +OV » O, and V » T 1 +OV. This leads to consider the O-module con- +structed as the direct sum of the tensor powers, i.e. +T ‚ +OV :“ +8 +à +k“1 +T k +OV “ O ‘ V ‘ pV bO Vq ‘ pV bO V bO Vq ‘ ¨ ¨ ¨ +Proposition A.2.1. The O-module T ‚ +OV comes equipped with a graded unital O-algebra structure, +which is induced by the canonical map +T k +OV ˆ T ℓ +OV ÝÑ T k +OV bO T ℓ +OV » T k`ℓ +O +V, +that is extended by bilinearity to all T ‚ +OV. We denote this product by b. The unit element is 1 P O » +T 0 +OV, in particular, we have 1 b v “ 1 ¨ v “ v for every v P V. +A.2.1 +T ‚ +OV as a co-algebra +In this section, we consider an O-module V which can be possibly graded. However, the construction +is independent whether the module is graded or not. +The notion of co-algebra structure is important in the context of the thesis, since it allows dealing +with infinite dimension objects. It appears all along the thesis. For this concept, see Definition 1.2.1. +A natural way to construct a co-product (see e.g [JLL, Kas12]) structure 1 on T ‚ +OV +∆: T ‚ +OV ÝÑ T ‚ +OV +â +T ‚ +OV +is to define it on elements v P V » T 1 +OV of polynomial-degree 1, and on the unit element 1 P O » T 0 +OV +and extend it to a (degree 0) O-algebra morphism to the whole T ‚ +OV, namely for v1 b ¨ ¨ ¨ b vk P T k +OV, +∆pv1 b ¨ ¨ ¨ b vkq “ ∆pv1q ¨ ¨ ¨ ∆pvkq. +The one which is defined by +v ÞÑ v b 1 ` 1 b v +1 ÞÑ 1 b 1 +endows T ‚ +OV with a co-associative (co-commutative) co-algebra structure, that is, it satisfies the axioms +of Definition 1.2.1. Indeed, the maps +∆ b id, id b ∆: T ‚ +OV b T ‚ +OV Ñ T ‚ +OV b T ‚ +OV b T ‚ +OV +1here b is an "outer" tensor product, it should not be confused with the internal tensor product in T ‚ +OV that denotes +also its graded algebra structure. + +APPENDIX A. TENSOR ALGEBRA +165 +are algebra morphisms, so are p∆ b idq ˝ ∆ and pid b ∆q ˝ ∆, therefore it suffices to check that they +coincide on V. Let us check that. For v P V, +p∆ b idq ˝ ∆pvq “ ∆ b idpv b 1 ` 1 b vq +“ ∆pvqb1 ` ∆p1qbv +“ pv b 1 ` 1 b vq b 1 ` p1 b 1q b v +“ pv b 1q b 1 ` p1 b vq b 1 ` p1 b 1q b v +“ pid b ∆q ˝ ∆pvq, +by associativity of b. +By extending ∆ to an algebra morphism, one gets explicit expressions as follows. For v b w P T 2 +OV, +one has +∆pv b wq “ ∆pvq∆pwq +“ pv b 1 ` 1 b vqpw b 1 ` 1 b wq +“ pv b wq b 1 ` v b w ` p´1q|v||w|w b v ` 1 b pv b wq. +We have used Formula (A.2) and 1 b v “ 1 ¨ v “ v, each time b is the tensor symbol in T ‚ +OV. If V +is not graded, i.e., concentrated in degree zero, there is no sign p´1q|v||w|. More generally, for every +v1, . . . , vn P V, +∆pv1 b ¨ ¨ ¨ b vnq “ +n´1 +ÿ +i“1 +ϵpσq +ÿ +σPSpi,n´iq +vσp1q b ¨ ¨ ¨ b vσpiq +â +vσpi`1q b ¨ ¨ ¨ b vσpnq, +(A.3) +where σ P Spi, n ´ iq is a pi, n ´ iq-shuffle and ϵpσq is the Koszul sign associated to the n-uplet +v1, . . . , vn. See Section 1.1 for more details. + +APPENDIX B +Homological algebra +The goal of this chapter is to introduce some important results on homological algebra, which are used +in this thesis. Although, these are classical notions in commutative algebra, I think it is important to +make a brush-up on them for the readability of the thesis. Most of the notions of this chapter can be +found in [Cha14, Eis95, Hid89, Mic07], I also have learned a lot from the Lecture notes [Las18]. +B.1 +Complexes of modules +We recall that a module over O is like a vector space in the sense that all the axioms still hold, except +that the underlying field is replaced by O. In this section, when O “ K, the reader may replace +"module" by "vector space". +For us, a Z-graded module over O is a module V endowed with a direct sum decomposition V “ +‘iPZVi of O-modules. +We simply say "O-modules" for graded modules which are concentrated in +degree zero. For every i P Z, elements of Vi are said to be of degree i. Let V and W be graded +Z-modules over O, a O-linear map L: V Ñ W is said to be homogeneous or a morphism of Z-graded +O-modules of degree |L| :“ ℓ P Z, if LpVkq Ď Vk`ℓ for all k P Z. +The set of all O-linear maps +of degree ℓ from V to W form an O-module that we denote by Homℓ +OpV, Wq. +This implies that +HomOpV, Wq :“ À +ℓPZ Homℓ +OpV, Wq is a graded module. Also, this graded module comes equipped +with natural graded Lie bracket given by the graded commutator, namely +rF, Gs :“ F ˝ G ´ p´1q|F||G|G ˝ F +(B.1) +for any homogeneous elements F, G P HomOpV, Wq. It is easily checked that the bracket satisfies +1. rF, Gs “ ´p´1q|F||G|rG, Fs +(graded skew-symmetry) +2. p´1q|F||H|rF, rG, Hss`p´1q|H||G|rH, rF, Gss`p´1q|G||F|rG, rH, Fss “ 0, +(graded Jacobi identity) +for homogeneous O-linear maps F, G, H P HomOpV, Wq. +166 + +APPENDIX B. HOMOLOGICAL ALGEBRA +167 +Definition B.1.1. A complex of O-modules pV‚, dq is a graded module V “ À +iPZ Vi together with a +squared to zero O-linear map d: V Ñ V of degree `1 called the differential map. In other words, it is +a sequence +¨ ¨ ¨ ÝÑVi´1 +d +ÝÑ Vi +d +ÝÑ Vi`1ÝÑ ¨ ¨ ¨ +(B.2) +O-linear maps such that d2 “ d ˝ d “ 0. +1. For every i P Z, elements of Vi are called cochains of degree i. +2. We say that (B.2) is bounded below/above if Vi “ 0 for i ď n{i ě n, for some n P Z. +3. A subcomplex of a complex pV, dq a collection of O-modules pV1 +i Ď ViqiPZ such that dpV1 +iq Ă Vi`1 +for each i P Z. In particular pV1, d1 “ d|V1q is a complex of O-modules. +In particular, it induces a complex +` +V{V1, d +˘ +called the quotient complex where for i P Z, +pV{V1qi :“ Vi{V1 +i and +d: Vi{V1 +i ÝÑ Vi`1{V1 +i`1 +is determined uniquely by the universal property of the quotient. +Remark B.1.2. Let pV, dq be a complex of O-modules. Denote by di : Vi Ñ Vi`1 the restriction of d +to Vi. Then d2 “ 0 means that di ˝ di´1 for each i P Z. In particular, Imdi´1 Ď ker di for every i P Z. +This leads us to the next definition. +Definition B.1.3. Let pV, dq be a complex of O-modules. For each i P Z, +1. a i-cocycle of pV, dq is an element of kerpVi +d +ÝÑ Vi`1q; +2. a i-coboundary of pV, dq is an element of ImpVi´1 +d +ÝÑ Viq; +3. the i-th cohomology group of pV, dq is the quotient HipVq :“ kerpVi +d +ÝÑ Vi`1q +ImpVi´1 +d +ÝÑ Viq +. +The complex pV, dq is exact at i if HipVq “ t0u. It is said to be exact or acyclic if it is exact at every +degree i P Z. +Definition B.1.4. Let pV, dVq and pW, dWq be complexes of O-modules. +1. A chain map or complex of O-modules morphism between the complexes pV, dq and pW, dq is a +O-linear map L: V Ñ W of degree 0, which commutes with the differentials, that is a collection +of O-linear map L‚ : V‚ ÝÑ W‚, such that the following diagram commutes +¨ ¨ ¨ +� Vi +Li +� +dV � Vi`1 +Li`1 +� +� ¨ ¨ ¨ +¨ ¨ ¨ +� Wi +dW � Wi`1 +� ¨ ¨ ¨ +(B.3) +i.e. dW ˝ Li “ Li`1 ˝ dV for every i P Z. + +APPENDIX B. HOMOLOGICAL ALGEBRA +168 +2. A homotopy between two chain maps K‚, L‚ : V‚ ÝÑ W‚ is the datum thi : Vi ÝÑ Wi´1uiě1 of +O-linear maps, that satisfies for each i P Z, Ki ´ Li “ dW ˝ hi ` hi´1 ˝ dV. These maps are +displayed in the following diagram as +¨ ¨ ¨ +� Vi´1 +Ki´1´Li´1 +� +dV +� Vi +hi +� +Ki´Li +� +dV � Vi`1 +Ki`1´Li`1 +� +hi`1 +� +� ¨ ¨ ¨ +¨ ¨ ¨ +� Wi´1 +dW +� Wi +dW � Wi`1 +� ¨ ¨ ¨ +(B.4) +(a) When there is a homotopy between two chain maps, L‚, K‚ : V‚ ÝÑ W‚, we often write +L „ K. One can check that „ is indeed an equivalence relation. +(b) Two complexes of O-modules pV, dVq and pW, dWq are said to be homotopy equivalent, if +there exist chain maps L‚ : V‚ ÝÑ W‚ and K‚ : W‚ ÝÑ V‚ such that L ˝ K „ idW‚ and +K ˝ L „ idV‚. Likewise, one can check that homotopy equivalence between complexes of +O-modules is an equivalence relation. +Remark B.1.5. Note that in particular, if L‚ : V‚ ÝÑ W‚ is a chain map that is an O-linear iso- +morphism, then its inverse is also a chain map. Thus, they define a homotopy equivalence between +pV, dVq and pW, dWq. +Remark B.1.6. Let L‚ : V‚ ÝÑ W‚ be a chain map. For every i P Z, we have +L +ˆ +kerpVi +dV +ÝÑ Vi`1q +˙ +Ď kerpWi +dW +ÝÑ Wi`1q, and L +ˆ +ImpVi´1 +dV +ÝÑ Viq +˙ +Ď ImpWi´1 +dW +ÝÑ Wiq. +L induces naturally a well-defined O-linear map HpLq: HpVq Ñ HpWq, rvs ÞÑ rLpvqs: for every +v P kerpVi +dV +ÝÑ Vi`1q and v0 P Vi´1 +HpLqprv ` dVpv0qsq “ rLpv ` dVpv0qqs “ rLpvq ` dW ˝ Lpv0qqs “ rLpvqs “ HpLqprvsq. +Notice that +1. homotopic chain maps induce the same map on cohomology groups. +2. if the complexes of O-modules pV, dVq and pW, dWq are homotopy equivalent through L, then +HpLq: HpVq Ñ HpWq is an isomorphism. +Proposition B.1.7. If pV1, d1q is an acyclic subcomplex of a complex pV, dq, then +H‚pV{V1q » H‚pVq. +Proof. The projection p‚ : V ÝÑ V‚{V1 +‚ is a chain map from pV, dq and pV{V1, dq. We claim p‚ induces +an isomorphism on the cohomology groups. To show injectivity of Hppq: H‚pVq ÝÑ H‚pV{V1q, let +e P ker d such that ppeq “ e P Im d, i.e. there is u P V|e|´1 such that e “ dpuq. It follows that +e ´ du P V1 +|e|. This implies, +d1pe ´ duq “ dpe ´ duq “ 0. +(B.5) +By Exactness of pV1, d1q, +ùñ e ´ du “ d1pvq, +(for some v P V1 +|e|´1 Ă V|e|´1) +ùñ e “ dpu ` vq P Im d. + +APPENDIX B. HOMOLOGICAL ALGEBRA +169 +This proves injectivity. Surjectivity goes as follows: let e P Vi such that dpeq “ 0. +dpeq “ 0 ùñ dpeq P V1 +i`1 +We have, d1pdpeqq “ d ˝ dpeq “ 0. By exactness of pV1, d1q, we can write dpeq “ d1pvq for some v P V1 +i. +This implies that, e ´ v “ u P ker d. Therefore, +Hppqprusq “ rppuqs “ rppeq ´ ppvqqs “ rppeqs “ res. +This completes the proof. +The Chevalley-Eilenberg complex +The following example of complex is important. We have used it several times in this thesis. Especially +in Chapter 9 to define obstruction classes. Let us recall the definition. We refer the reader e.g. to +[Wag10] for more details. +Let pg, r¨ , ¨sgq be a Lie algebra and V a K-vector space. +Definition B.1.8. A representation or action of g on V is a Lie algebra morphism +ν : pg, r¨ , ¨sgq ÝÑ pEndpV q, r¨ , ¨sq +(B.6) +where pEndpV q, r¨ , ¨sq denotes the vector space EndpV q of endomorphisms of V together with the Lie +bracket r¨ , ¨s which is the commutator: rα, βs “ α ˝ β ´ β ˝ α, @ α, β P EndpV q. In this case, V is then +called a g-module (w.r.t to ν). In the literature, the action ν is often denoted by ¨. +Equation (B.6) means that for all x, y P g, +νprx, ysgq “ rνpxq, νpyqs “ νpxq ˝ νpyq ´ νpyq ˝ νpxq. +Example B.1.9. Here are two important examples of g-modules that we often use. +• The adjoint action: g acts on itself by the Lie bracket, i.e. νpxqpyq :“ rx, ysg for all x, y P g. +Indeed, ν : g ÝÑ Endpgq, x ÞÝÑ rx, ¨ sg “: adx is a Lie algebra morphism: +adrx,ysg “ rrx, ysg, ¨ sg +“ ´rry, ¨ sg, xsg ´ rr¨ , xsg, ysg, +pby identity of Jacobiq +“ rx, ry, ¨ sgsg ´ ry, rx, ¨ sgsg +“ adx ˝ ady ´ ady ˝ adx “ radx, adys. +• The trivial action: i.e. K is a g-module through the action νpxqpλq :“ 0 for all x P g and all +λ P K. +‘ +Now let us recall the definition of the Chevalley-Eilenberg complex. + +APPENDIX B. HOMOLOGICAL ALGEBRA +170 +Definition B.1.10. Let ν : g ÝÑ EndpV q be an action of g on V . The Chevalley-Eilenberg complex +of g valued in V is the complex +¨ ¨ ¨ ÝÑHomKp^i´1g, V q dCE +ÝÑ HomKp^ig, V q dCE +ÝÑ HomKp^i`1g, V qÝÑ ¨ ¨ ¨ +(B.7) +whose i-th cochains space is defined to be HomKp^ig, V q, the vector space of i-linear skew-symmetric +linear maps from g ˆ ¨ ¨ ¨ ˆ g +looooomooooon +i-times +to V , under the convention HomKp^0g, V q » V . The differential map is +defined for µ P HomKp^ig, V q by +´ +dCEµ +¯ +px1, . . . , xi`1q “ +i`1 +ÿ +k“1 +p´1qk´1νpxkqpµpx1, . . . , pxk, . . . , xi`1qq +` +ÿ +1ďkălďi`1 +p´1qk`lµprxk, xlsg, x1, . . . , pxkl, . . . , xi`1q +where pxk means xk is missing in the k-th place also pxkl means that xk, xl are missing in the k-th and +l-th place respectively. +Remark B.1.11. +1. It follows from the identity of Jacobi that dCE ˝ dCE “ 0. +2. For all x, y, z P g we have, +• dCEpxq “ νpxq. +• For µ P HomKpg, V q, +` +dCEµ +˘ +px, yq “ νpxqpµpyqq ´ νpyqpµpxqq ´ µprx, ysgq. +• For η P HomKp^2g, V q, +´ +dCEη +¯ +px, y, zq “ νpxqpηpy, zqq ´ νpyqpηpx, zqq ` νpzqpηpx, yqq +´ ηprx, ysg, yq ` ηprx, zsg, yq ´ ηpry, zsg, xq. +Remark B.1.12. When g is of finite dimension, the Chevalley-Eilenberg complex (B.7) is canonically +isomorphic to the complex p^‚g˚ b V, dq and for ξ “ ξ1 ^ ¨ ¨ ¨ ^ ξk P ^kg˚, +dpξ b vq “ +nÿ +i“1 +pξ ^ ξiq b pξi ¨ vq ´ dgpξq b v +(B.8) +where ξ1, . . . , ξn is a basis of g. In the formula (B.8), dg is the Chevalley-Eileberg differential of g w.r.t +the trivial action on K. +B.1.1 +Operations on complexes +We have used and adapted the following lemma many times in this thesis, e.g. Section 4.3.1. + +APPENDIX B. HOMOLOGICAL ALGEBRA +171 +Lemma B.1.13. Let pV‚, dVq and pW‚, dWq be complexes of O-modules. Then, the pair pHom‚ +OpV, Wq , Bq +is a complex of O-modules, where the differential map is given by +BpFq :“ dW ˝ F ´ p´1q|F|F ˝ dV, +(B.9) +for all homogeneous element F P HomOpV, Wq. +Proof. The O-linear map, B: HomOpV, Wq Ñ HomOpV, Wq, is clearly of degree `1, since the maps +dV and dW are. Moreover, for every F P HomOpV, Wq +B2pFq “ BpdW ˝ F ´ p´1q|F|F ˝ dVq +“ dW ˝ dW ˝ F +loooooomoooooon +“0 +´p´1q|F|`1(((((( +dW ˝ F ˝ dV ´ p´1q|F| +˜ +(((((( +dW ˝ F ˝ dV ´ p´1q|F|`1 F ˝ dV ˝ dV +looooomooooon +“0 +¸ +“ 0. +Remark B.1.14. It’s worth it to notice that +1. the cocycles F of pHom‚ +OpV, Wq, Bq are those that satisfy dV ˝ F “ p´1q|F|F ˝ dW. In particular, +the chain maps between pV, dVq and pW, dWq are the 0-cocyles of pHom‚ +OpV, Wq , Bq. +2. two chain maps F‚, G‚ : V‚ ÝÑ W‚ are homotopic if and only if G ´ F is 0-coboundary of +pHom‚ +OpV, Wq , Bq, that is, there exists a O-linear map H P Hom´1 +O pV, Wq of degree ´1 such +that +F ´ G “ dW ˝ H ` H ˝ dV. +3. when pV‚, dVq “ pW‚, dWq, we have +` +Hom‚ +OpV, Vq, B “ rdV, ¨ s +˘ +. +Direct sum and tensor product of complexes +Let pV‚, dVq and pW‚, dWq be complexes of O-modules. Then, the tensor product V‚ bO W‚ together +with the grading +pV bO Wqk “ +à +redi`j“k +Vi bO Wj +for k P Z, comes equipped with a differential map classically defined by +B “ dV b id ` id b dW, +namely, +Bpv b wq “ dVpvq b w ` p´1q|v|v b dWpwq, +for all homogeneous elements v, w P V bO W, is complex of O-modules. The operator B is indeed of +degree `1. It is easily checked that B2 “ 0. + +APPENDIX B. HOMOLOGICAL ALGEBRA +172 +Bi-complex +Definition B.1.15. A bi-complex or double complex is a collection of O-modules V “ pVi,jqi,jPZ +together with two families of O-linear maps +1. dh +i,j : Vi,,j Ñ Vi`1,j, such that dh +i,j ˝ dh +i´1,j “ 0, +for i, j P Z called horizontal differential map +2. dv +i,j : Vi,j Ñ Vi,j`1, such that dv +i,j ˝ dv +i,j´1 “ 0, +for all i, j P Z called vertical differential map +that obey for all i, j P Z the identity +dh +i,j`1 ˝ dv +i,j “ dv +i,j ˝ dh +i`1,j. +In this thesis we only consider first quadrant bi-complexes that is, Vi,j “ 0 for all i P Zď0 and j P N0. +These can be represented as the commutative diagram +... +... +... +Ò +Ò +Ò +¨ ¨ ¨ +Ñ +Vi,j +dh +Ñ +Vi,j +dh +Ñ +Vi,j +Ñ +0 +dv Ò +dv Ò +dv Ò +¨ ¨ ¨ +Ñ +Vi,j +dh +Ñ +Vi,j +dh +Ñ +Vi,j +Ñ +0 +dv Ò +dv Ò +dv Ò +¨ ¨ ¨ +Ñ +Vi,j +dh +Ñ +Vi,j +dh +Ñ +Vi,j +Ñ +0 +Ò +Ò +Ò +0 +0 +0 +"-2 column" +"-1 column" +"last column" +(B.10) +One associate to the bi-complex (B.10) the so-called total complex which is defined by the anti- +diagonals of (B.10), namely +´ +Tr :“ À +i`j“r Vi,j +¯ +r with total differential D: Tr Ñ Tr`1 defined by +Dpτijq :“ dhpτijq ´ p´1qrdvpτijq, for τij P Vi,j. +Indeed, +D2 “ pdh ´ p´1qr`1dvq ˝ pdh ´ p´1qrdvq +“ pdhq2 +loomoon +“0 +´ + +p´1qrdh ˝ dv ´ ((((((( +( +p´1qr`1dv ˝ dh ´ pdvq2 +loomoon +“0 +“ 0. +Proposition B.1.16 (Acyclic Assembly Lemma [Cha14]). Let pVi,jqi,jPZ be a first quadrant bi-complex +like in the notation above such that the rows are exact, then the total complex pT‚, Dq is exact. +Mapping cone. Let pV, dVq and pW, dWq be two complexes and L: V‚ Ñ W‚ a chain map. The +mapping cone is the complex pC, Bq whose degree i is given by Vi´1 ‘ Wi and whose differential is +defined as +B “ +˜ +´dV +0 +´L +dW +¸ + +APPENDIX B. HOMOLOGICAL ALGEBRA +173 +That is, the differential is given on elements pv, wq P V ‘ W by +Bpv, wq “ p´dVpvq, dWpwq ´ Lpvqq. +The mapping cone pC, Bq is exact if and only if L: V‚ Ñ W‚ is a quasi-isomorphism, [Cha14], Cor. +1.5.4 p.19. +B.2 +Resolutions of a module +Definition B.2.1. An O-module V is said to be projective if it fulfills the following: given a O-linear +map L: V : Ñ Z, every surjective O-linear map J : W Ñ Z admits a O-linear map rL: V Ñ W such +that the following diagram commutes +V +� +� +W +� Z +� 0 +(B.11) +Example B.2.2. Free modules are projective modules (see e.g. Proposition 5.1.2 of [Mic07]). +Definition B.2.3. Let A be a O-module. A free/projective resolution (or resolution by free/projective +modules) of A is an exact complex pV‚, dq +¨ ¨ ¨ ÝÑV´i´1 +d +ÝÑ V´i +d +ÝÑ V´i`1 +d +ÝÑ ¨ ¨ ¨ +d +ÝÑ V´2 +d +ÝÑ V´1 +π +ÝÑ A ÝÑ 0 +(B.12) +such that the V´i’s are free/projective modules. +Remark B.2.4. The complex (B.12) may be of infinite length. When the length is finite, we say that +we have a finite free/projective resolution. In that case, the sequence +0ÝÑV´n +d +ÝÑ V´n`1 +d +ÝÑ V´n`2 +d +ÝÑ ¨ ¨ ¨ +d +ÝÑ V´2 +d +ÝÑ V´1 +π +ÝÑ A ÝÑ 0 +(B.13) +is exact in every degree. +In particular, the sequence 0 Ñ V´n +dÑ Vn´1 is exact at ´n. +For +instance, the map V´n +dÑ Vn´1 injective. Thus, V´n » ImpV´n +dÑ Vn´1q. By exactness, we have that +V´n » ImpV´n +dÑ Vn´1q “ kerpV´n`1 +dÑ V´nq. +Proposition B.2.5. Every O-module A admits a free/projective resolution. +Proof. Firstly, notice that every module is isomorphic to a quotient of a free module: to see this, +choose a set of generators tvi P A | i P Iu of the module A so that V “ +ÿ +iPI +Ovi. The O-linear map +π: +à +iPI +O Ñ A, pfjqjPJĎI ÞÑ +ÿ +jPJ +fjvj, J is finite +is surjective. +By the first isomorphism theorem, it follows that A » pÀ +iPI Oq{ ker π. +Now put +À +iPI O “: V´1. This yields an exact sequence +0 ÝÑ ker π ãÝÑ V´1 +π +ÝÑ AÝÑ0. +But ker π does not need to be a free O-module, but there is a free O-module V´2 together with a +surjective O-linear map V´2 +π1 � � ker π such that V´2{ ker π1 » ker π. This is added to the previous +sequence as follows + +APPENDIX B. HOMOLOGICAL ALGEBRA +174 +0 +� ker d1 � � +� V´2 +π1 � � +d1 +� � +ker π � � +� V´1 +π +� � A . +Once again, ker d1 does not need to be a free O-module, therefore we can continue the procedure. +Inductively, assume that we have constructed a O-linear map, dn : V´n´1 Ñ V´n, for n ě 2. Thus, +there exists a free O-module V´n´2 together with a surjective O-linear map πn`1 : V´n´2 Ñ V´n´1 +such that V´n´2{ ker πn`1 » ker dn. Joining this to the previous sequence, one get +0 +� ker dn`1 � � +� V´n´2 +πn`1� � +dn`1 +� � +ker dn � � +� V´n´1 +dn � � V´n +� ¨ ¨ ¨ +� V´2 +d1 +� V´1 +π +� � A +(B.14) +By construction, we have kerpdnq “ Impdn`1q. Therefore, we have built an exact sequence up to +length n ` 1. This completes the proof. +One shall notice that the process (B.14) may be continued forever without reaching a free kernel. +Here is an important operation on complexes. The following Proposition states the localization in +the sense of item 2 of Section 3.2.1 preserves exactness. More precisely, +Proposition B.2.6 ([Sta22], Section 10.9 or [And]). Let A be an O-module and S Ă O a multiplicative +subset. For any free resolution of A +¨ ¨ ¨ +d +� V´3 +d +� V´2 +d +� V´1 +π +� A , +the complex +¨ ¨ ¨ +S´1d� S´1V´3 +S´1d � S´1V´2 +S´1d � S´1V´1 +S´1π � S´1A +is a free resolution of S´1A by S´1O-modules. +It is well-known that a submodule of a finitely generated module is not finitely generated in +general. The following assertion guarantees this for finitely generated modules over Noetherian rings +(see Proposition 1.4 of [Eis95], Page 28). +Proposition B.2.7. If O is Noetherian as a ring, then all submodules of an O-module V are finitely +generated if and only if V is finitely generated. +The following theorem assures existence of free resolution of finite length in a special case. +Theorem B.2.8 (Hilbert Syzygy Theorem). [Pee, Eis04] Assume O “ Crx1, . . . , xds. Any finitely +generated (graded) O-module A admits a finite graded free resolution by finitely generated O-modules +¨ ¨ ¨ +d +� V´3 +d +� V´2 +d +� V´1 +π +� A +of length N ď d ` 1. + +APPENDIX B. HOMOLOGICAL ALGEBRA +175 +Tor complex +Let pV‚, dV, πq be a projective resolution of an O-module A. For any O-module B, we consider the +complex +¨ ¨ ¨ +� V´3 bO B +dVbid� V´2 bO B +dVbid� V´1 bO B +dVbid � A bO B . +We define for i ě 0 +Tor´ipA, Bq :“ H´ipV‚ bO Bq. +Here are some properties of Tor [Las18]. +Proposition B.2.9. The functor Tor satisfies +1. Tor0pA, Bq » A bO B. +2. If A is projective, then Tor´ipA, Bq “ 0 for every i ě 1. +3. If B is flat, then Tor´ipA, Bq “ 0 for every i ě 1. +4. For all i, Tor´ipA, Bq “ Tor´ipB, Aq. +5. The construction of TorpA, Bq is independent of the choice of the resolution, i.e. +any other +projective resolution of A or B yields the same Tor groups. +Minimal resolutions +Detailed definitions on local rings can be found in [Mas62]. +Definition B.2.10. +1. A ring R is said to be a local ring if it has a "unique maximal ideal", i.e., +a proper ideal m Ă R such that m contains every other ideal of R. +2. A local ring R with maximal ideal m is called regular if m can be generated by n elements, where +n “ dim R is the krull dimension1 of R. +Assume now that R is a local ring with maximal ideal m, and let K :“ R{m. Geometrically, local +rings correspond to germs of functions on a manifolds or affine variety at a point. +We assume that pR, mq is a local Noetherian commutative ring. Here is an important lemma that +uses definition of local rings. +Lemma B.2.11 (Nakayama). Let V be a finitely generated R-module such that r “ dimpV{mVq ă 8. +Then, any basis of the vector space V{mV lifts to a (minimal) generating set for V as a R-module. In +particular, V can be generated by r elements. +Proof. [Eis04] Lemma 1.4, p. 6. +Remark B.2.12. By Nakayama Lemma, a local ring R is regular if the dimension of the R{m-vector +space m{m2 is dim R. +1the Krull dimension of R is by definition the supremum of lengths of all chains of prime ideals in R [Hid89], p. 30 + +APPENDIX B. HOMOLOGICAL ALGEBRA +176 +One can construct free resolutions of finitely generated modules over R like in the proof of Propo- +sition B.2.5 by taking a minimal set of homogeneous generators at each step. This can be formalized +as follows, +Definition B.2.13. A projective resolution +¨ ¨ ¨ +d +� V´3 +d +� V´2 +d +� V´1 +� +π +� A +� 0 +(B.15) +of a finitely generated R-module A is said to be minimal if the differential map satisfies dpV´iq Ď +mV´i`1 for all i ě 2. +Proposition B.2.14. [Eis04] Every finitely generated R-module A admits a minimal free resolution. +Proof. The proof goes just like in Proposition B.2.5, one just need to take minimal set of generators +in the construction. By doing so, it suffices to check that dpV´2q Ă V´1: Let r “ dimpA{mAq. By +Nakayama Lemma, we can take V´1 “ Rr. One has a short exact sequence, +0 ÝÑ dpV´2q “ ker π ãÝÑ V´1 ÝÑ A ÝÑ 0. +It induces an exact sequence +dpV´2q{m dpV´2q ãÝÑ V´1{mV´1 ÝÑ A{mA ÝÑ 0, +by tensorizing with K “ R{m. Since by construction V´1{mV´1 » A{mA » Kr, the image of the +map dpV´2q{m dpV´2q ãÝÑ V´1{mV´1 is zero, i.e., dpV´2q Ď mV´1. Likewise, by Noetheriality of R, +ker π Ă V´1 is finitely generated. One can repeat the procedure by starting with a minimal set of +generators of ker π and so on. The proof continues by recursion. +Remark B.2.15. Minimal resolutions and the Tor complex: given a minimal projective resolution as +in (B.15), its quotient by the maximal ideal m corresponds to the complex +¨ ¨ ¨ +� V´3 bO K +dbid � V´2 bO K +dbid � V´1 bO K +πbid � A bO K +� 0 +whose cohomology compute Tor‚pA, Kq. By minimality of pV‚, d, πq one has d b id ” 0, therefore +Tor´ipA, Kq » V´ibO K for every i ě 2. In particular, the ranks of the V´i’s for i ě 2 are independent +of the choices made in the construction of the minimal projective resolution. +Koszul complex +Assume that V is a free O-module of finite rank n, which is concentrated in degree ´1. Here, we +denote by Ź‚ V the graded symmetric algebra of V. Let F : V Ñ O be a O-linear map. Given a free +basis e1, . . . , en of V, F is completely determined by n-uplet pf1, . . . , fnq Ă O. +Definition B.2.16. [Eis95, Hid89] The Koszul complex associated to pf1, . . . , fnq is the complex +0 ÝÑ +n +ľ +V +d +ÝÑ +n´1 +ľ +V +d +ÝÑ ¨ ¨ ¨ ÝÑ +2 +ľ +V +d +ÝÑ V +F +ÝÑ O. +(B.16) +whose differential d: Ź‚ V +d +ÝÑ Ź‚´1 V is the unique derivation of Ź‚ V such that, d|V “ F. In other +words, +dpei1 ^ ¨ ¨ ¨ ^ eikq “ +kÿ +j“1 +p´1qj´1Fpeijqei1 ^ ¨ ¨ ¨ ˆeij ¨ ¨ ¨ ^ eik. +(B.17) + +APPENDIX B. HOMOLOGICAL ALGEBRA +177 +It is easily checked that this gives a well-defined complex. +Definition B.2.17. An ordered sequence of elements f1, . . . , fr P O is called a regular sequence on a +module V if the following conditions are satisfied +1. fi is not a zero divisor on the quotient V{xf1, . . . , fi´1yV, +2. xf1, . . . , fryV ‰ V. +Proposition B.2.18. If pf1, . . . , fkq is regular sequence on O, then the Koszul complex (B.16) has no +cohomology in degree less equal to ´1. +Proof. Theorem 16.5. of [Hid89]. +Example B.2.19. For O “ Krx1, . . . , xds be the polynomial ring in d indeterminates, the Koszul +complex which is associated to px1, . . . , xdq induces a free resolution of K » O{xx1, . . . , xdy. +B.3 +Geometric resolutions of a singular foliation +Let us start with this definition. +Definition B.3.1. Let F be a singular foliation on M. A complex of vector bundles over F consists +of a triple pE‚, d‚, ρq, where +1. E‚ “ pE´iqiě1 is a family of vector bundles over M, indexed by negative integers. +2. dpi`1q P HompE´i´1, E´iq is a vector bundle morphism over the identity of M called the differ- +ential map +3. ρ: E´1 ÝÑ TM is a vector bundle morphism over the identity of M called the anchor map with +ρpΓpE´1qq “ F. +such that +¨ ¨ ¨ +� E´i´1 +dpi`1q � +� +E´i +dpiq � +� +Ei´1 +� +� +dp2q� E´1 +ρ +� +� +TM +� +M +M +M +M +M +(B.18) +which form a (chain) complex, i.e. +dpiq ˝ dpi`1q “ 0 and ρ ˝ dp2q “ 0. +Remark B.3.2. Two main cohomology groups can be associated to a complex of vector bundles over +F: +1. Cohomology at the level of sections. +The complex of vector bundles (B.18) induces a +complex of sheaves of modules over functions. More explicitly, for every open subset U Ă M, +there is a complex: +¨ ¨ ¨ ÝÑΓUpE´i´1q dpi`1q +ÝÑ ΓUpE´iq dpiq +ÝÑ ΓUpE´i`1qÝÑ ¨ ¨ ¨ dp2q +ÝÑ ΓUpE´1q +ρ +ÝÑ FU Ă XpUq. + +APPENDIX B. HOMOLOGICAL ALGEBRA +178 +Since Impdpi`1qq Ď ker dpiq for every i P N, we are allowed to define the quotient spaces, +H´ipE‚, Uq “ +$ +’ +’ +’ +’ +’ +’ +’ +’ +’ +& +’ +’ +’ +’ +’ +’ +’ +’ +’ +% +ker +´ +ΓUpE´1q +ρ +ÝÑF +¯ +Im +ˆ +ΓUpE´2q +dp2q +ÝÑΓUpE´1q +˙ +for i “ 1 +ker +˜ +ΓUpE´iq +dpiq +ÝÑΓUpEi`1q +¸ +Im +ˆ +ΓUpE´i´1q +dpi`1q +ÝÑ ΓUpE´iq +˙ +if i ě 2. +They are modules over functions on U. For each i ě 1, H´ipE‚, Uq is called the i-th cohomology +of pE‚, d‚, ρq at the level of sections on U. +(a) We way then pE‚, d‚, ρq is a geometric resolution of F if for every open set U Ă M and +i ě 1, if it induces an exact complex on the level of sections, that is +H´ipE‚, U1q “ t0u +for every open subset U1 Ă U. +(b) A geometric resolution pE‚, d‚, ρq of F is said to be minimal at a point m P M if for each +i ě 2 the linear map dpiq +|m : E´i ÝÑ E´i`1|m vanishes. +2. Cohomology at an arbitrary point m P M. Also, the complex of vector bundles (B.18), at +an arbitrary point m P M, restricts to a complex of vector spaces +¨ ¨ ¨ ÝÑE´i´1|m +dpi`1q +|m +ÝÑ E´i|m +dpiq +|m +ÝÑ E´i`1|mÝÑ ¨ ¨ ¨ +dp2q +|m +ÝÑ E´1 +ρ|m +ÝÑ TmM. +We can look at the quotient vector spaces: +H´ipE‚, mq “ +$ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +& +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +% +ker +ˆ +E´1|m +ρ|m +ÝÑTmM +˙ +Im +˜ +E´2|m +dp2q|m +ÝÑ E´1|m +¸ +for i “ 1 +ker +˜ +E´i|m +dpiq|m +ÝÑ Ei`1|m +¸ +Im +˜ +E´i´1|m +dpi`1q|m +ÝÑ +E´i|m +¸ +if i ě 2. +Here we call H´ipE‚, mq the i-th cohomology of pE‚, d‚, ρq at the point m. +It is important to notice the following: even if pE‚, d‚, ρq is a geometric resolution of F, there +are no reasons for pE‚, d‚, ρq to be exact at m. For example, if the resolution is minimal at some +point m P M, then H´ipE‚, mq » E´i|m for each i ě 2 and H´1pE‚, mq » kerpρ|mq. +Remark B.3.3. Most of the definitions and operations on chain complexes of modules are adapted +in a obvious manner to complexes of vector bundles over M, both on the level of the sections or at a +point. See also [LLS20, Lav17] for more. + +APPENDIX B. HOMOLOGICAL ALGEBRA +179 +On existence of geometric resolutions +Geometric resolutions of a singular foliation F as defined in Remark B.3.2(a) are not guaranteed in +all contexts. However, there always exists a projective resolution of F as a O-module (see B.2). But +these resolutions do not induce always a geometric resolution of F, since the projective modules of a +projective resolution may not correspond to vector bundles because they may not be locally finitely +generated. When the latter condition is satisfied, the Serre-Swan theorem [Swa62, Mor13] states that +there is a one-to-one correspondence between locally finitely generated projective modules and sec- +tions of vector bundles. Under the assumptions of the latter theorem, geometric resolutions of singular +foliations are exactly projective resolutions at the sections level in the category of chain complexes by +O-modules, since sections of vector bundles over M are projective O-modules. +The following proposition summarizes some contexts where geometric resolutions exist, see [LLS20] +for their proofs. +Proposition B.3.4. +1. Any algebraic singular foliation2 on Kd admits geometric resolutions by trivial vector bundles +and of length ď d ` 1. +The same holds for a real analytic of holomorphic singular foliation, but only in a neighborhood +of a point. +2. A locally real analytic singular foliation on a manifold of dimension d admits a geometric reso- +lution of length ď d ` 1 on any relatively compact open subset of M. +Here we have some examples of geometric resolutions of singular foliations. +Example B.3.5. Let F0 “ tX P XpV q | Xp0q “ 0u be the singular foliation made of all vector fields +vanishing at the origin of a vector space V (e.g. think of CN or RN). The contraction by the Euler +vector field ÝÑ +E “ +N +ÿ +i“1 +xi +B +Bxi +gives rise to a complex of trivial vector bundles +¨ ¨ ¨ ÝÑ ^3 T ˚V +ιÝÑ +E +ÝÑ ^2T ˚V +ιÝÑ +E +ÝÑ T ˚V +ιÝÑ +E +ÝÑ C ˆ V “: C, +(B.19) +whose complex on the sections level is pΩ‚pV q, ιÝÑ +E q. Here px1, . . . , xNq are the canonical coordinates +on V . The latter is the Kozul complex associated to the coordinate functions x1, . . . , xN of V . Since +the xi’s form a regular sequence, it is well known that pΩ‚pV q, ιÝÑ +E q is exact. +The following complex of vector bundles over V +¨ ¨ ¨ ÝÑ ^3 T ˚V b TV +ιÝÑ +E +bid +ÝÑ ^2T ˚V b TV +ιÝÑ +E +bid +ÝÑ T ˚V b TV +ιÝÑ +E +bid +ÝÑ C b TV. +(B.20) +is a geometric resolution of F0 since +´ +Ω‚pV q b XpV q, ιÝÑ +E b id +¯ +is also exact (here ΩipV q :“ +Γp^iT ˚V q stands for the sheaf of i-forms on V ). +2A singular foliation which is generated by polynomial vector fields on Kd. + +APPENDIX B. HOMOLOGICAL ALGEBRA +180 +More generally, the construction we have made in (B.20) is still possible by contracting with any +vector field X “ +N +ÿ +i“1 +Xi B +Bxi +P XpV q. The latter yields a complex of vector bundles that covers the +singular foliation FX generated by the Xi B +Bxj ’s. For instance, if X is a polynomial vector field and +pX1, . . . , XNq form a regular sequence, we get a geometric resolution of FX. +Example B.3.6. Let F2 “ I2 +0XpK2q Ă F0 be the sub-singular singular foliation made of vector fields +vanishing at order 2 at the origin of K2, where I2 +0 Ă OpK2q is the ideal generated by the monomials +x2, xy, y2. Note that the ideal I2 +0 admits a free resolution of the form +0 ÝÑ OpK2q ‘ OpK2q +δ1 +ÝÑ OpK2q ‘ OpK2q ‘ OpK2q +δ0 +ÝÑ I2 +0 ÝÑ 0, +(B.21) +where for all f, g, h P OpK2q, +δ0pf, g, hq “ x2f ` xyg ` y2h and δ1pf, gq “ pxf, xf ´ yg, xgq. +The free resolution (B.21) has to take the form +0 ÝÑ ΓpI´2q +δ1 +ÝÑ ΓpI´1q +δ0 +ÝÑ I2 +0 ÝÑ 0, +for sum trivial vector bundles I´1, I´2 on K2. Thus, the following complex +0 ÝÑ I´2 b TK2 δ1bid +ÝÑ I´1 b TK2 δ0bid +ÝÑ I2 +0 b TK2 ÝÑ 0 +(B.22) +is a geometric resolution of F2. Note that I´1 can be identified with the tivial vector bundle S2ppK2q˚q. +More generally, let Fk be the singular foliation made of vector fields vanishing at order k at the +origin of a vector space V of dimension N over R or C. 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On sheaves of Lie-Rinehart algebras. 2020. https://arxiv.org/pdf/2010. +15463.pdf. +[vIR94] +Šafarevič Igor Rosticlavovič. Basic algebraic geometry . 1, [Varieties in projective space] / +Igor R. Shafarevich ; [translator Miles Reid]. Springer-Verlag, Berlin Heidelberg New York +Paris, 2nd revised and expanded edition edition, 1994. +[Vit15] +Luca Vitagliano. On the strong homotopy associative algebra of a foliation. Commun. +Cont. Math., 2015. https://doi.org/10.1142/S0219199714500266. +[Vor04] +Theodore Voronov. Higher derived brackets for arbitrary derivations. Travaux Mathéma- +tiques, (XVII):163–186, 2004. +[Vor05] +Theodore Voronov. Higher derived brackets and homotopy algebras. J. Pure Appl. Algebra, +202(1-3):133–153, 2005. https://doi.org/10.1016/j.jpaa.2005.01.010. +[Vor10] +Theodore Voronov. Q-manifolds and higher analogs of Lie algebroids. In XXIX Workshop +on Geometric Methods in Physics, volume 1307 of AIP Conf. Proc., pages 191–202. Amer. +Inst. Phys., Melville, NY, 2010. +[Wag10] +Friedrich Wagemann. Introduction to Lie algebra cohomology with a view towards BRST +cohomology. Lecture notes, Université Nantes, 2010. +[ZA18] +Marco Zambon and Iakovos Androulidakis. Singular subalgebroids. arXiv, 2018. https: +//arxiv.org/abs/1805.02480. +[Zam18] +Marco Zambon. Holonomy transformations for Lie subalgebroids. arXiv, 2018. https: +//arxiv.org/abs/2103.10409. +[Še01] +Pavol Ševera. Some title containing the words “homotopy” and “symplectic”, e.g. this one. +arXiv, 2001. https://arxiv.org/abs/math/0105080. +Université de Lorraine, CNRS, IECL, F-57000 Metz, France. + + diff --git a/edE_T4oBgHgl3EQf1hxM/content/tmp_files/load_file.txt b/edE_T4oBgHgl3EQf1hxM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d331f7d23c9e61c382414a641c203b26d52c0e8c --- /dev/null +++ b/edE_T4oBgHgl3EQf1hxM/content/tmp_files/load_file.txt @@ -0,0 +1,10211 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf,len=10210 +page_content='Université de Lorraine UNIVERSAL HIGHER LIE ALGEBRAS OF SINGULAR SPACES AND THEIR SYMMETRIES Auteur Ruben Louis Directeur Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Camille Laurent-Gengoux Thèse présentée et soutenue publiquement le 12 novembre 2022 pour l’obtention du Doctorat de l’Université de Lorraine (mention Mathématiques) Membres du jury : Directeur de thèse : M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Camille Laurent-Gengoux Professeur, Université de Lorraine, Metz Président de jury : M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Robert Yuncken Professeur, Université de Lorraine, Metz Rapporteurs : M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Marco Zambon Professeur, KU Leuven, Leuven Mme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Chenchang Zhu Professeure, Université de Göttingen, Göttingen Examinateurs : Mme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Claire Debord Professeure, Université de Paris, Paris M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Pol Vanhaecke Professeur, Université de Poitiers, Poitiers Membre invité : M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Rajan Mehta Professeur, Smith College, Massachusetts arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='08335v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='DG] 19 Jan 2023 UNIVERSITE IAEM DELORRAINEInstitut ELIECARTAN1 ABSTRACT This thesis breaks into two main parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let me describe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We show that there is an equivalence of categories between Lie-Rinehart algebras over a commu- tative algebra O and homotopy equivalence classes of negatively graded acyclic Lie 8-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, this result makes sense of the universal Lie 8-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This extends to a purely algebraic setting the construction of the universal Q-manifold of a locally real analytic singular foliation of [LLS20, Lav17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, to any ideal I Ă O preserved by the anchor map of a Lie-Rinehart algebra A, we associate a homotopy equivalence class of negatively graded Lie 8-algebroids over complexes computing TorOpA, O{Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Several explicit examples are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The second part is dedicated to some applications of the results on Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We associate to any affine variety a universal Lie 8-algebroid of the Lie-Rinehart algebra of its vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We study the effect of some common operations on affine varieties such as blow-ups, germs at a point, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We give an interpretation of the blow-up of a singular foliation F in the sense of Mohsen [Moh21] in terms of the universal Lie 8-algebroid of F, in fact an almost Lie algebroid over a geometric resolution of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We introduce the notion of longitudinal vector fields on a graded manifold over a singular foliation, and study their cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We prove that the cohomology groups of the latter vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We study symmetries of singular foliations through universal Lie 8-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More pre- cisely, we prove that a weak symmetry action of a Lie algebra g on a singular foliation F (which is morally an action of g on the leaf space M{F) induces a unique up to homotopy Lie 8-morphism from g to the Differential Graded Lie Algebra (DGLA) of vector fields on a universal Lie 8-algebroid of F (such morphim is known under the name "L8-algebra action" in [MZ12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We deduce from this general result several geometrical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For instance, We give an example of a Lie algebra action on an affine sub-variety which cannot be extended on the ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Last, we present the notion of bi-submersion towers over a singular foliation and lift symmetries to those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Keywords: Homotopy algebras, Lie 8-algebroids, dg-manifolds, singular foliations, algebraic geom- etry, singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2 ACKNOWLEDGEMENTS I wish to express my gratitude to God and to several people for their help on the accomplishment of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' First, I would like to express my gratitude to my supervisor Camille Laurent-Gengoux for ac- cepting me as his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' He was patient, attentive to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' He was a great support for me and the one on which I could lean when I feel lost in my ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' He encouraged me in my research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' He gave me confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' He knew how to motivate me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' He gave me valuable advice and suggested directions to take, articles to read in order to lead better my research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' He transmitted to me knowledge and the ethics of a researcher, he is therefore my mathematic father.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I also thank him for the time he devoted to me and for his comments and his suggestions throughout of this Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I feel very lucky for this opportunity I had to work with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' My gratitude also goes to Chenchang Zhu and Marco Zambon for accepting to be reporters for my thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I also thank Claire Debord, Pol Vanhaecke, Robert Yuncken, and Rajan Amit Mehta for taking part in the jury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I would like to acknowledge the full financial support for this Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='D from Région Grand Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Likewise, I thank the managers and staff of Université de Lorraine for their collaboration and their help in the administrative procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I particularly would like to warmly thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Philippe Bonneau for helping me find accommodation and settling me comfortably in Metz to begin my PhD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thanks to the CNRS MITI 80Prime project GRANUM also to Franco-German PHC project Procope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I would like to thank the Institut Henri Poincaré for hosting me in november 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I want to thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lavau for his advice and comments on my paper "Symmetries of singular foliations".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I would like to thank C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ospel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Vanhaecke and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Salnikov for giving the possibility to present the results on Lie-Rinehart algebras at the “Rencontre Poisson à La Rochelle, 21-22 October 2021”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I was glad to meet my mathematic grand father Claude Roger at this conference and I would like to thank him for his articles on the Lie-Rinehart algebras that he gave me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, I thank the organizers of the Geometry Seminar at the Aristotle University of Thessaloniki, in particular Panagiotis Batakidis for inviting me in January 28th, 2022 to give a talk on my work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I would like to thank Iakovos Androulidakis with whom I had the honor of discussing my work at the "Foliations, pseudodifferential operators and groupoids" school, Göttingen, Germany in February 28th-March 4th, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I also would like to thank Leonid Ryvkin for always being ready to discuss with me when I needed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I am also grateful to the organizers of Poisson 2022 Advanced School at CRM (Centre de Recerca Matemàtica) Barcelona, for giving this precious opportunity to be in charge of the problems sessions for the Lecture "Singular Foliations".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Likewise, I would like to thank the organizers of Poisson 2022 Conference - ICMAT, Madrid for giving the possibility to present my results at the Poster session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Last, I would like to express sincere gratitude to Université d’État d’Haïti and more precisely the department of mathematics of École Normale Supérieure (ENS), for giving a golden opportunity to meet mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I would like to mention a few names of ENS professors who have contributed to this achievement: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Bérard Cenatus, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lesly Dejean, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3 Antonine Phigareau, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Pierre Timothe, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='B Antenord, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' OLguine Yacinthe, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Oscar Walguen, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Steeve Germain, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Aril Milce, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Yvesner Marcelin and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' K Innocent and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dieuseul Predelus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I specially would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Achis Chery and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Pol Vanhaecke for supporting me by writing letters in my favor to obtain this grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4 Je tiens à remercier Anderson Augusma ainsi que Dor Dieunel pour m’avoir aidé à vérifier les erreurs typographiques dans le manuscrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Je remercie chaleureusement Wanglaise Fateon qui m’a également aidé à relire mes articles et à identifier les coquilles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' La rencontrer a été l’une des meilleures choses qui me soient arrivées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Enfin, je dédie cette thèse à mes parents, Madame et Monsieur Wisner Louis qui ont toujours été présents pour m’encourager au tout début et tout au long de ces années de thèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Je remercie mon seul et unique frère et petit frère Benjamin Louis pour ses blagues drôles qui m’ont aidé à ne pas devenir fou dans la foulée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Je t’aime mon frère.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' À tous mes proches !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Contents I Lie-Rinehart algebras ” Acyclic Lie 8-algebroids 13 1 Preliminaries 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Graded symmetric algebras .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Graded symmetric co-algebras and their morphisms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Co-morphisms and co-derivations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Richardson-Nijenhuis bracket and co-derivations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Basic constructions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 On free resolutions of length ď 2 and Lie-Rinehart algebras .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 48 4 Main results of Part I 52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Presentation of the problem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 64 5 CONTENTS 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 Examples of universal Lie 8-algebroids of Lie-Rinehart algebras .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 Restriction to ϕ “ 0 of vector fields annihilating ϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': 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varieties 81 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Background on affine varieties and some constructions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 116 8 Symmetries of singular foliations through Lie 8-algebroids 123 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Definitions and examples .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 123 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 A Lie 8-morphism lifting a weak symmetry of a foliation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 166 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Operations on complexes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 170 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Resolutions of a module .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 173 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 Geometric resolutions of a singular foliation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 177 Introduction The recent studies about Lie 8-algebras or Lie 8-groups, their morphisms and their -oids equivalent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie 8-algebroids [Cam19, Vor05, Vor10] and “higher groupoids” [SS20]) is usually justified by their use in various fields of theoretical physics and mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie 8-algebras or -oids often appear where, at first look, they do not seem to be part of the story, but end up to be needed to answer natural questions, in particular questions where no higher-structure concept seems a priori involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Among examples of such a situation, let us cite deformation quantization of Poisson manifolds [Kon03] and many recent developments of BV operator theory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Cam17], deformations of coisotropic submanifolds [CF04], integration problems of Lie algebroids by stacky-groupoids [TZ06], complex submanifolds and Atiyah classes [CSX16, Kap99, LGSX21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The list could be longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For instance, it appears in theory of singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Singular foliations arise frequently in differ- ential or algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here following [AS09, AZ14, Cer79, Deb01, LLS20] we define a singular foliation on a smooth, complex, algebraic, real analytic manifold M with sheaf of functions O to be a subsheaf F: U ÝÑ FpUq of the sheaf of vector fields X, which is closed under the Lie bracket and locally finitely generated as an O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Hermann’s theorem [Her62], this is enough to induce a partition of the manifold M into embedded submanifolds of possibly different dimensions, called leaves of the singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Singular foliations appear for instance as orbits of Lie group actions with possibly different dimensions or as symplectic leaves of a Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When all the leaves have the same dimension, we recover the usual “regular foliations”[DHH86, LGLR22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In [LLS20]-[Lav17], it is proven that “behind” several singular foliations F there is a natural ho- motopy class of Lie 8-algebroids, called universal Lie 8-algebroid of F, and that the latter answers natural basic questions about the existence of “good” generators and relations for a singular folia- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='The first part of this thesis is mainly an algebraization of [LLS20]-[Lav17], that allows to enlarge widely the classes of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 are similar to the main theorems Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9 in [LLS20]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 equips any free O-resolution of a Lie-Rinehart algebra A with a Lie 8-algebroid structure (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' in [LLS20] was a statement for geometric resolutions of locally real analytic singular foliation on an open subset with compact closure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is a sort of homotopy transfer theorem, except that no existing homotopy transfer theorem applies in the context of generic O-modules (for instance, [Cam19] deals only with projective O-modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The difficulty 8 CONTENTS 9 is that we cannot apply the explicit transfer formulas that appear in the homological perturbation lemma because there is in general no O-linear section of A to its projective resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 states that any Lie 8-algebroid structure that terminates in A comes equipped with a unique up to homotopy Lie 8-algebroid morphism to any structure as in the first item (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' in [LLS20] was a similar statement for Lie 8-algebroids whose anchor takes values in a given singular foliation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As in [LLS20], an immediate corollary of the result is that any two Lie 8-algebroids as in the first item are homotopy equivalent in a unique up to homotopy manner, defining therefore a class canonically associated to the Lie-Rinehart algebra, that deserve in view of the second item to be called “universal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' While [LLS20] dealt with Lie 8-algebroids over projective resolutions of finite length and finite dimension, we work here with Lie 8-algebroids over any free resolution -even those of infinite length and of infinite dimension in every degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, since we are in a context where taking twice the dual does not bring back the initial space, we can not work with Q-manifolds (those being the “dual” of Lie 8-algebroids): it is much complicated to deal with morphisms and homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By doing so, several limitations of [LLS20] are overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' While [LLS20] only applied to singular foliations which were algebraic or locally real analytic on a relatively compact open subset, the present thesis associates a natural homotopy class of Lie 8-algebroids to any Lie-Rinehart algebra, and in particular a) to any singular foliation on a smooth manifold, (finitely generated or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This construction still works with singular foliations in the sense of Stefan-Sussmann for instance, or to any involutive C8pMq-module in ΓpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' b) to any affine variety, to which we associate its Lie-Rinehart algebra of vector fields), and more generally to derivations of any commutative algebra, c) to singular Lie algebroids in the sense of Androulidakis and Zambon [AZ18], d) to unexpected various contexts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Poisson vector fields of a Poisson manifold, seen as a Lie-Rinehart algebra over Casimir functions, or symmetries of a singular foliation, seen as a Lie-Rinehart algebra over functions constant on the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These Lie 8-algebroids are constructed on O-free resolutions of the initial Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' They are universal in some sense (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2), and they also are in particular unique up to homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A similar algebraization of the main results of [LLS20], using semi-models category, appeared re- cently in Yaël Frégier and Rigel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Juarez-Ojeda [FJO18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are strong similarities between our results and theirs, but morphisms and homotopies in [FJO18] do not match ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is highly possi- ble, however, that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 could be recovered using their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Luca Vitagliano [Vit15] also constructed Lie 8-algebra structures out of regular foliations, which are of course a particular case CONTENTS 10 of Lie-Rinehart algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These constructions do not have the same purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For regular foliations, our Lie 8-algebroid structure is trivial in the sense that it is a Lie algebroid, while his structures become trivial when a good transverse submanifold exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lars Kjeseth [Kje01b, Kje01a] also has a notion of resolutions of Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But his construction is more in the spirit of Koszul-Tate resolution: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' in [Kje01b] defines Lie-Rinehart algebras resolutions as resolutions of their Chevalley-Eilenberg coalgebra, not of the Lie-Rinehart algebra itself as a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It answers a differ- ent category of questions, related to BRST and the search of cohomological model for Lie-Rinehart algebra cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These results can be used to understand the geometry of singular foliations such as their symme- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, Let pM, Fq be a foliated manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A global symmetry of a singular foliation F on M is a diffeomorphism φ: M ÝÑ M which preserves F, that is, φ˚pFq “ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The image of a leaf through a global symmetry is again a leaf (not necessarily the same leaf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For G a Lie group, a strict symmetry action of G on a foliated manifold pM, Fq is a smooth action G ˆ M ÝÑ M that acts by global symmetries [GZ21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Infinitesimally, it corresponds to a Lie algebra morphism g ÝÑ XpMq between the Lie algebra pg, r¨ , ¨sgq of G and the Lie algebra of symmetries of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A strict symmetry action of G on M goes down to the leaf space M{F, even though the latter space is not a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The opposite direction is more sophisticated, since a strict symmetry action of G on M{F does not induce a strict action over M in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, it makes sense to consider linear maps ϱ: g ÝÑ XpMq that satisfy rϱpxq, Fs Ă F for all x P g, and which are Lie algebra morphisms up to F, namely, ϱprx, ysgq ´ rϱpxq, ϱpyqs P F for all x, y P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter linear maps are called “weak symmetry actions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These actions induce a “strict action”on the leaf space i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a Lie algebra morphism g ÝÑ XpM{Fq, whenever M{F is a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us emphasize on the following observation: An infinitesimal action of a Lie algebra g on a manifold M is a Lie algebra morphism g ÝÑ XpMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Replacing M by a Lie 8-algebroid pE, Qq, one expects to define them as Lie 8-algebra morphisms g ÝÑ XpE, Qq, the latter space being a DGLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Various results about those are given in Mehta-Zambon [MZ12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, these authors give several equivalent definitions and interpretations of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In view of [LLS20, Lav17] it is shown that behind every singular foliation or more generally any Lie- Rinehart algebras [LGL22b] there exists a Lie 8-algebroid structure which is unique up to homotopy called the universal Lie 8-algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is a natural question: what does a symmetry of a singular foliation F induce on an universal Lie 8-algebroid of F?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 of this thesis gives an answer to that question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It states that any weak symmetry action of a Lie algebra on a singular foliation F can be lifted to a Lie 8-morphism valued in the DGLA of vector fields on an universal Lie 8-algebroid of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Such Lie 8-morphisms were studied by Mehta and Zambon [MZ12] as "L8-algebra actions".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This goes in the same direction as [GZ21] who already underlined Lie-2-group structures associated to strict symmetry action of Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Furthermore, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 says this lift is unique modulo homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This result gives several geometric consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is an elementary question: can a Lie algebra action g Ñ XpWq on an affine variety W Ă Cd be extended to a Lie algebra action g Ñ XpCdq on Cd?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Said differently: it is trivial that any vector field on W extends to Cd, but can this extension be done in such a manner that it preserves the Lie bracket?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Although no “8-oids” appears in the question, which seems to be a pure algebraic geometry question, we claim that the answer goes through Lie CONTENTS 11 8-algebroids and singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, the idea is then to say that any g-action on W induces a weak symmetry action on the singular foliation IW XpCdq of all vector fields vanishing on W (here IW is the ideal that defines W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, we know that it is possible to lift any weak symmetry action of singular foliation into a Lie 8-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But is it possible to build such a Lie 8-morphism where the polynomial-degree ´1 of the second order Taylor coefficient is zero?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are cohomological obstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In some specific cases, obstruction classes appear on some cohomology, although in general the obstruction is rather a Maurer-Cartan-like equations that may or may not have solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We show that both questions are in fact related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is another interesting question where we would like to apply our results: Can we desingularize a singular affine variety W Ď Cd by making use of the universal Lie 8-algebroid of the singular foliation F “ XpWq of vector fields tangent W?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2, we use the geometric resolution of the singular foliation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the resolution on which the universal Lie 8-algebroid is built) to recover several notions of resolution of singularities: on being due to Nash [LU81] and a second one to Mohsen [Moh21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But the meaning of these spaces are unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We would like to relate them with the higher brackets of the universal Lie 8-algebroid, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to understand the role of the 3-ary bracket in this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the last chapter of the thesis, we introduce the notion of "bi-submersion towers" over singular foliations that we denote by TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter notion as the name suggests is a family of "bi-submersions" which are built one over the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The concept of bi-submersion over singular foliations has been introduced in [AS09] and it is used in K-theory [AS19] or differential geometry [AS11, AZ14, GZ19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We show that such a bi-submersion tower over a singular foliation F exists if F admits a geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Provided that it exists, we show in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25 that any infinitesimal action of a Lie algebra g on the singular foliation F lifts to the bi-submersion tower TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This lift looks like the beginning of a kind of Lie 8-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We wonder if we can continue the construction in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25 to a Lie 8-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The thesis is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In chapter 1, we recall some basics on graded co-algebra structures on the graded symmetry algebra and their morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We also review the notion of co-derivations and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 2, we introduce the notion of Lie 8-algebroid on an arbitrary commutative unital algebra O over a field of characteristic zero, and also define their morphisms in terms of co-derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define the notion of homotopy between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Some technical Lemmas and Propositions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 3, we fix notations and review definitions, examples, and give main properties of Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Besides, we construct a Lie 2-algebroid structure over any Lie-Rinehart algebra that admits a free resolution of length less or equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 4, we state and prove the main results of the first part of the thesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the equivalence of categories between Lie-Rinehart algebras and homotopy classes of free acyclic Lie 8-algebroids, which justifies the name universal Lie 8-algebroid of a Lie-Rinehart algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, we describe the universal Lie 8-algebroids of several Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The complexity reached by the higher brackets in these examples should convince us that it is not a trivial structure, even for relatively simple Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Chapter 5 is devoted to the applications of the results of the previous chapters to affine varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recall some basics definitions and theorems on affine varieties W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We present three main constructions of Lie-Rinehart associated to W and relate their universal Lie 8-algebroids together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, some examples of the universal Lie 8-algebroids are given, such as blow-ups, vector CONTENTS 12 fields vanishing on a complete intersection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 6, we recall the notion of Q-manifolds and apply the results on Lie-Rinehart algebras to recover the universal Lie 8-algebroids of a singular foliation [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This chapter ends with a result on the cohomology of longitudinal vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 7, we recall the definition of Androulidakis and Skandalis isotropy Lie algebra of a singular foliation at a point and recall from [LLS20] its relation with the Universal Lie 8-algebroids of a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We end with a blow-up procedure for a singular foliation inspired by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Mohsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 8, we study symmetries of singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We present some definitions and facts on weak symmetry actions of Lie algebras on singular foliations and give some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We state the main results of the second part of the thesis and present their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 9, we define an obstruction class for extending a Lie algebra action on an affine variety to ambient space and also give some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 10, we look at symmetries of bi-submersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Afterwards, we introduce the notion of bi-submersion towers over a singular foliation and point out some observations related to their symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Finally, in Appendix A, we recall the definition of tensor algebra and fix notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Appendix B, we recall some general facts on homological algebra and give some geometric constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Part I Lie-Rinehart algebras ” Acyclic Lie 8-algebroids 13 CHAPTER 1 Preliminaries This chapter sets the ground for the whole thesis, especially chapters 2, 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It fixes terminologies, conventions and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For more details, we also refer the reader to Appendix A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Throughout this thesis, K is a field of characteristic zero, and O is an associative commutative unital K-algebra unless otherwise mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, an O-module E is seen as K-vector space in the natural way, λ¨e :“ pλ¨1Oq¨e, where 1O ” 1 is the unit of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the sequel, we will drop the notation "¨".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Geometrically, O can be understood as the algebra of smooth functions on a manifold M, or on an open subset U Ă M of a complex manifold, or the coordinate ring of an affine variety W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Graded symmetric algebras For E “ ‘iPZEi be a Z-graded module, we denote by |x| P Z the degree of a homogeneous element x P E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by Ä‚ E and call (reduced) graded symmetric algebra of E over O the quotient T ‚ OE{xx bO y ´ p´1q|x||y|y bO xy of the tensor algebra (see Appendix A) over O of E, namely T ‚ OE :“ 8 à k“1 E bO ¨ ¨ ¨ bO E looooooomooooooon k times by the ideal generated by x bO y ´ p´1q|x||y|y bO x, with x, y arbitrary homogeneous elements of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This quotient is a graded commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote its product by d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Similarly, we denote by S‚ KpEq and call (reduced) graded symmetric algebra of E over the field K the quotient of the tensor algebra (over K) of E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', T ‚ KE :“ 8 à k“1 E bK ¨ ¨ ¨ bK E looooooomooooooon k times 14 CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' PRELIMINARIES 15 by the ideal generated by x bK y ´ p´1q|x||y|y bK x, with x, y arbitrary homogeneous elements of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by ¨ the product in S‚ KpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The algebras Ä‚ E and S‚ KpEq come equipped with two different “grading” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us make them explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define the degree of x “ x1 d ¨ ¨ ¨ d xn P Än E or x “ x1 ¨ ¨ ¨ ¨ ¨ xn P Sn KpEq by |x1 ¨ ¨ ¨ ¨ ¨ xn| “ |x1 d ¨ ¨ ¨ d xn| “ |x1| ` ¨ ¨ ¨ ` |xn| for any homogeneous x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn P E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' With respect to this degree, Ä‚ E and S‚ KpEq are graded commutative algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The second grading is called "polynomial-degree".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The polynomial-degree of x1 d¨ ¨ ¨dxn P Än E or x1¨¨ ¨ ¨¨xn P Sn KpEq is defined to be n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have the following polynomial-degree decomposition Ä‚ E “ ‘kě1 Äk E and S‚ KpEq “ ‘kě1Sk KpEq, where Äk E and Sk KpEq stand for the O-module of elements of polynomial-degree k and the K-vector space of elements of polynomial-degree k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Convention 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For E a graded O-module, the set of elements of polynomial-degree k and degree d in Ä‚ E (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Sk KpEq) shall be denoted by Äk E|d (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Sk KpEq|d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For example, k ä E|d “ à i1`¨¨¨`ik“d Ei1 bO ¨ ¨ ¨ bO Eik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any homogeneous elements x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xk P E and σ P Sk a permutation of t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ku, the Koszul sign ϵpσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xkq is defined by: xσp1q d ¨ ¨ ¨ d xσpkq “ ϵpσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xkq x1 d ¨ ¨ ¨ d xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We often write ϵpσq for ϵpσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For i, j P N, a pi, jq-shuffle is a permutation σ P Si`j such that σp1q ă .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ă σpiq and σpi ` 1q ă .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ă σpi`jq, and the set of all pi, jq-shuffles is denoted by Spi, jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More generally, a pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ikq- shuffle is a permutation σ P Si1`¨¨¨`ik such that σp1q ă ¨ ¨ ¨ ă σpi1q σpi1 ` 1q ă ¨ ¨ ¨ ă σpi1 ` i2q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' σpi1 ` i2 ` ¨ ¨ ¨ ` ij´1 ` 1q ă ¨ ¨ ¨ ă σpi1 ` i2 ` ¨ ¨ ¨ ` ijq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' σpi1 ` i2 ` ¨ ¨ ¨ ` ik´1 ` 1q ă ¨ ¨ ¨ ă σpi1 ` i2 ` ¨ ¨ ¨ ` ikq The set of all pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ikq-shuffles is denoted by Spi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ikq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' PRELIMINARIES 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Graded symmetric co-algebras and their morphisms Graded co-algebra structures are the dual version of graded algebra structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', they are obtained by reversing all arrows and permutes the order of composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recall the definition of a co- algebra structure on an O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g [JLL, Kas12, Man] or Chapter 8 of [Man05b] for more details on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A coassociative graded cocommutative co-algebra structure on an graded O-module C “ ‘iPZCi is a linear map of degree zero called (graded) coproduct ∆: C ÝÑ C b C with ∆pCkq Ă à i`j“k Ci b Cj is such that ∆pCkq has non-zero1 intersection with only finitely many spaces Ci b Cj p∆ b idq ˝ ∆ “ pid b ∆q ˝ ∆: C Ñ C b C b C, called graded coassociativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' τ ˝ ∆ “ ∆, called graded cocommutativity, where τ : C b C Ñ C b C is defined by τpa b bq “ p´1q|a||b|b b a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Equivalently, the following diagrams commute C ∆ � ∆ � C b C ∆bid � C b C idb∆ � C b C b C C ∆ � ∆ � C b C τ � C b C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The pair pC, ∆q is called graded O-co-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that pC, ∆q is counital if there is a morphism of graded vector spaces u: C Ñ K such that pu b idq ˝ ∆ “ pid b uq ˝ ∆ “ id i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the diagram below commutes C id � ∆ � ∆ � C b C idbu � C b C ubid� K b C » C » C b K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pC, ∆q and pC1, ∆1q be two co-algebras over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, the tensor product pC b C1, pidC b τ b idC1q ˝ ∆ b ∆1q is a co-algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The polynomial ring Krts in one indeterminate t is a co-algebra with the coproduct ∆ given on monomials tn, n ě 0 by: ∆ptnq “ nÿ k“0 ˆn k ˙ tk b tn´k and extends by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The graded symmetric algebra is also an example of co-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1Note that this condition is automatically satisfied when the module is concentrated in positive degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' PRELIMINARIES 17 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let E be a O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Both Ä‚ E and S‚ KpEq admit natural co-algebra structures with respect to the deconcatenation ∆ defined by: ∆px1 d ¨ ¨ ¨ d xnq “ n´1 ÿ i“1 ÿ σPSpi,n´iq ϵpσqxσp1q d ¨ ¨ ¨ d xσpiq b xσpi`1q d ¨ ¨ ¨ d xσpnq (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) for every x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn P E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For small n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The formula of Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) on homogeneous elements x, y, z P E means that ∆pxq “ 0 ∆px d yq “ x b y ` p´1q|x||y|y b x ∆px d y d zq “ x d y b z ` p´1q|x||z|x d z b y ` p´1q|x|p|z|`|y|qy d z b x ` x b y d z ` p´1q|x||y|y b x d z ` p´1q|z|p|x|`|y|qz b x d y Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' According to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, the coproduct ∆ is graded with respect to the positive grading given by the "polynomial-degree" of monomials in the symmetric algebras Ä‚ E and S‚ KpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider M “ Rd with global coordinates px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The algebra of differential forms pΩpMq, ^q over M has a co-algebra structure ∆Ω : ΩpMq Ñ ΩpMq b ΩpMq given as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Namely, ∆pdx1 ^ ¨ ¨ ¨ ^ dxnq “ n´1 ÿ i“1 ÿ σPSpi,n´iq ϵpσqdxσp1q ^ ¨ ¨ ¨ ^ dxσpiq b dxσpi`1q ^ ¨ ¨ ¨ ^ dxσpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This matches (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) if we consider E “ Ω1pMq is concentrated in degree `1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Co-morphisms and co-derivations We recall the definitions of co-morphisms, co-derivations of graded co-algebra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pC1, ∆1q and pC, ∆q be two graded co-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A morphism of co-algebras or co-morphism from pC1, ∆1q to pC, ∆q is a linear map Φ: C1 ÝÑ C of degree 0 such that the following diagram commutes C1 ∆1 � Φ � C ∆ � C1 b C1 ΦbΦ � C b C (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) In formula: ∆ ˝ Φ “ pΦ b Φq ˝ ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ: C1 ÝÑ C be a co-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Φ-co-derivation of degree l is a linear map H: C1 ÝÑ C of degree l such that the following diagram commutes, C1 ∆1 � H � C ∆ � C1 b C1 HbΦ`ΦbH � C b C (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' PRELIMINARIES 18 that is, the so-called co-Leibniz identity is satisfied: ∆ ˝ H “ pH b Φ ` Φ b Hq ˝ ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) When C1 “ C and Φ “ id we speak of a co-derivation (of degree l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easily checked that the composition of two co-morphisms C2 Ψ Ñ C1 ΦÑ C is again a co-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by CoDerpCq the linear space of co-derivations of pC, ∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The set CoDerpCq is a graded Lie sub-algebra of HompC, Cq when equipped with the graded commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For two co-derivations H1, H2 P CoDerpCq one has, ∆ ˝ H1 ˝ H2 “ pH1 b id ` id b H1q ˝ pH2 b id ` id b H2q ˝ ∆ “ pH1 ˝ H2 b id ` H1 b H2 ` p´1q|H1||H2|H2 b H1 ` id b H1 ˝ H2q ˝ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, we obtain a similar equation for p´1q|H1||H2|∆ ˝ H2 ˝ H1 by changing the roles of H1 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Subtracting both terms, one get the co-Leibniz identity for rH1, H2s “ H1˝H2´p´1q|H1||H2|H2˝H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us describe Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 in the context of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let E1 and E be graded O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A linear map Φ: S‚ KpE1q Ñ S‚ KpEq is said to be of polynomial-degree/degree r P Z, if it sends polynomials of Sk KpEq to those of Sk´r K pEq/elements of S‚ KpEq|d to S‚ KpEq|d`r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In formulas, for very k ě 0, (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' d P Z) one has Φ ´ Sk KpEq ¯ Ď Sk´r K pEq, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Φ ` S‚ KpEq|d ˘ Ď S‚ KpEq|d`r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any linear map Φ: S‚ KpE1q Ñ S‚ KpEq of degree l can be decomposed with respect to the polynomial-degree as formal sum: Φ “ ÿ rPZ Φprq (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) where, for all k P N0, Φprq : S‚ KpE1q Ñ S‚´r K pEq is a linear map of polynomial-degree r and of degree l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It should be understood that Φprqpvq “ 0 for all v P Sk KpE1q with k ď r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, the decomposition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) makes sense in particular when the polynomial-degrees of Φ are lower bounded that is, there is an integer N P Z, such that Φprq “ 0 for all r ă N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, for every v P Sk KpEq, ÿ rPZ Φprqpvq “ ÿ Nďrăk Φprqpvq, is a finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) becomes Φ “ ÿ rěN Φprq, for some N P Z, since the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='s of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that a linear map Φ: S‚ KpE1q Ñ S‚ KpEq is of polynomial-degree N if and only if ΦpNq is the unique non-zero term, namely Φprq “ 0, for r ‰ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, for the composition of two linear maps S‚ KpE2q Ψ Ñ S‚ KpE1q ΦÑ S‚ KpEq, we have pΦ ˝ Ψqprq “ ÿ i`j“r Φpiq ˝ Ψpjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) for every r P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, although the sum is infinite, it becomes finite when applied to a given element in S‚ KpE1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' PRELIMINARIES 19 Co-morphisms on symmetric algebras Let us have a discussion on co-morphisms from the symmetric graded co-algebras pS‚ KpE1q, ∆1q and pS‚ KpEq, ∆q, where ∆1 and ∆ are the respective coproducts like in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given any linear map F : S‚ KpE1q Ñ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Denoting by Fr : Sr`1 K pE1q Ñ E for r P N0, the restriction of F to Sr`1 K pE1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The linear map F can be extended to a unique co-morphism ¯F : pS‚ KpE1q, ∆1q Ñ pS‚ KpEq, ∆q by taking for k P N the component on Sk KpEq to be, for any homogeneous x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn P E1 ÿ i1`¨¨¨`ik“n ÿ σPSpi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',irq ϵpσq 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' k ź j“1 Fijpxσpi1`¨¨¨`ij´1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xσpi1`¨¨¨`ijqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) where Spi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ikq is the set of pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ikq-shuffles, with i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ik P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us compute ¯F : ` Sk KpE1q, ∆1˘ ÝÑ pS‚ KpEq, ∆q for k “ 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x, y, z P E1, ¯Fpxq “ F0pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ¯Fpx ¨ yq “ F1px ¨ yq ` F0pxq ¨ F0pyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ¯Fpx ¨ y ¨ zq “ F2px ¨ y ¨ zq ` F0pxq ¨ F1py ¨ zq ` p´1q|x||y|F0pyq ¨ F1px ¨ zq ` p´1qp|x|`|y|q|z|F0pzq ¨ F1px ¨ yq ` F0pxq ¨ F0pyq ¨ F0pzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can easily check that ∆ ˝ ¯F “ p ¯F b ¯Fq ˝ ∆1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g ∆ ˝ ¯Fpxq “ ∆ ˝ F0pxq “ 0 “ p ¯F b ¯Fq ˝ ∆1pxq On one side, ∆ ˝ ¯Fpx ¨ yq “ ∆ ˝ F1px ¨ yq ` ∆pF0pxq ¨ F0pyqq “ F0pxq b F0pyq ` p´1q|x||y|F0pyq b F0pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On the other side, p ¯F b ¯Fq ˝ ∆1px, yq “ p ¯F b ¯Fqpx b y ` p´1q|x||y|y b xq “ p ¯F b ¯Fqpx b yq ` p´1q|x||y|p ¯F b ¯Fqpy b xq “ F0pxq b F0pyq ` p´1q|x||y|F0pyq ¨ F0pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following proposition claims that every co-morphism from S‚ KpE1q to S‚ KpEq is of the form described in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a proof, see [JLL, Kas12, Man05b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let E1, E be two graded O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any morphism of graded vector space F : SKpE1q ÝÑ E there exists an (unique) morphism of graded co-algebras ¯F : pSKpE1q, ∆1q ÝÑ pSKpEq, ∆q that satisfies pr ˝ ¯F “ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here pr is the canonical projection onto the term of polynomial-degree 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', pr: S‚ KpEq Ñ S1 KpEq » E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following facts are crucial, and will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' PRELIMINARIES 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14, a co-morphism Φ: S‚ KpE1q Ñ S‚ KpEq is entirely determined by the collection indexed by r P N0 of maps called its r-th Taylor coefficients: Φr : Sr`1 K pE1q Φ ÝÑ S‚ KpEq pr ÝÑ E, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) Notice that the component Φprq of polynomial-degree r ě 0 of Φ coincides with r-th Taylor coefficient Φr on Sr`1 K pE1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In fact we have, Φr “ ppr ˝ Φprqq|Sr`1 K pE1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By item 1, a co-morphism Φ: S‚ KpE1q Ñ S‚ KpEq is completely determined by its components of polynomial-degree r ě 0, hence it admits a polynomial decomposition of the form: Φ “ ÿ rě0 Φprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Similar results as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14 hold for Φ-co-derivations or for co-derivations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' see Corollary VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='34 in [Man05b] or [JLL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a co-morphism Φ: S‚ KpE1q Ñ S‚ KpEq a Φ-co- derivation H of degree l is entirely determined by the Taylor coefficients defined as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) Hr : Sr`1 K pE1q Ñ E, r P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For k P N0 and x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn P E1, the component on Sk KpEq of Hpx1 ¨ ¨ ¨ ¨ ¨ xnq takes the form ÿ i1`¨¨¨`ik“n ÿ σPSpi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ikq ϵpσq 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='Hi1pxσp1q, ¨ ¨ ¨ , xσpi1qq ¨ k´1 ź j“1 Φij´1pxσpi1`¨¨¨`ij´1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xσpi1`¨¨¨`ijqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) Also, H has a decomposition into polynomial-degree of the form : H “ ÿ rě0 Hprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Every linear map H : S‚ KpEq Ñ E of degree l admits an unique extension to a co-derivation ¯H : S‚ KpEq Ñ S‚ KpEq of degree l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter is given explicitly by the formula ¯Hpx1 ¨ ¨ ¨ ¨ ¨ xnq “ nÿ i“1 ÿ σPSpi,n´iq ϵpσqHpxσp1q ¨ ¨ ¨ ¨ ¨ xσpiqq ¨ xσpi`1q ¨ ¨ ¨ ¨ ¨ xσpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) The following lemma-definition is helpful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17 (Definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that a co-algebra morphism Φ: S‚ KpE1q Ñ S‚ KpEq is O-multilinear when the equivalent conditions below are satisfied: (i) For every n ě 0, the n-th Taylor coefficient Φpnq : Sn`1 K pE1q ÝÑ E of Φ is O-multilinear (ii) There exists an induced co-algebra morphism ΦO : Ä‚ E1 ÝÑ Ä‚ E making the following diagram commutative : S‚ KpE1q � Φ � S‚ KpEq � Ä‚ E1 ΦO � Ä‚ E When it is the case we still write Φ for ΦO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The equivalence is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Similarly, the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) shows that the Taylor coefficients of H are O-multilinear if and only if H induces a ΦO-co-derivation HO : Ä‚ E1 ÝÑ Ä‚ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' PRELIMINARIES 21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Richardson-Nijenhuis bracket and co-derivations We need to use the Richardson-Nijenhuis bracket to express our results and explain some proofs in the coming chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This bracket was defined mainly to understand Lie algebra structures on a vector space and their deformations [NR66] as well as Poisson structures and their generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recall the definition in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let E be an O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For k P N0, homomorphisms of degree j from Sk`1 K pEq to E shall be, by definition, the space Homj K ´ Sk`1 K pEq , E ¯ :“ ‘mPZHomK ´ Sk`1 K pEq |m´j, Em ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader for example to [KMS93, KS04] for the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Richardson-Nihenhuis bracket is a K-bilinear map r ¨ , ¨ sRN : Homi K ´ Sp`1 K pEq , E ¯ b Homj K ´ Sq`1 K pEq , E ¯ Ñ Homi`j K ´ Sp`q`1 K pEq , E ¯ which is defined on homogeneous elements P P Hom‚ K ´ Sp`1 K pEq , E ¯ and R P Hom‚ K ´ Sq`1 K pEq , E ¯ by rP, RsRN “ P ˝ R ´ p´1q|P||R|R ˝ P, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) with the understanding that P ˝ R is defined on x0 ¨ ¨ ¨ ¨ xp`q P Sp`q`1 K pEq by pP ˝ Rqpx0 ¨ ¨ ¨ ¨ xp`qq :“ ÿ σPSpq`1,pq ϵpσqPpRpxσp0q ¨ ¨ ¨ xσpqqq ¨ xσpq`1q ¨ ¨ ¨ ¨ xσpp`qqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) The bracket is extended by bilinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The bracket r¨ , ¨sRN should not be confused with the graded commutator r¨ , ¨s of the graded vector space HomK pS‚ KpEq, S‚ KpEqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In fact, the graded commutator of two elements P, R P Hom‚ K pS‚ KpEq, S‚ KpEqq is rP, Rs “ P ˝R´p´1q|P||R|R˝P, but here P ˝R is the usual composition of homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easy to check that Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The graded vector space Hom‚ K pS‚ KpEq , Eq together with the Richardson-Nihenhuis bracket r ¨ , ¨ sRN is a graded Lie algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a given i P Z, and a given P “ ř kě0 Pk with Pk P Homi K ´ Sk`1 K pEq, E ¯ , we denote by ¯P P CoDerpS‚ KpEqq the unique co-derivation of degree i with Taylor coefficients pPkqkPN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The relation between the Richardson-Nijenhuis bracket an co-derivations is described in the following lemma: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map P ÞÑ ¯P is a graded Lie algebra morphism that is, for every P, R of degrees l, r as above, we have Ğ rP, RsRN “ r ¯P, ¯Rs “ ¯P ˝ ¯R ´ p´1qlr ¯R ˝ ¯P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Linearity follows from uniqueness of the extension ¯P described in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, for P, R as above, one should have pr ˝ p ¯P ` ¯Qq “ P ` R “ pr ˝ p Ğ P ` Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By unicity, we must have Ğ P ` R “ ¯P ` ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Likewise, one has for any λ P K, Ď λP “ λ ¯P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' PRELIMINARIES 22 To show that it is a graded Lie algebra morphism, it suffices to check that pr ˝ r ¯P, ¯Rs “ rP, RsRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x “ x0 ¨ ¨ ¨ ¨ xr P Sr`1 K pEq ` pr ˝ r ¯P, ¯Rs ˘ r pxq “ P ˝ ¯Rpxq ´ p´1q|P||R|R ˝ ¯Ppxq (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) By using Formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12), we obtain for example P ˝ ¯Rpxq “ kÿ j“0 ÿ σPSpj`1,k´jq ϵpσqPpRjpxσp0q ¨ ¨ ¨ ¨ ¨ xσpjqq ¨ xσpj`1q ¨ ¨ ¨ ¨ ¨ xσpkqq “ ÿ i ` j “ k i, j ě 0 ÿ σPSpj`1,iq ϵpσqPipRjpxσp0q ¨ ¨ ¨ ¨ ¨ xσpjqq ¨ xσpj`1q ¨ ¨ ¨ ¨ ¨ xσprqq “ ÿ i ` j “ r i, j ě 0 pPi ˝ Rjqpxq with the understanding that Pipx0 ¨ ¨ ¨ ¨ xmq “ 0 for m ‰ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As a consequence, the right-hand side of Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) becomes ÿ i ` j “ r i, j ě 0 ´ pPi ˝ Rjqpxq ´ p´1q|P||R|pRj ˝ Piqpxq ¯ “ ÿ i ` j “ r i, j ě 0 rPi, RjsRNpxq “ prP, RsRNqr pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conclusion: In this chapter, we recalled the co-algebra language, in particular comorphisms, Richardson- Nijenhuis bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' An important point is not to confuse the tensor products bK and bO, hence the graded symmetric algebra S‚ KpEq and S‚ OpEq, which is denoted by S‚pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This introduction is completed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2 Lie 8-algebroids and their morphisms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Definitions In this section, we present the notion Lie 8-algebroid over the algebra O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We also give geometrical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to Appendix B for some generalities on graded modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By definition, Lie 8-algebras on a graded vector space V are co-derivations of degree ´1 squaring to 0 of the graded symmetric algebra SpV q (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g see [LS93, Ryv16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie 8-algebroids generalize Lie 8-algebras and Lie algebroids, but the situation is more sophisticated, because the 2-ary bracket is not O-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us make some preparations before introducing the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A derivation of O is a K-linear map X : O Ñ O that satisfies the so-called Leibniz identity Xpfgq “ Xpfqg ` fXpgq for all f, g P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The set of all derivations of O inherits a Lie K-algebra structure from the Lie algebra EndKpOq, whose Lie bracket is the commutator, that is, rX, Y s “ X ˝ Y ´ Y ˝ X, for all X, Y derivations of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by DerpOq the Lie algebra of derivations of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, for X P DerpOq, Xrfs stands for the derivation X applied to f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Geometrically, derivations of O can be understood as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' when O “ C8pMq is the algebra of functions of a smooth manifold M, as vector fields on a smooth manifold M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 23 CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' when O is the algebra of holomorphic functions on an open subset U of Cd, as vector fields on U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' when O “ OW is the coordinate ring of an affine variety W, as vector fields on the affine variety W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 dg-almost Lie algebroids over O Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Lav17, LLS20] A (negatively graded) almost differential graded Lie algebroid pE‚, ℓ1, ℓ2, ρq over O is a complex pE, ℓ1, ρq : ¨ ¨ ¨ ℓ1 ÝÑ E´3 ℓ1 ÝÑ E´2 ℓ1 ÝÑ E´1 ρ ÝÑ DerpOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of projective O-modules equipped with a graded symmetric degree `1 K-bilinear bracket ℓ2 “ r¨ , ¨s: d2 E Ñ E such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ℓ2 satisfies the Leibniz identity with respect to ρ: E´1 ÝÑ DerpOq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ℓ2px, fyq “ fℓ2px, yq ` ρpxqrfsy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) for all x P E´1, y P E and f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ℓ1 is degree `1-derivation of ℓ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all x P E´i, y P E: ℓ1pℓ2px, yqq ` ℓ2pℓ1pxq, yq ` p´1qiℓ2px, ℓ1pyqq “ 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρ is a morphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all x, y P E´1 ρpℓ2px, yqq “ rρpxq, ρpyqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-linear map ρ is called the anchor map, and ℓ1 the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any almost graded Lie-algebroid pE‚, ℓ1, ℓ2, ρq define the Jacobiator Jacpx, y, zq “ ℓ2pℓ2px, yq, zq ` p´1q|y||z|ℓ2pℓ2px, zq, yq ` p´1q|x|p|y|`|z|qℓ2pℓ2py, zq, xq, x, y, z P E (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) For any triple of elements x, y, z P E´1 of degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Equivalently: Jacpx, y, zq “ ℓ2pℓ2px, yq, zq` ö px, y, zq, where ö px, y, zq means that we sum on the circular permutations of x, y, z with Koszul signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is graded symmetric of degree ´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By axiom 3, the Jacobiator takes values in the kernel of the anchor map, since ρpJacpx, y, zqq “ ρpℓ2pℓ2px, yq, zqq` ö px, y, zq “ rρpℓ2px, yqq, ρpzqs` ö px, y, zq “ rrρpxq, ρpyqs, ρpzqs` ö px, y, zq “ 0, since r¨ , ¨s satisfies Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easy to see that the bracket ℓ2 goes to the quotient and endows, E´1 ker ρ and E´1 ℓ1pE´2q with a Lie K-algebra bracket ¯ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 25 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map px, y, zq ÞÑ Jacpx, y, zq is O-trilinear: indeed, it is clear if x, y, z P Eď´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By graded symmetry, we need to verify this point when the at least one of the variables is of degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let f P O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Case 1: x P E´1 and y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' z P Eď´2 Jacpx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq “ ℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq ` p´1q|z||y|ℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ` p´1q|z|`|y|ℓ2pℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq “ fℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` p´1q|z||y|pℓ2pfℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ` ℓ2pρpxqrfsz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yqq ` p´1q|z|`|y|fℓ2pℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq` p´1q|z|`|y|p´1q|z|`|y|`1ρpxqrfsℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq “ fJacpx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ((((((((((( ( p´1q|z||y|ρpxqrfsℓ2pz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ´ ((((((( ( ρpxqrfsℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq Case 2: x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y P E´1 and z P Eď´2 Jacpx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq “ ℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq ` p´1q|z|ℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ` p´1q|z|`1ℓ2pℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq “ fℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ρpℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yqqrfsz ` p´1q|z|ℓ2pfℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ρpxqrfsz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yqq` p´1q|z|`1ℓ2pfℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ρpyqrfsz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq “ fJacpx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ρpℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yqqrfsz ` p´1q|z|pp´1q|z|(((((((( ρpyqrfs ℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ((((((( ( ρpxqrfsℓ2pz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ` p´1q|z|ρpyqrρpxqrfsszq ` (((((((((((((((( p´1q|z|`1pp´1q|z|ρpxqrfs ℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zqq ` ((((((( ( ρpyqrfsℓ2pz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq ` p´1q|z|ρpxqrρpyqrfsszq “ fJacpx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ((((((((((((( pρpℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yqq ´ rρpxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρpyqsqrfsz “ fJacpx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Case 3: for x, y, z P E´1, the proof is identical to case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A differential graded Lie algebra (DGLA) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g [Man05a]) is a graded vector space g “ à jPZ gj over K together with a bilinear map called graded Lie bracket r¨ , ¨s : gi ˆ gj Ñ gi`j and a differential map d: gi Ñ gi´1 such that for all homogeneous elements v1, v2, v3 P g has the properties graded commutativity: rv1, v2s “ ´p´1q|v1||v2|rv2, v1s, Jacobi’s identity: p´1q|v1||v3|rv1, rv2, v3ss ` p´1q|v2||v1|rv2, rv3, v1ss ` p´1q|v3||v2|rv3, rv1, v2ss “ 0, Leibniz’s identity: drv1, v2s “ rdv1, v2s ` p´1q|v1|rv1, dv2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' DGLAs are examples of almost differential graded Lie algebroid (ADGLA) with O “ K (and gj “ 0 for j ě 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We take E´i :“ gir1s for i ě 1, ℓ1pxq :“ ´dpxq, and ℓ2px, yq :“ p´1q|x|rx, ys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Of course, here ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that ADGLAs are more general than DGLAs, since it does not impose Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Hue03] An almost-Lie algebroid over a manifold M is a triple pA Ñ M, r¨ , ¨sA , ρAq made of a vector bundle A Ñ M, a skew-symmetric bracket r¨ , ¨sA : ΓpAq ˆ ΓpAq Ñ ΓpAq, fulfilling the Leibniz identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all a, b P ΓpAq, f P C8pMq ra, fbsA “ fra, bsA ` ρApaqrfsb, CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 26 and a vector bundle morphism ρ: A Ñ TM, that satisfies, ρpra, bsAq “ rρApaq, ρApbqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that ρA is the anchor of pA Ñ M, r¨ , ¨sA , ρAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When r¨ , ¨sA satisfies Jacobi’s identity, we speak of a Lie algebroid over M [Mac05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Almost Lie algebroids over a manifold M give examples of almost differential graded Lie algebroid over O “ C8pMq with E´1 “ ΓpAq, E´i “ 0 for i ‰ 1, ℓ2 “ r¨ , ¨sA, and the anchor is ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Lie 8-algebroids over O Lie 8-algebroids over manifolds were introduced (explicitly or implicitly) by various authors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [SSS12], [Vor10], and [Še01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer to Giuseppe Bonavolontà and Norbert Poncin for a complete overview of the matter [BP13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It extends the notion of almost differential graded Lie algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [LS93] A negatively graded Lie 8-algebroid over O is a collection E “ pE´iqiě1 of projective O-modules, equipped with: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a collection of linear maps ℓi : Si KpEq ÝÑ E of degree `1 called i-ary brackets 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a O-linear map E´1 ÝÑ DerpOq called anchor map satisfying the following axioms : piq the higher Jacobi identity: nÿ i“1 ÿ σPSi,n´i ϵpσq ℓn´i`1pℓipxσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xσpiqq, xσpi`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xσpnqq “ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) for all n ě 1 and homogeneous elements x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn P E, piiq for i ‰ 2, the bracket ℓi is O-linear, while for i “ 2, ℓ2px, fyq “ ρpxqrfs y ` fℓ2px, yq for all x, y P E, f P O , where, by convention, ρE is extended by zero on E´i for all i ě 2, piiiq ρ ˝ ℓ1 “ 0 on E´2, pivq ρ is a morphism of brackets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', ρpℓ2px, yqq “ rρpxq, ρpyqs for all x, y P E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote it by pE‚, ℓ‚, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From now on, we simply say “Lie 8-algebroid” for “negatively graded Lie 8- algebroid”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us explain these axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 that: CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The higher Jacobi identity is equivalent to nÿ i“1 rℓi, ℓn`1´isRN “ 0, for all positive integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' See Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19 a definition of r¨ , ¨sRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For n “ 1, higher Jacobi identity yields ℓ1 ˝ ℓ1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, ¨ ¨ ¨ ℓ1 ÝÑ E´3 ℓ1 ÝÑ E´2 ℓ1 ÝÑ E´1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) is a complex of projective O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Higher Jacobi identity for n “ 2 reads ℓ1pℓ2px, yqq ` ℓ2pℓ1pxq, yq ` p´1q|x|ℓ2px, ℓ1pyqq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The third and the fourth axiom are consequences of item piq, and piiq if O has no zero divisors1 on E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, for all x P E´2 and y P E´1 and f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Higher Jacobi identity specialized at n “ 2 and Leibniz identity read ℓ1pℓ2px, fyqq “ ℓ2pℓ1pxq, fyq “ fℓ2pℓ1pxq, yq ` ρpℓ1pxqqrfsy Using Leibniz identity again, the left-hand side of the equation above also reads ℓ1pℓ2px, fyqq “ fℓ1pℓ2px, yqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence: ρpℓ1pxqqrfsy “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If O has no zero divisors on E´1, then ρpℓ1pxqqrfs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since x and f are arbitrary, we have ρ ˝ ℓ1|E´2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' by writing higher Jacobi for n “ 3 on elements x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' z P E´1 and f P O while using Leibniz identity we get 0 “ ℓ1pℓ3px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzqq ` ℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq ´ ℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ` ℓ2pℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq “ fℓ1pℓ3px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zqq ` fℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ρpℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yqqrfsz ´ ℓ2pfℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ´ ℓ2pρpxqrfsz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ` ℓ2pfℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq ` ℓ2pρpyqrfsz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq “ f pℓ1pℓ3px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fzqq ` ℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ´ ℓ2pℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ` ℓ2pℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xqq loooooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooooon “0 `ρpℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yqqrfsz ` ρpyqrρpxqrfssz ´ ρpxqrρpyqrfssz ` ((((((( ( ρpyqrfsℓ2px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq ` ((((((( ( ρpyqrfsℓ2pz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xq ´ ((((((( ( ρpxqrfsℓ2pz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ´ ((((((( ( ρpxqrfsℓ2py,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies, pρpℓ2px, yqqrfs ´ rρpxq, ρpyqsrfsqz “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, when O has no zero divisors on E´1, we obtain that ρpℓ2px, yqq “ rρpxq, ρpyqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie 8-algebroid is said to be acyclic if the complex (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) has no cohomology in degree ď ´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for every f P O the linear map E´1 f � E´1 given by multiplication by f, is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 28 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When O “ K we have only the axiom piq of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8, since the other axioms are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, we recover the axioms that define Lie 8-algebras [Sta92, LS93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Geometrically, we are in the case where the projective modules pE´iqiě1 of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 are finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Serre-Swan theorem [Swa62] assures for every i ě 1, E´i “ ΓpE´iq for some vector bundle E´i Ñ M over a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 is rewritten word by word as follows: A (finitely generated) negatively graded Lie 8-algebroid pE, pℓkqkě1, ρq over a manifold M is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a collection of vector bundles E “ pE´iqiě1 over M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' together with a sheaf of Lie 8-algebra structures pℓkqkě1 over the sheaf of sections of E 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' that comes with a vector bundle morphism ρ: E´1 ÝÑ TM, called the anchor, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' such that the k-ary-brackets are all O-multilinear except when k “ 2 and at least one of the arguments is of degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The 2-ary bracket satisfies the Leibniz identity ℓ2px, fyq “ ρpxqrfsy ` fℓ2px, yq, x P ΓpE´1q, y P ΓpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Lie 8-algebroids as homological co-derivations When it comes to manipulating morphisms of Lie 8-alegbroids, it quickly becomes quite tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, it is very useful to dualize in order to make the notion of morphisms clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the finite dimensional case [Vor10], it is usual to see it as a derivation of the symmetric algebra of the dual, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' as a Q-manifold (see Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The duality finite rank Lie 8-algebroids and Q-manifolds is especially efficient to deal with morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In infinite dimension case, we cannot dualize Lie 8-algebroids in the sense of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Voronov [Vor10, BP13] anymore, since the identification of SpV q˚ with SpV ˚q does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' What I do to deal with this problem in the infinite case, is to stay in the world of squared to zero co-derivations and impose some particular additionnal properties on their Taylor coeficients, related to O-linearity and the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now we give an alternative description of Lie 8-algebroids in terms of co-derivation (extending the usual [Vor05, Vor10, BP13] correspondence between Lie 8-algebroids and Q-manifolds in the finite rank case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A co-derivation Q P CoDerpSKpEqq of degree `1 is said to be an homological co- derivation or co-differential when Q2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given such a co-derivation Q, the triplet pSKpEq, ∆, Qq is then called a differential graded co-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following lemma is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A co-derivation Q P CoDerpSKpEqq of degree `1 is an homological co-derivation if and only if pr ˝ Q2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10, the bracket rQ, Qs “ Q ˝ Q ´ p´1q|Q||Q|Q ˝ Q “ 2Q2 is a co-derivation, since co-derivations of a graded co-algebra is closed under the graded commutator bracket r¨ , ¨s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, Q2 “ 1 2rQ, Qs is a co-derivation, and it is completely determined by pr ˝ Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The composition of two co-derivations is not a co-derivation in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, the composition of a co-derivation with itself is not a co-derivation unless it is of odd degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a good understanding of the next proposition, see notations of Taylor coefficients in Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a collection E “ pE´iqiě1 of projective O-modules, there is a one-to- one correspondence between Lie 8-algebroid structures pE‚, ℓ‚, ρq on E and pairs pQE, ρq made of an homological co-derivation QE : S‚ KpEq Ñ S‚ KpEq and a O-linear morphism, ρ: E´1 Ñ DerpOq called the anchor, such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for k ‰ 1 the k-th Taylor coefficient Qpkq E : Sk`1 K pE1q ÝÑ E of QE is O-multilinear, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all x, y P E and f P O, we have, Qp1q E px ¨ fyq “ fQp1q E px ¨ yq ` ρpxqrfs y, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρ ˝ Qp0q E “ 0 on E´2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρ ˝ Qp1q E px ¨ yq “ rρpxq, ρpyqs, for all x, y P E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The correspondence consists in assigning to a Lie 8-algebroid pE‚, ℓ‚, ρq “ pℓ1, ℓ2, ℓ3, ¨ ¨ ¨ q the co- derivation QE whose k-th Taylor coefficient is the k-ary bracket ℓk`1 for all k P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Take the i-th Taylor coefficient of co-derivation QE that satisfies the requirements 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 as, pr ˝ Qpiq E “ ℓi`1 : Si`1 K pEq Ñ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22 we have 2Q2 E “ rQE, QEs “ ÿ ně1 ÿ i`j“n`1 Ğ rℓi, ℓjsRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, by uniqueness of the extension as co-derivation, Q2 E “ 0 is equivalent to 0 “ nÿ i“1 rℓi, ℓn`1´isRN P Hom2 K pSn KpEq , Eq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all integer n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if O “ K, we recover the equivalence between Lie 8-algebras and co-differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From now, when relevant, we sometime denote an underlying structure of Lie 8-algebroid pE‚, ℓ‚, ρq on E by pE, QE, ρq instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that QE : S‚ KpEq ÝÑ S‚ KpEq does not induce a co-derivation on Ä‚ E unless ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us make the correspondence given by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 explicit on the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 30 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Differential Graded Lie Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We come back to Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any differential graded Lie algebra pg “ ‘PZgi, r¨ , ¨s , dq is Lie 8-algebroid with trivial anchor, where the unary bracket ℓ1 and the binary bracket ℓ2 are obtained by adding a sign to the differential map d and the graded skew-symmetric Lie bracket r¨ , ¨s respectively, and the other brackets ℓk for k ě 3 vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For k ě 3, the k-th Taylor coefficient of the corresponding co-derivation Qg is zero, hence it can be written as Qg “ Qp0q g ` Qp1q g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every homogeneous monomial x1 ¨ ¨ ¨ xk P Skg Qp0q g px1 ¨ ¨ ¨ xkq “ kÿ i“1 p´1qp|x1|`¨¨¨`|xi´1|q|xi|ℓ1pxiq ¨ x1 ¨ ¨ ¨ pxi ¨ ¨ ¨ xk, Qp1q g px1 ¨ ¨ ¨ xkq “ ÿ 1ďiăjďk p´1qp|x1|`¨¨¨`|xj´1|q|xj|`p|x1|`¨¨¨`|xi´1|q|xi|ℓ2pxi, xjq ¨ x1 ¨ ¨ ¨ rxi ¨ ¨ ¨ pxj ¨ ¨ ¨ xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 say that pg “ ‘PZgi, r¨ , ¨s , dq is a DGLA if and only if Q2 g “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' or pQp0q g q2 “ pQp1q g q2 “ 0 and Qp0q g ˝ Qp1q g ` Qp1q g ˝ Qp0q g “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 Morphisms of Lie 8-algebroids and homotopies This section extends Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 of [LLS20] to the infinite dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Morphisms of Lie 8-algebroids Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie 8-algebroid morphism (or Lie 8-morphism) from a Lie 8-algebroid pE1, QE1, ρ1q to a Lie 8-algebroid pE, QE, ρq, is a co-morphism Φ: S‚ KpE1q ÝÑ S‚ KpEq such that Φ ˝ QE1 “ QE ˝ Φ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' all Taylor coefficients are O-multilinear, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' which satisfies ρ ˝ Φ0 “ ρ1 on E1 ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Above, Φ0 “ Φp0q |E1 : E1 Ñ E is the 0-th Taylor coefficient of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is equivalent to saying Φ is O-multilinear in the sense of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Recall from [LS93] that Lie 8-algebra morphisms from pS‚ KpEq, QEq to pS‚ KpE1q, QE1q are defined to be co-algebra morphisms Φ: S‚ KpE1q ÝÑ S‚ KpEq that satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 adds two additional assumptions to turn a Lie 8-algebra morphism into a Lie 8-algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a Lie 8-algebroid morphism Φ: S‚ KpE1q ÝÑ S‚ KpEq, the 0-th Taylor coefficient Φ0 : pE1, ℓ1 1q ÝÑ pE, ℓ1q of Φ is the chain map, that is, the following diagram commutes ¨ ¨ ¨ � E1 ´3 Φ0 � ℓ1 1 � E1 ´2 Φ0 � ℓ1 1 � E1 ´1 Φ0 � ρ1 � DerpOq id � ¨ ¨ ¨ � E´3 ℓ1 � E´2 ℓ1 � E´1 ρ � DerpOq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 31 namely, Φ0 ˝ ℓ1 1 “ ℓ1 ˝ Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, by going a bit further, it satisfies for all x, y P E1 Φ0 ˝ ℓ1 2px, yq ` Φ1 ˝ ℓ1 1px d yq “ ℓ1 ˝ Φ1px d yq ` ℓ2pΦ0pxq, Φ0pyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq and pB, r¨ , ¨sB , ρBq be two Lie algebroids over a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie algebroid morphism φ: A ÝÑ B over the identity [Mac05] induces a Lie 8-algebroid morphism S‚pφq: S‚pΓpAqq Ñ S‚pΓpBqq in the section level: where corresponding co-algebra morphism S‚pφq is defined as a1 ¨ ¨ ¨ ¨ an ÞÑ φpa1q ¨ ¨ ¨ ¨ φpanq, for all a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , an P A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie 8-morphisms of Differential Graded Lie Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us consider pg, r¨ , ¨sg , dgq and ph, r¨ , ¨sh , dhq two differential graded Lie algebras (see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6), where Qg and Qh de- note their respective associated co-derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ: pS‚pgr1sq, Qgq ÝÑ pS‚phr1sq, Qhq be a Lie 8-morphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' one has Φ ˝ Qg “ Qh ˝ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) In particular, Φ0 ˝dg “ dh ˝Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us write down what the restriction of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) to low Taylor coefficients: Let x, y P g be two homogeneous elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has, Φpx ¨ yq “ Φ1px ¨ yq ` Φ0pxq ¨ Φ0pyq, Φpxq “ Φ0pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A direct computation of the LHS of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) applied to x ¨ y gives, Φ ˝ Qgpx ¨ yq “ Φ ´ ´dgpxq ¨ yq ´ p´1qp|x|´1qp|y|´1qdgpyq ¨ x ` p´1q|x|rx, ysg ¯ “ ´Φ1pdgpxq ¨ yq ´ Φ0pdgpxqq ¨ Φ0pyq ´ p´1qp|x|´1qp|y|´1q pΦ1pdgpyq ¨ xq ´ Φ0pdgpyqq ¨ Φ0pxqq ` p´1q|x|Φ0prx, ysgq Also, the RHS of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) applied to x ¨ y gives, Qh ˝ Φpx ¨ yq “ QhpΦ1px ¨ yq ` Φ0pxq ¨ Φ0pyqq “ ´dhpΦ1pxqq ¨ y ´ dhpΦ0pxqq ¨ Φ0pyq ´ p´1qp|x|´1qp|y|´1qdhpΦ0pyqq ¨ Φ0pxq ` p´1q|x|rΦ0pxq, Φ0pyqsh Since both sides are equal, and Φ0 ˝ dg “ dh ˝ Φ0 we obtain, Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqsh “ p´1qp|x|´1q|y|qΦ1pdgpyq ¨ xq ´ p´1q|x| pΦ1pdgpxq ¨ yq ´ dhpΦ1px ¨ yqqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Homotopies In this section, we define homotopy between Lie 8-morphisms, and between Lie 8-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This extends [LLS20] from finite dimensional Q-manifolds to arbitrary Lie 8-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 32 A practical definition Let us consider pΩ‚pra, bsq, ^, ddRq the differential graded algebra made of the forms on ra, bs together with the wedge product and the Rham differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If we denote by t the coordinate on ra, bs, we have Ω‚pra, bsq “ C8pra, bsq ‘ C8pra, bsq dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We equip Ω‚pra, bsq with a co-associative co-algebra structure ∆: Ω‚pra, bsq Ñ Ω‚pra, bsq bΩ‚pra,bsq Ω‚pra, bsq, given by ∆Ωp1q “ 1 b 1, where we extend by Ω‚pra, bsq- linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE1, QE1, ρ1q and pE, QE, ρq be Lie 8-algebroids over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The triplet ` S‚ KpEq bK Ω‚pra, bsq, ¯∆ “ pid b τ b idq ˝ ∆ b ∆Ω, B “ QE1 b id ` id b ddR ˘ is a differential graded co-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One should understand that for α P C8pra, bsq and for an homogeneous element v P S‚ KpEq, pQE b id ` id b ddRqpv b αq “ QEpvq b α ` p´1q|v|v b α1dt also, pQE b id ` id b ddRqpv b αdtq “ QEpvq b αdt ` v b ddRpαdtq loooomoooon “0 “ QEpvq b αdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Clearly, B2 “ 0, see the definition of tensor product of complexes, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check that B is indeed a co-derivation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t the co-product ¯∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Take α P C8pra, bsq and a homogeneous element v P S‚ KpEq, we will use the Sweedler notation, ∆pvq “ vp1q b vp2q, to avoid a long useless text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On one hand, ¯∆ ˝ Bpv b αq “ pid b τ b idq ˝ ∆ b ∆ΩpQEpvq b α ` p´1q|v|v b α1dtq “ id b τ b id ´ ∆ ˝ QEpvq b α b 1 ` p´1q|v|∆pvq b α1dt b 1 ¯ “ pid b τ b idq ˝ ´ pQE b id ` id b QEq ˝ ∆pvq b α b 1 ` p´1q|v|∆pvq b α1dt b 1 ¯ “ QEpvp1qq b α b vp2q b 1 ` p´1q|vp1q|vp1q b α b QEpvp2qq b 1` p´1q|v|`|vp2q|vp1q b pα1dtq b vp2q b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On the other hand, pB b id ` id b Bq ˝ ¯∆pv b αq “ pB b id ` id b Bq ˝ pid b τ b idqp∆pvq b α b 1q “ pB b id ` id b Bqpvp1q b α b vp2q b 1q “ Bpvp1q b αq b vp1q b 1 ` p´1q|vp1q|vp1q b α b Bpvp2q b 1q “ ´ QEpvp1q b α ` p´1q|vp1q|vp1q b α1dtq ¯ b vp2q b 1` p´1q|vp1q|vp1q b α b QEpvp2qq b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 33 Both sides are obviously equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Likewise, a straightforward computation shows that ¯∆ ˝ Bpv b αdtq “ pB b id ` id b Bq ˝ ¯∆pv b αdtq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence ¯∆ ˝ B “ pB b id ` id b Bq ˝ ¯∆ on S‚ KpEq bK Ω‚pra, bsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Elements of degree k of S‚ KpEq bK Ω‚pra, bsq are K-linear combinations of elements of the form v b α and w b βdt, with α, β P C8pra, bsq and v P S‚ KpEq|k and w P S‚ KpEq|k´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter can be seen as maps, t P ra, bs ÞÑ αptqv b 1 and t P ra, bs ÞÑ βptqw b dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, elements of degree k of this complex can be considered as element of SKpEq bK Ω‚pra, bsq that depend on t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' as t P ra, bs ÞÑ vt b 1 ` wt b dt, with vt P S‚ KpE1q|k and wt P S‚ KpE1q|k´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following is a temporary definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It will be generalized later to another more practical for gluing homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A homotopy that joins two Lie 8-algebroid morphisms Φ, Ψ: S‚ KpE1q Ñ S‚ KpEq is the datum made of an interval ra, bs Ă R and a chain map pS‚ KpE1q, QE1q H ÝÑ pS‚ KpEq bK Ω‚pra, bsq, QE b id ` id b ddRq v ÞÝÑ ´ t P ra, bs ÞÑ Jtpvq b 1 ´ p´1q|v|Htpvq b dt ¯ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' which is a co-algebra morphism, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and that coincides with Φ and Ψ at t “ a and b respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all v P S‚ KpE1q, one has Hpvqpaq “ Φpvq b 1 and Hpvqpbq “ Ψpvq b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the definition above, the map H induces for every t P ra, bs two different O- multilinear maps $ & % Jt : S‚ KpE1q ÝÑ S‚ KpEq Ht : S‚ KpE1q ÝÑ S‚ KpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since H is of degree 0, one of the maps is of degree zero and the other one of degree ´1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By using respectively the property of co-algebra morphisms and chain map property, we obtain the following for every t P ra, bs: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ¯∆ be the co-product on S‚ KpEq bK Ω‚pra, bsq like in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have, ¯∆ ˝ Hpvq “ H b H ˝ ∆1pvq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let v “ v1 ¨ ¨ ¨ vn P S‚ KpE1q, a direct computation of gives us ∆ b ∆Ω ˝ Hpvqptq “ ∆ ˝ Jtpvq b 1 b 1 ´ p´1q|v|∆ ˝ Htpvq b dt b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us use the Sweedler notation just like in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 to compute H b H ˝ ∆1pvqptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have, H b H ˝ ∆1pvqptq “ H b Hpvp1q b vp2qqptq “ Hptqpvp1qq b Hpvp1qqptq “ ´ Jtpvp1qq b 1 ´ p´1q|vp1q|Htpvp1qq b dt ¯ b ´ Jtpvp2qq b 1 ´ p´1q|vp2q|Htpvp2qq b dt ¯ CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 34 Hence, pid b τ b idq´1 ˝ pHbHq ˝ ∆1pvq “ pJt b Jtq ˝ ∆1pvq b 1 b 1 ` ((((((((((((( Ht b Ht ˝ ∆1pvq b dt b dt ` p´1q|v| ` Ht b Jt ˝ ∆1pvq ` Jt b Ht ˝ ∆1pvq ˘ b dt ˆ 1 By equating both sides, we obtain equations that say that Jt is a co-morphism and Ht a Jt-co- derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and for any homogeneous element v P S‚ KpE1q we have: in one side H ˝ QE1pvqptq “ Jt ˝ QE1pvq b 1 ´ p´1q|v|`1Htpvq ˝ QE1pvq b dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to the other side2, pQE b id ` id b ddRq ˝ Hpvqptq “ pQE b id ` id b ddRq ˝ ´ Jtpvq b 1 ´ p´1q|v|Htpvq b dt ¯ “ QE ˝ Jtpvq b 1 ` p´1q|v| dJt dt pvq b dt ´ p´1q|v|QE ˝ Htpvq b dt By equating both sides we see that Jt and Ht satisfy the following condition $ & % Jt ˝ QE1pvq “ QE ˝ Jtpvq dJt dt pvq “ QE ˝ Htpvq ` Ht ˝ QE1pvq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) In addition, we have for v P E1 ´1 that dJp0q t dt pvq “ Qp0q E ˝ Hp0q t pvq ` Hp0q t ˝ Qp0q E1 looooomooooon “0 pvq and Jp0q t pvq “ Φp0qpvq ` ż t a ℓ1 ˝ Hp0q s pvqds “ Φp0qpvq ` ℓ1 ˝ ż t a Hp0q s pvqds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) This implies that ρ ˝ Jp0q t pvq “ ρ ˝ Φp0qpvq “ ρ1pvq, since ρ ˝ ℓ1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Gluing homotopies as defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 is however not so easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can reformulate the definition of homotopies between Lie 8-morphisms in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Before we do this, we would like to fix some vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recall some facts on vector-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let V be a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Unless a topology on V is chosen, the notion of V -valued continuous or differentiable or smooth function, and the concept of a limit on an interval I “ ra, bs Ă R do not make any sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, we can always define the following notion 2It should be understood that e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g, pQE b id ` id b ddRqpαptqv b 1q “ pQEpvq b α ` v b ddRαq ptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 35 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A vector-valued map γ : I ÝÑ V is said to be a piecewise rational path on I if there exists a finite increasing sequence a “ t0 ď ¨ ¨ ¨ ď tN “ b of gluing points, such that for all i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , N ´ 1 the restriction γi of γ to rti, ti`1s is of the form γiptq “ nÿ j“1 βi jptqvj, for some n P N where for every j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n, vj P V and βi j : I Ñ R a real rational function on rti, ti`1s that has no pole on rti, ti`1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that γ is continuous on I, if for all i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , N ´ 1, γi and γi`1 coincide at the gluing point ti`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When V is a space of linear maps between the vector spaces S and T, we shall say that a V -valued map Ξ: I Ñ V is a piecewise rational (continuous) if the map pt P I ÞÑ Ξtpsqq is a piecewise rational (continuous) T-valued path for all s P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define the limit of γ at u P rti, ti`1s to be equal to nÿ j“1 lim tÑu βi jptqvj when, for every j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n, βi j admits a limit a u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can assume that tv1, v2, v3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' u is a basis of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recall the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is a very classical fact that the integral of a piecewise-C1 function β : I ÝÑ R on a compact interval I “ ra, bs Ă R which is subordinate to a subdivision a “ t0 ă ¨ ¨ ¨ ă tN “ b admits primitives which are piecewise-C1 on the same subdivision a “ t0 ă ¨ ¨ ¨ ă tN “ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' An important fact is that continuous piecewise-C1 functions β : I ÝÑ R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' piecewise-C1 functions which are also continuous (even at the junction points) admit piecewise continuous derivatives β1ptq, and βpbq ´ βpaq “ şb a β1ptqdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 has this important consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A piecewise rational continuous function γ : I Ñ V is differentiable at every point which is not a gluing point, and the latter is again piecewise rational on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conversely, every piecewise rational functions admit a piecewise rational continuous primitive, unique up to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The derivative of γ can be defined in the usual way using the item 3 of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Likewise, for their primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof is then immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can now give the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A family pJtqtPI of co-algebra morphisms are said to be piecewise rational con- tinuous if its Taylor coefficients Jpnq t are piecewise rational continuous for all n P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For such a family pJtqtPI, a family pHtqtPI made of Jt-co-derivations is said to be piecewise rational if all its Taylor coefficients are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 36 We are now ready to define formulate the definition of homotopies as follows and extend Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='53 in [LLS20] to the infinite rank case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ and Ψ be Lie 8-algebroid morphisms from pE1, QE1, ρ1q to pE, QE, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A homotopy between or that joins Φ and Ψ is a pair pJt, HtqtPra,bs consisting of: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a piecewise rational continuous path t ÞÑ Jt valued in Lie 8-algebroid morphisms between S‚ KpE1q and S‚ KpEq satisfying the boundary condition: Φa “ Φ and Φb “ Ψ, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a piecewise rational path t ÞÑ Ht, with Ht a Jt-co-derivations of degree ´1 from S‚ KpE1q to S‚ KpEq, such that the following equation: dJt dt “ QE ˝ Ht ` Ht ˝ QE1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) holds for every t Psa , br where it is defined (that is, not a gluing point for the Taylor coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, for every v P Sďn K pE1q, dJt dt pvq “ QE ˝ Htpvq ` Ht ˝ QE1pvq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) for all t which is not a gluing point of the Taylor coefficient of Jpkq t , Hpkq t for k “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When these conditions are satisfied, we say that Φ and Ψ are homotopy equivalent, and we write Φ „ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This clearly extends (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the above definitions, it is not required that the gluing points of the various Taylor coefficients Jpnq t or Hpnq t to be the same for all n P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following Proposition shows that the notion of homotopy given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14 implies the usual notion of homotopy between chain maps (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ and Ψ be Lie 8-algebroid morphisms from pE1, QE1, ρ1q to pE, QE, ρq which are homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, Ψ ´ Φ “ QE ˝ H ` H ˝ QE1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) for some O-linear map H : S‚ KpE1q ÝÑ S‚ KpEq of degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Take pJt, HtqtPra,bs a homotopy between Φ and Ψ as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By definition, t ÞÑ Jt is piecewise rational continuous, therefore it is continuous on ra, bs (even at the junctions points), we can use Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 and write Φb ´ Φa “ ż b a dJt dt dt Ψ ´ Φ “ ż b a pQE ˝ Ht ` Ht ˝ QE1q dt “ QE ˝ ˆż b a Ht dt ˙ ` ˆż b a Ht dt ˙ ˝ QE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, the O-multilinear map H :“ şb a Ht dt satisfies Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 37 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that two Lie 8-algebroids pE1, QE1, ρ1q to pE, QE, ρq over O are homotopy equivalent or homotopic if there exists two Lie 8-algebroid morphisms pE1, QE1, ρ1q Φ � pE, QE, ρq Ψ � such that Φ˝Ψ: pE, QE, ρq Ñ pE, QE, ρq and Ψ˝Φ: pE1, QE1, ρ1q Ñ pE1, QE1, ρ1q are homotopy equivalent to the identity map of respective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following proposition justifies Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ be a Lie 8-algebroid morphism from pE1, QE1, ρ1q to pE, QE, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all t P ra , bs, let Hpnq t : Sn`1 K pE1q Ñ E be a family O-multilinear piecewise rational maps indexed by n P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There exists a unique piecewise rational continuous family of co-algebra morphisms Jt such that (a) Ja “ Φ (b) pJt, Htq is a solution of the differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12), where Ht is the Jt-co-derivation whose n-th Taylor coefficient is Hpnq t for all n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, for all t P ra, bs, pJs, HsqsPra,ts is a homotopy that joins Φ and Jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We claim that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) is a differential equation that can be solved recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In polynomial-degree zero, it reads, dJp0q t dt “ Qp0q E ˝ Hp0q t ` Hp0q t ˝ Qp0q E1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) and Jp0q t “ Φp0q ` ż t a ´ Qp0q E ˝ Hp0q s ` Hp0q s ˝ Qp0q E1 ¯ ds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) is defined for all t P ra, bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, d dtJpn`1q t : Sn`2pE1q Ñ E is an algebraic expression of Qp0q E , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qpn`1q E , Qp0q E1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qpn`1q E1 Jp0q t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Jpnq t , Hp0q t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Hpn`1q t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But Jpn`1q t does not appear in the pn ` 1q-th Taylor coefficient of QE ˝Ht `Ht ˝QE1 by Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12, there exists a unique piecewise rational continuous solution Jpn`1q t such that Jpn`1q a “ Φpn`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The construction of the Taylor coeffi- cients of the co-algebra morphisms Jt then goes by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Recursion formulas also show that Jt is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' that Jt is a O-multilinear chain map for all t P ra, bs: For the same reason as in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11), Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) implies that ρ ˝ Jp0q t |E1 “ ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The function given by Λkptq “ pQE ˝ Jt ´ Jt ˝ QE1qpkq for all t P ra, bs, k P N0 are differentiable at all points t except for a finitely many t P ra, bs and are piecewise rational con- tinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map dJt dt is a Lie 8-morphism because Q2 E “ 0 and Q2 E1 “ 0, hence Λ1ptq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By continuity, Λkptq is constant over the interval ra, bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since Ja “ Φ is a Lie 8-algebroid morphism, we have Λkpaq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, Λkptq “ 0 and, QE ˝ Jt “ Jt ˝ QE1, for all t P ra, bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 38 Let us show that homotopy in the sense above defines an equivalence relation „ between Lie 8-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A pair pJt, Htq is a homotopy between Lie 8-algebroid morphisms Ja and Jb if and only if for all rational function, g: ra, bs Ñ rc, ds without poles on ra, bs, the pair pJgptq, g1ptqHgptqq is a homotopy between Jgpaq and Jgpbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let g: ra, bs Ñ rc, ds be a rational function without poles on ra, bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A straightforward compu- tation gives: dJt dt “ QE ˝ Ht ` Ht ˝ QE1 (by definitionq ñ dJ dt pgptqq “ QE ˝ Hgptq ` Hgptq ˝ QE1 (by replacing t by gptq) ñ dJgptq dt “ QE ˝ ` g1ptqHgptq ˘ ` ` g1ptqHgptq ˘ ˝ QE1 (by multiplying both sides by g1ptq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The last equation means that pJgptq, g1ptqHgptqq is a homotopy between Jgpaq and Jgpbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The backward implication is obvious, it suffices to consider a “ c, b “ d and g “ id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Homotopy between Lie 8-morphisms is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In addition, it is compatible with composition, that is, if Φ, Ψ: S‚ KpE1q Ñ S‚ KpEq are homotopic Lie 8-algebroid morphisms and ˆΦ, ˆΨ: S‚ KpEq Ñ S‚ KpE2q are homotopic Lie 8-algebroid morphisms, then, so are their compositions ˆΦ ˝ Φ and ˆΨ ˝ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We first show that this notion of homotopy is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ, Ψ and Ξ: S‚ KpE1q ÝÑ S‚ KpEq be three Lie 8-morphisms of algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ‚ Reflexivity: The pair pJt “ Φ, Ht “ 0qtPr0,1s defines a homotopy between Φ and Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ‚ Symmetry: Let pJt, HtqtPr0,1s be a homotopy between Φ to Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20 with gptq “ 1 ´ t, we obtain a homotopy between Ψ and Φ via the pair pΦ1´t, ´H1´tqtPr0,1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ‚ Transitivity: Assume Φ„Ψ and Ψ„Ξ and let pJt, H1,tqtPr0, 1 2 s be a homotopy between Φ and Ψ and let p ¯Jt, H2,tqtPr 1 2 ,1s be a homotopy between Ψ and Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By gluing Jt and ¯Jt, respectively H1t and H2,t we obtain a homotopy p ˜Jt, HtqtPr0,1s between Φ and Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We then show it is compatible with composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us denote by pJt, Htq the homotopy between Φ and Ψ, and p ˆJt, ˆHtq the homotopy between ˆΦ and ˆΨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We obtain, d ˆJt ˝ Jt dt “ d ˆJt dt ˝ Jt ` ˆJt ˝ d ˆJt dt “ QE2 ˝ ´ ˆHt ˝ Jt ` ˆJt ˝ Ht ¯ ` ´ ˆHt ˝ Jt ` ˆJt ˝ Ht ¯ ˝ QE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, ˆΦ ˝ Φ and ˆΨ ˝ Ψ are homotopic via the pair p ˆJt ˝ Jt, ˆHt ˝ Jt ` ˆJt ˝ Htq which is easily checked to satisfy all axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We conclude this section with a lemma that will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that this is the lemma that forces to extend Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8, for it would not be true anymore in the smooth setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pJt, HtqtPrc,`8r be a homotopy such that for all n P N0 and for every t ě n, Hpnq t “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then the n-th Taylor coefficient Jpnq t is constant on rn, `8r and the co-algebra morphism J8 whose n-th Taylor coefficient is Jpnq t for any n P N0 and t P rn, `8r is a Lie 8-algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, for g : ra, brÑ rc, `8r a rational function with no pole on ra, br and such that lim tÑb gptq “ `8, the pair pJgptq, g1ptqHgptqq is a homotopy between Jc and J8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 39 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the n-th Taylor coefficient of the Jt-co-derivation dJpnq t dt “ pQE ˝Ht `Ht ˝QE1qpnq depends only on Hpiq t for i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n ´ 1, we have by assumption dJpnq t dt “ 0 for all t ě n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As a consequence, Jpnq t is constant on rn, `8r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19 that J8 is a Lie 8-algebroid morphism since for every n P N0 and t P rn, `8r pQE ˝ J8 ´ J8 ˝ QE1qpnq “ ÿ i`j“n pQpiq E ˝ Jpiq t ´ Jpjq t ˝ Qpjq E1 q “ 0 (since Jt is a Lie 8-algebroid morphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove the last part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By assumption, there exists a ď bn ď b such that for all t P rbn, bs, we have gptq ě n, so that Jpnq gptq “ Jpnq 8 and g1ptqHpnq gptq “ 0 on rbn, bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The function Jpnq t (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hpnq t ) being piecewise rational continuous (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' piecewise rational) on rc, ns, the same holds for Jpnq gptq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' g1ptqHpnq gptq) on ra, bns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By gluing with a constant function J8 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' with 0), we see that all Taylor coefficients of Jgptq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' g1ptqHgptq) are piecewise rational continuous (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' piecewise rational) with finitely many gluing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22 explains how to glue infinitely many homotopies, at least when for a given n P N0, only finitely of them affects the n-th Taylor coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This would not be possible using only Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 More technical lemmas and propositions Let us state and prove these technical assertions for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE1, QE1, ρ1q and pE, QE, ρq be a Lie 8-algebroid over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ: S‚ KpE1q Ñ S‚ KpEq be a Lie 8-algebroid morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For H: S‚ KpE1q Ñ S‚ KpEq a O-multilinear Φ-co-derivation of degree k ď ´1, H ˝ QE1 ´ p´1qkQE ˝ H is a O-multilinear Φ-co-derivation of degree k ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We first check that H ˝ QE ´ p´1qkQE1 ˝ H is a Φ-co-derivation: ∆ ˝ H ˝ QE1 “ pH b Φ ` Φ b Hq ˝ ∆1 ˝ Q1 E, by definition of H “ pH b Φ ` Φ b Hq ˝ pQE1 b id ` id b QE1q ˝ ∆1, by definition of Q1 E “ pH ˝ QE1 b Φ ` p´1qkΦ ˝ QE1 b H ` H b Φ ˝ QE1 ` Φ b H ˝ QE1q ˝ ∆1 Subtracting a similar equation for p´1qk∆ ˝ QE ˝ H and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6), one obtains the Φ-co-derivation property for H ˝ QE1 ´ p´1qkQE1 ˝ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We now check that H ˝ QE1 ´ p´1qkQE ˝ H is O-multilinear, for which it suffices to check that its Taylor coefficients are O-multilinear by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn P E1 be homogeneous elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume xi P E1 ´1 (if we have more elements of degree ´1 the same reasoning holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To verify O-multilinearity it suffices to check that for all f P O: pr ˝ pH ˝ QE1 ´ p´1qkQE ˝ Hqpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ,xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xnq “ fpr ˝ pH ˝ QE1 ´ p´1qkQE ˝ Hqpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Only the terms where the 2-ary bracket with a degree ´1 element on one-side and f on the other side may forbid f to go in front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are two such terms: ϵpxi, xj, xIijqHpn´1qpℓ1 2pxi, fxjq, xIijq and ´ p´1qkϵpxi, xIiqℓ2pΦ0pxiq, fHpn´1qpxIiqq CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 40 where xIi and xIij stand for the list x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn where xi and xi, xj are missing, respectively, and Hpn´1q is the pn ´ 1q-th Taylor coefficient of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ρ ˝ Φ0 “ ρ1, in both terms ρpxiqrfs appears, and these two terms add up to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If the degree of H is non-negative, then H ˝ QE1 ´ p´1qkQE ˝ H may not be O- multilinear anymore, since there may exist extra terms where the anchor map appears, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' terms of the form ℓ2pHpxIjq, fΦp0qpxjqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ: S‚ KpE1q Ñ S‚ KpEq be a co-algebra morphism such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Φ is O-multilinear, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρ ˝ Φ0 “ ρ1 on E1, If for every 3 n P N0, pΦ ˝ QE1 ´ QE ˝ Φqpiq “ 0 for each 0 ď i ď n, then the map SKpE1q Ñ SKpEq given by: pΦ ˝ QE1 ´ QE ˝ Φqpn`1q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is a Φp0q-co-derivation of degree `1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is O-multilinear, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and the induced Φp0q-co-derivation ´Ä‚ E1, Qp0q E1 ¯ ÝÑ ´Ä‚ E, Qp0q E ¯ satisfies: Qp0q E ˝ pΦ ˝ QE1 ´ QE ˝ Φqpn`1q “ pQE ˝ Φ ´ Φ ˝ QE1qpn`1q ˝ Qp0q E1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A straightforward computation yields: ∆pΦ ˝ QE1 ´ QE ˝ Φq “ pΦ b Φq ˝ ∆1 ˝ QE1 ´ pQE b id ` id b QEq ˝ ∆ ˝ Φ “ ppΦ ˝ QE1 ´ QE ˝ Φq b Φ ` Φ b pΦ ˝ QE1 ´ QE ˝ Φqq ˝ ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now, ∆ preserves polynomial-degree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ∆ : Sn KpEq ÝÑ ‘i`j“nSi KpEq b Sj KpEq and so does ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Taking into account the assumption pΦ ˝ QE1 ´ QE ˝ Φqpiq “ 0 for every 0 ď i ď n, we obtain: ∆ ˝ pΦ ˝ QE1 ´ QE ˝ Φqpn`1q “ ´ pΦ ˝ QE1 ´ QE ˝ Φqpn`1q b Φp0q ` Φp0q b pΦ ˝ QE1 ´ QE ˝ Φqpn`1q¯ ˝ ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' All the other terms disappear for polynomial-degree reasons Hence, pΦ ˝ QE1 ´ QE ˝ Φqpn`1q is a Φp0q-co-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove that it is O-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It suffices to check O-linearity of TΦ :“ Φ ˝ QE1 ´ QE ˝ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us choose homogeneous elements x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xN P E1 and let us assume that xi P E1 ´1 is the only term of degree ´1: The proof in the case where there is more than one such an homogeneous element of degree ´1 is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We choose j ‰ i and we compute TΦpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fxj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xNq for some f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The only terms in the previous expression which are maybe non-linear in f are those for which the 2-ary brackets of a term containing fxj with xi or Φ0pxiq appear (since Φ and all other brackets 3Φ ˝ QE1 ´ QE ˝ Φ being a Φ-co-derivation, its component of polynomial-degree i is zero for 0 ď i ď n if only if its i-th Taylor coefficient is zero for 0 ď i ď n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 41 are O-linear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are two such terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The first one appears when we apply QE1 first, and then Φ: this forces Φ pℓ1 2pxi, fxjq, xIijq to appear, and the non-linear term is then: ϵpσiqρ1pxiqrfs ΦpxIiq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17) with σi the permutation that let i goes in front and leave the remaining terms unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is a second term that appears when one applies Φ first, then QE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since it is a co-morphism, Φpx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xNq is the product of several terms among which only one is of degree ´1, namely the term ϵpx, σiqΦ0pxiqΦpfxIiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Applying QE to this term yields the non-linear term ϵpσiqρpΦ0pxiqqrfs ΦpxIiq, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18) where Ii and Iij are as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ρ ˝ Φ0 “ ρ1, we see that the terms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18) containing an anchor add up to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check that pΦ ˝ QE1 ´ QE ˝ Φqpn`1q is a chain map, in the sense that it satisfies item 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Considering again TΦ :“ Φ ˝ QE1 ´ QE ˝ Φ, we have that T pkq Φ “ 0, for all k “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since TΦ ˝ QE1 “ QE ˝ TΦ, one has 0 “ pTΦ ˝ QE1 ` QE ˝ TΦqpn`1q “ T pn`1q Φ ˝ Qp0q E1 ` Qp0q E ˝ T pn`1q Φ ` ÿ i`j“n`1 i,jě1 ´ T piq Φ ˝ Qpjq E1 ` Qpjq E ˝ T piq Φ ¯ loooooooooooooooomoooooooooooooooon 0 By consequent, the O-linear map pΦ ˝ QE1 ´ QE ˝ Φqpn`1q satisfies item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ, Ξ : S‚ KpE1q ÝÑ S‚ KpEq be O-linear Lie 8-algebroid morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let n P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If Ξpiq “ Φpiq for every 0 ď i ď n, then pΞ ´ Φqpn`1q : S‚ KpE1q ÝÑ S‚ KpEq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is a Φp0q-co-derivation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is O-multilinear 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and the induced Φp0q-co-derivation ´Ä‚ E1, Qp0q E1 ¯ ÝÑ ´Ä‚ E, Qp0q E ¯ satisfies: Qp0q E ˝ pΞ ´ Φqpn`1q “ pΞ ´ Φqpn`1q ˝ Qp0q E1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xk P E1, one has: ∆pΞ ´ Φqpx1 d ¨ ¨ ¨ d xkq “ kÿ j“1 ÿ σPSpj,k´jq ϵpσqΞpxσp1q d ¨ ¨ ¨ d xσpjqq b Ξpxσpj`1q d ¨ ¨ ¨ d xσpkqq ´ kÿ j“1 ÿ σPSpj,k´jq ϵpσqpΦpxσp1q d ¨ ¨ ¨ d xσpjqq b Φpxσpj`1q d ¨ ¨ ¨ d xσpkqq “ ppΞ ´ Φq b Φ ` Ξ b pΞ ´ Φqq ˝ ∆1px1 d ¨ ¨ ¨ d xkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ∆ has polynomial-degree 0 and pΞ ´ Φqpiq “ 0 for all 0 ď i ď n, we obtain ∆pΞ´Φqpn`1qpx1d¨ ¨ ¨dxkq “ ´ pΞ ´ Φqpn`1q b Φp0q ` Φp0q b pΞ ´ Φqpn`1q¯ ˝∆1px1d¨ ¨ ¨dxkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19) CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE 8-ALGEBROIDS AND THEIR MORPHISMS 42 This proves the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since both Φ and Ξ are O-multilinear, pΞ ´ Φqpn`1q is O-multilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since, Φ and Ξ are Lie 8-morphisms: pΞ ´ Φq ˝ QE1 ´ QE ˝ pΞ ´ Φq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20) By looking at the component of polynomial-degree n`1, one obtains, pΞ´Φqpn`1q ˝Qp0q E1 ´Qp0q E ˝pΞ´ Φqpn`1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves the third item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conclusion: This chapter recapitulates classical definition of Lie 8-algebroids, and in particular describes it as co-derivation, which is not usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Morphisms and homotopies are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' About homotopies, two versions are given: one uses smooth maps, and is way easier (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Unfortunately, to allow infinitely many gluings, we have to introduce a more complicated notion of homotopy, which uses continuous locally C1-maps (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 3 Lie-Rinehart algebras and their morphisms Except for Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, this section is essentially a review of the literature on the subject, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Hue04, Hue98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Definitions Lie-Rinehart algebras are the algebraic encoding of the notion of Lie algebroids over a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We owe this concept to [Rin63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie-Rinehart algebra over an algebra O is a triple pA, r¨, ¨sA, ρAq with A an O- module, r¨, ¨sA a Lie algebra bracket on A, and ρA : A ÝÑ DerpOq an O-linear Lie algebra morphism called anchor map, satisfying the so-called Leibniz identity: ra, fbsA “ ρApaqrfs b ` fra, bsA for all a, b P A, f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let η: O ÝÑ O1 be an algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie-Rinehart algebra morphism over η is a Lie algebra morphism φ: A ÝÑ A1 such that for every a P A and f P O: (a) φpfaq “ ηpfqφpaq (b) ηpρApaqrfsq “ ρA1pφpaqrηpfqs, When O “ O1 and η “ id, we say that φ is a Lie-Rinehart algebra morphism over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A submodule B Ď A is a said to be a Lie-Rinehart subalgebra of A if it carries a Lie-Rinehart algebra structure over O whose Lie bracket and anchor map are the restriction of the bracket r¨ , ¨sA and the anchor ρA respectively to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie-Rinehart algebras over O form a category that we denote by Lie-Rhart-alg/O Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie-Rinehart algebra is said to be a Lie algebroid if A is a projective O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' See Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 for the relation with usual Lie algebroid as vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 43 CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS 44 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every Lie 8-algebroid pE‚, ℓ‚, ρq over O, the quotient space E´1 ℓ1pE´2q comes equipped with a natural Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The 2-ary bracket ℓ2 goes to quotient to E´1 ℓ1pE´2q to define a Lie algebra since for all x P E´2 and y P E´1 we have ℓ2pℓ1pxq, yq “ ℓ1pℓ2px, yqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, the Jacobi identity holds, since ℓ2pℓ2px, yq, yq` ö px, y, zq “ ´ℓ1pℓ3px, y, zqq @x, y, z P E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In addition, the anchor map ρ goes to quotient to a Lie algebra morphism E´1 ℓ1pE´2q Ñ DerpOq, since ρ ˝ ℓ1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This Lie-Rinehart algebra is called the basic Lie-Rinehart algebra of pE‚, ℓ‚, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To summarize, every Lie 8-algebroid induces a Lie-Rinehart algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The opposite direction, that is, wondering whether a Lie-Rinehart algebra pA, r¨ , ¨sA , ρq over O is the basic Lie-Rinehart algebra of some almost differential graded Lie algebroid or more generally a Lie 8-algebroid over O is part of the questions that I discussed in this thesis (see Chapter 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This extends the main results of [LLS20] from locally real analytic finitely generated singular foliations (see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) to arbitrary Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us fix some vocabulary that will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that an almost differential graded Lie algebroid pE‚, ℓ1, ℓ2, ρq or a Lie 8-algebroid pE‚, ℓ‚, ρq over O 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' covers (through a hook π) a Lie-Rinehart algebra pA, r¨ , ¨sA, ρAq, if there exists a morphism of brackets π: E´1 Ñ A such that π ˝ ρA “ ρ and πpE´1q “ A, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' terminates in A through the hook π, when πpE´1q Ď A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' According to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3, any Lie 8-algebroid pE‚, ℓ‚, ρq covers its basic Lie-Rinehart algebra through the hook π which is given by the projection π: E´1 ÝÑ E´1{ℓ1pE´2q, since π respects the brackets and πpE´1q “ E´1{ℓ1pE´2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE1 ‚, ℓ1 ‚, ρ1q and pE‚, ℓ‚, ρq and be Lie 8-algebroids over O that terminate in a Lie-Rinehart algebra pA, r¨ , ¨sA, ρAq through the hooks π1 and, π respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that a Lie 8-algebroid morphism Φ: S‚ KpE1q Ñ S‚ KpEq is over A, if π ˝ Φ0 “ π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We will need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE1 ‚, ℓ1 ‚, ρ1q and pE‚, ℓ‚, ρq be Lie 8-Lie algebroids that terminate in some Lie- Rinehart algebra pA, r¨ , ¨sA , ρAq through hooks π1 and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pJt, HtqtPra,bs be a homotopy that joins Ja and Jb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If Ja is Lie 8-algebroid morphism that terminates at A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', π ˝ Jp0q a |E1 “ π1), then so is the 8-algebroid morphism Jt for all t P ra, bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is a direct consequence of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16), since Qp0q E |E “ ℓ1 : E´2 Ñ E´1 and Qp0q E1 |E1 “ ℓ1 1 : E1 ´2 Ñ E1 ´1 are valued in the kernels of π and, π1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Last, O-multilinearity of Jt follows from the O-multilinearity of QE ˝Ht`Ht˝QE1, which is granted by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS 45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Algebraic and geometric examples Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every commutative K-algebra O, the Lie algebra A “ DerpOq of derivations of a commutative algebra O is a Lie-Rinehart algebra over O, with the identity as an anchor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This Lie-Rinehart algebra is a terminal object in the category Lie-Rhart-alg/O: for every Lie-Rinehart pA, r¨ , ¨sA , ρAq the anchor A ρA ÝÑ DerpOq is obviously a Lie-Rinehart algebra morphism over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, vector fields on a smooth or Stein manifold or an affine variety are examples of Lie-Rinehart algebras over their respective natural algebras of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let M be a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Serre-Swan Theorem, Lie algebroids over M are precisely Lie-Rinehart algebras over C8pMq of the form pΓpAq, r¨, ¨s, ρq where A is a vector bundle over M and ρ : A Ñ TM is a vector bundle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Sections of a Lie algebroid that have values in the kernel of the anchor map form a Lie-Rinehart algebra KerpρAq for which the anchor map is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The next two examples are part of our the main source of motivation to study Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The second is more general than the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [AS09, AZ14, Cer79, Deb01, LLS20, LGLR22] A singular folia- tion on a smooth, real analytic, or complex manifold M or Zariski open subset U Ď Cd is a subsheaf F Ď XpMq that fulfills the following conditions Stability under Lie bracket: rF, Fs Ď F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Locally finitely generateness: every m P M admits an open neighborhood U together with a finite number of vector fields X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xr P XpUq such that for every open subset V Ď U the vector fields X1|V, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xr|V generates F on V as a C8pVq-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are several other ways to define singular foliations on a manifold M [AZ13, Cer79, Daz85, Deb01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' All these definitions have in common to define then as Lie-Rinehart sub-algebras F of the Lie-Rinehart algebra XpMq of vector fields on M (or compactly supported vector fields XcpMq on M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here are some important consequences of the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Singular foliation admits leaves: there exists a partition of M into submanifolds called leaves such that for all m P M, the image of the evaluation map F Ñ TmM is the tangent space of the leaf through m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When F coincides with the space of vector fields tangent to all leaves at all points, we shall speak of a “Stefan-Sussman singular foliation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Singular foliations are self-preserving : the flow of vector fields in F, whenever defined, preserves F [AS09, GY18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13, F :“ ρpΓpE´1qq is a singular foliation in the sense above, called the basic singular foliation of pE, pℓkqkě1, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say, then the Lie 8-algebroid pE, pℓkqkě1, ρq is over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [AZ18, Zam18, ZA18] Let pA, r¨ , ¨sA , ρAq be a Lie algebroid over a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A singular subalgebroid B of A, is a C8pMq-submodule of ΓpAq that is locally finitely generated and involutive i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' stable under the Lie bracket r¨ , ¨sA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This notion is a generalization of singular foliations in the sense of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4, since singular foliations on M are singular subalgebroids of TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS 46 A singular subalgebroid B is an example of Lie-Rinehart algebra: its bracket and anchor are the restrictions of r¨ , ¨sA and ρA to B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a singular foliation F on a manifold M, consider S :“ tX P XpMq | rX, Fs Ď Fu (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' infinitesimal symmetries of F) and C :“ tf P C8pMq | Y rfs “ 0, for all Y P Fu (that can be thought of as functions constant along the leaves of F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The quotient S F is Lie-Rinehart algebra over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 (Poisson manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [LGPV13] We recall that a Poisson manifold is a manifold M together with a biderivation t¨ , ¨u on its algebra of smooth functions C8pMq that satisfies Jacobi’s identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' pC8pMq, t¨ , ¨uq is Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The biderivation t¨ , ¨u is compatible with the product of functions in the following sense: for all f, g, h P C8pMq tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g, hu “ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='tg, hu ` g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='tf, hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is equivalent to giving a bivector field π P Γp^2TMq such that the Schouten-Nijenhuis bracket with itself vanishes, that is, rπ, πsSN “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every a Poisson manifold pM, πq, the operator δπ : Γp^‚TMq Ñ Γp^‚`1TMq, P ÞÑ ´rP, πsSN defines a complex ¨ ¨ ¨ � Γp^p´1TMq δp´1 π � Γp^pTMq δp π � Γp^p`1TMq � ¨ ¨ ¨ , since rπ, πsSN “ 0 implies δπ ˝ δπ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every p P N0, the quotient Hp πpMq :“ ker δp π Imδp´1 π is called the p-th Poisson cohomology of pM, πq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define A :“ H1 πpMq to be the first Poisson cohomology of π and O :“ H0 πpMq “ Caspπq to be the algebra of Casimir functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The bracket of vector fields makes A a Lie-Rinehart algebra over Caspπq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Basic constructions Let pA, r¨ , ¨sA , ρAq a Lie-Rinehart algebra over an algebra O and I Ă O be an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The submodule IA is a Lie-Rinehart subalgebra of A and its anchor is given by the restriction of ρA over IA Ă A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This follows easily from rfa, gbsA “ fgra, bsA ` fρApaqrgs b ´ gρApbqrfs a for all a, b P A, f, g P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider a Lie-Rinehart algebra pA, r¨, ¨sA, ρAq over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every Lie-Rinehart ideal I Ă O, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' any ideal such that ρApAqrIs Ă I the quotient space A{IA inherits a natural Lie-Rinehart algebra structure over O{I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We call this Lie-Rinehart algebra the restriction w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t the Lie-Rinehart ideal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the context of affine varieties, when I is the ideal of functions vanishing on an affine sub-variety W, we shall denote A IA by i˚ W A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS 47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Localizing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t a multiplicative subset S Ă O (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' S Ď Ozt0u is closed under multiplication and 1 P S) is a very powerful tool in commutative algebra and in algebraic geometry to deal with "global" problems by reducing them to "local" ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recall the construction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' see [Sta22], Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9 or [And] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (a) Let V be a O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The localization of V at S is defined as follows: consider the equivalence relation on S ˆ V that is given by ps, vq „ ps1, v1q ðñ Du P S, ups1v ´ sv1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The class of an element ps, vq P S ˆ V by v s, and by S´1V :“ S ˆ V{„ the set of equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, the localization of O at S is the localization of O as a O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, S´1O is a K-algebra together with the addition and multiplication defined by f s ` g s1 :“ fs1 ` sg ss1 and f s g s1 :“ fg ss1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Similarly, S´1V is a S´1O-module with addition and scalar product multiplication defined in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, we have S´1V » S´1O bO V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) A derivation D P DerpOq admits a unique extension to a derivation S´1D P DerpS´1Oq which given for ps, fq P S ˆ O by the classical formula pS´1Dq ˆf s ˙ :“ Dpfqs ´ Dpsqf s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρq be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The localization module S´1A “ S´1ObOA comes equipped with a natural structure of Lie-Rinehart algebra over the localization algebra S´1O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The new anchor map is defined by a s P S´1A ÞÑ 1 sS´1pρApaqq P DerpS´1Oq, and the new Lie algebra bracket is given by „1 sa, 1 ub ȷ S´1A “ 1 sura, bsA ´ ρApaqrus su2 b ` ρApbqrss s2u a for a, b P A, ps, uq P S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The localization map A ãÑ S´1A is a Lie-Rinehart algebra morphism over the localization map O ãÑ S´1O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since localization exists, the notion of sheaf of Lie- Rinehart algebras [Vil20] over a projective variety, or a scheme, makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Algebra extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume that the algebra O has no zero divisor, and let O be its field of fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any subalgebra ˜O with O Ă ˜O Ă O such that ρpaq is for any a P A valued in derivations of O that preserves ˜O, there is a natural Lie-Rinehart algebra structure over ˜O on the space ˜O bO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Blow-up at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us consider a particular case of the previous construction, when O is the algebra Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If the anchor map of a Lie-Rinehart algebra A over O takes values in CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS 48 vector fields on CN vanishing at the origin, then for all i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , N, the polynomial algebra OUi generated by x1 xi , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xi´1 xi , xi, xi`1 xi , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xN xi satisfies the previous condition, and OUi bO A comes equipped with a Lie-Rinehart algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Geometrically, this operation corresponds to taking the blow-up of CN at the origin, then looking at the i-th natural chart Ui on this blow-up: OUi are the polynomial functions on Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The family OUi bO A (for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , N) is therefore an atlas for a sheaf of Lie-Rinehart algebras (in the sense of [Vil20]) on the blow-up of CN at the origin, referred to as the blow-up of A at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 On free resolutions of length ď 2 and Lie-Rinehart algebras In this section, we discuss the case when a Lie-Rinehart algebra A admits a free resolution of length 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In those cases, we claim that there are Lie algebra-like structures on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' See B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 for the notion of free resolution of modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These results will be soon generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I start with the following remark (owed to Marco Zambon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any Lie-Rinehart algebra pA, r¨ , ¨sA , ρAq is the image of an almost differential graded Lie algebroid pE´1, ρq concentrated in degree ´1: to see this, let tai P A | i P Iu be a set of generators of A (take all elements of A if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There exists elements uk ij P O, such that for given indices i, j, the coefficient uk ij is zero except for finitely many indices k, together with rai, ajsA “ ÿ kPI uk ijak @i, j P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) The coefficients uk ij can be chosen to satisfy the skew-symmetry condition uk ij “ ´uk ji: by skew- symmetry of the bracket r¨ , ¨sA on has rai, ajsA “ 1 2 prai, ajsA ´ raj, aisAq “ ÿ kPI 1 2puk ij ´ uk jiqak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, one can replace uk ij by 1 2puk ij ´ uk jiq if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now, choose E´1 to be the free O-module generated by the symbols peiqiPI together with the surjective map π: E´1 Ñ A, ei ÞÑ ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We now define: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' an anchor map by ρpeiq :“ ρApai), for all i P I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρ “ ρA ˝ π, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a skew-symmetric operation r¨ , ¨sE´1 on E as follows: rei, ejsE´1 “ ÿ kPI uk ijek for all i, j P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We extend by O-linearity and Leibniz identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction pE´1, r¨ , ¨sE´1, ρ) is an almost Lie algebroid over O whose image through the anchor map is A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In general, this bracket does not satisfy the Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If E´1 » A, this bracket is a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS 49 On free resolutions of length 1 Lie-Rinehart algebra A admits a free resolution of length 1 if and only if A is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, the almost Lie algebroid bracket r ¨, ¨sE´1 is a Lie algebroid bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In conclusion: free resolutions of length 1 admit a Lie algebroid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Free resolutions of length 2 Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra that admits a free resolution of length 2, namely an exact sequence of the form 0 � E´2 ℓ1 � E´1 ρ � � π � � A ρA� DerpOq (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) with E´1, E´2 free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) is a free resolution of A, the map π is in particular surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8, E´1 can be endowed with an almost Lie algebroid bracket, such that π is a morphism of brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can extend this bracket to sections of degree ´2 to obtain an almost differential graded Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us compute it: Let pe1 iqiPI1 and pejqiPI be a basis of E´2 and E´1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all i, j, k P I Y I1 we have 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' πprℓ1e1 i, ejsE´1q “ rπ ˝ ℓ1pe1 iq, πpejqs “ 0, (since π ˝ ℓ1 ” 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In other words, rℓ1pe1 iq, ejsE´1 P ker π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By exactness of the complex (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) there exists an element ∇e1 iej P E´2 such that πp∇e1 iejq “ rℓ1pe1 iq, ejsE´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) allows defining a bilinear map: E´1 b E´2 Ñ E´2 px, yq ÞÑ ∇xy by extending the ∇e1 iej’s by linearity and Leibniz identity with the understanding that the anchor map ρ vanishes on E´2 in order to have (a) ℓ1p∇xyq “ rℓ1pxq, ysE´1, @x E´2, y P E´1, (b) for all f P O: ∇xfy “ f∇xy ` ρpxqrfs y and ∇fxy “ f∇xy, for all x P E´1, y P E´2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remember that Jacpei, ej, ekq :“ rei, rej, eks2s2 ` rej, rek, eis2s2 ` rek, rei, ejs2s2 P ker π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By using again exactness of the complex (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) there is an element that we denote by rei, ej, eksE´1 P E´2 that satisfies ℓ1 ` rei, ej, eksE´1 ˘ “ Jacpei, ej, ekq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS 50 Thus, we can define a skew-symmetric trilinear map: r¨ , ¨ , ¨sE´1 : E´1 ^ E´1 ^ E´1 ÝÑ E´2 such that ℓ1prx, y, zsE´1q “ rx, ry, zs2s2 ` ry, rz, xs2s2 ` rz, rx, ys2s2, @x, y, z P E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following Proposition concludes the discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra that admits a free resolution of length 2 as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' E´1 admits an almost Lie algebroid structure pE´1, r¨ , ¨sE´1, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is a bilinear map: E´1 b E´2 Ñ E´2 px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' yq ÞÑ ∇xy and a skew-symmetric trilinear map: r¨ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ¨ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ¨sE´1 : E´1 ^ E´1 ^ E´1 ÝÑ E´2 such that for all f P O: (a) ∇xfy “ f∇xy ` ρpxqrfs y and ∇fxy “ f∇xy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all x P E´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y P E´2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) rfx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zsE´1 “ frx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zsE´1 for all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' z P E´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' such that the 2-ary bracket on E´1 ‘ E´2 defined by: rx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ys2 “ $ ’ ’ ’ ’ & ’ ’ ’ ’ % rx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ysE´1 for x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y P E´1 ∇xy for x P E´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y P E´2 ∇yx for x P E´2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y P E´1 0 for x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y P E´2 together with the 3-ary bracket on E´1 ‘ E´2 defined by rx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zs3 “ rx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' zsE´1 if x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' z P E´1 and zero otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' satisfies (a) for all x P E´2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' y P E´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ℓ1prx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ys2q ` rℓ1pxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ys2 “ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) (b) for all x, y, z P E´1 ℓ1prx, y, zs3q ` rx, ry, zs2s2 ` ry, rz, xs2s2 ` rz, rx, ys2s2 “ 0 (c) for all x, y P E´1 and z P E´2 rx, y, ℓ1pzqs3 ` rx, ry, zs2s2 ` ry, rz, xs2s2 ` rz, rx, ys2s2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' LIE-RINEHART ALGEBRAS AND THEIR MORPHISMS 51 The structure pE‚, ℓ1, ρ, r¨ , ¨s2, r¨ , ¨, ¨s3q described in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 is called Lie 2-algebroid [BC03, Lea17, SZ11], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a Lie 8-algebroid with E´i “ 0 for i ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A generalization of this construction on free resolutions of higher (even infinite) length is the object of the next Chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conclusion: We describe Lie-Rinehart algebras, give examples and several constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 is a pedagogical section, which presents an elementary case of the general case that we will study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4 Main results of Part I 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Presentation of the problem In order to understand the geometry of affine varieties or some similar problems related to singular foliations using the Lie algebra of their vector fields which is in fact Lie-Rinehart algebras, have mo- tivated us to understand Lie-Rinehart algebras in general from another point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have seen in the Chapter 3, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 that any Lie 8-algebroid over O induces a Lie- Rinehart algebra which we call its basic Lie-Rinehart algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this chapter, we are investigating the opposite direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', we study the following questions: given a Lie-Rinehart algebra A over O, can we find a Lie 8-algebroid over O whose basic Lie-Rinehart algebra is A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, in case where it exists, do we have uniqueness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now let us formalize that in a categorical language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by Lie-8-alg-oids/O the quotient category where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the objects are Lie 8-algebroids over O, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' arrows are homotopy equivalence classes of morphisms of Lie 8-algebroids over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and by Lie-Rhart-alg/O the category of Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We want to study the functor, Lie-8-alg-oids/O F ÝÑ Lie-Rhart-alg/O D!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ÐÝ The question now turn out to be: when does F admit a left/right inverse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the next section, we present in detail the main results related to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 52 CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Main results An existence Theorem The results below appeared in my first article [LGL22b] entitled "Lie-Rinehart algebra » acyclic Lie 8-algebroid" co-written with my supervisor C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Laurent-Gengoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This section extends the main results of [LLS20] from locally real analytic finitely generated sin- gular foliations to arbitrary Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It states that universal Lie 8-algebroids over a given Lie-Rinehart algebra exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It extends Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 in [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We are convinced that it may be deduced using the methods of semi-models categories as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 in [FJO18], but does not follow from a simple homotopy transfer argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any resolution of A by free O-modules ¨ ¨ ¨ ℓ1 ÝÑ E´3 ℓ1 ÝÑ E´2 ℓ1 ÝÑ E´1 π ÝÑ A (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) comes equipped with a Lie 8-algebroid structure whose unary bracket is ℓ1 and that covers A through the hook π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since any module admits free resolutions (see Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5), Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 implies that: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any Lie-Rinehart algebra A over O is the basic Lie-Rinehart algebra of an acyclic Lie 8-algebroid over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' While proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, we will see that if E´1 can be equipped with a Lie algebroid bracket (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a bracket whose Jacobiator is zero), then all k-ary brackets of the universal Lie 8-algebroid structure may be chosen to be zero on E´1: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, ℓ1, πq be a free resolution of a Lie-Rinehart algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If E´1 admits a Lie algebroid bracket r¨, ¨s such that π : E´1 Ñ A is a Lie-Rinehart morphism, then there exists a structure of universal Lie 8-algebroid pE‚, ℓ‚, ρq that covers A whose 2-ary bracket coincides with r¨, ¨s on E´1 and such that for every k ě 3 the k-ary bracket ℓk vanishes on Äk E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Universality of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 and its corollaries Here is our second main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is related to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 in [FJO18] (but morphisms are not the same), and extends Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9 in [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given, a) a Lie 8-algebroid pE1 ‚, ℓ1 ‚, ρ1q that covers A through a hook π1, and b) any acyclic Lie 8-algebroid pE‚, ℓ‚, ρq that covers A through a hook π, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' there exists a morphism of Lie 8-algebroids from pE1 ‚, ℓ1 ‚, ρ1q to pE‚, ℓ‚, ρq over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and any two such morphisms are homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is an immediate corollary of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any two acyclic Lie 8-algebroids that cover a given Lie-Rinehart algebra are ho- motopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This homotopy equivalence, moreover, is unique up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We will prove that the morphism that appears in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 can be made trivial upon choosing a “big enough” universal Lie 8-algebroid: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a Lie 8-algebroid structure pE1 ‚, ℓ1 ‚, ρ1q that terminates in A through a hook π1, then there exist an acyclic Lie 8-algebroid pE‚, ℓ‚, ρq that covers A through a hook π such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' E contains E1 as a subcomplex, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the Lie 8-algebroid morphism from E1 to E announced in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 can be chosen to be the inclusion map E1 ãÑ E (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a Lie 8-morphism where the only non-vanishing Taylor coefficient is the inclusion E1 ãÑ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following Corollary follows immediately from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let A be a Lie-Rinehart algebra over O and B be a Lie-Rinehart subalgebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any universal Lie 8-algebroid of B can be contained in a universal Lie 8-algebroid of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Induced Lie 8-algebroids structures on TorOpA, O{Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that an ideal I Ă O is a Lie-Rinehart ideal of A if ρApaqrIs Ă I for all a P A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any Lie-Rinehart ideal IO of A, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' we have rIA, AsA Ă IA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, the quotient space A{IA comes equipped with a natural Lie-Rinehart algebra structure over O{I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For pE‚, ℓ‚, ρq an acyclic Lie 8-algebroid that covers A, the quotient space E‚{I » O{I bO E‚ comes equipped with an induced Lie 8-algebroid structure: the n-ary brackets for n ‰ 2 go to quotient by linearity, while for n “ 2, the 2-ary bracket goes to the quotient in view of the relation ρEpE´1qrIs Ă I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, π goes to the quotient to a Lie-Rinehart algebra morphism E´1{I Ñ A{IA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let A be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every Lie-Rinehart ideal I Ă O, we call Lie 8-algebroid of I the quotient Lie 8-algebroid E‚{I, with pE‚, ℓ‚, ρq a universal Lie 8-algebroid that covers A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The complexes on which the Lie 8-algebroids of the ideal I are defined compute TorOpA, O{Iq by construction (see B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 55 Moreover, for any two universal Lie 8-algebroids of A, defined on E, E1 the homotopy equivalences Φ : E1 Ñ E and Ψ : E Ñ E1, whose existence is granted by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5, go to the quotient and induce an homotopy equivalences between O{I bO E‚ » E‚{I and O{I bO E1 ‚ » E1 ‚{I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following corollary is then an obvious consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let A be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let I Ă O be a Lie-Rinehart ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then any two Lie 8-algebroids of I are homotopy equivalent, and there is a distinguished class of homotopy equivalences between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Taking under account Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11, here is an alternative manner to restate this corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let A be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let I Ă O be a Lie-Rinehart ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then the complex computing Tor‚ OpA, O{Iq comes equipped with a natural Lie 8-algebroid structure over O{I, and any two such structures are homotopy equivalent in a unique up to homotopy manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When, in addition to being a Lie-Rinehart ideal, I is a maximal ideal, then K :“ O{I is a field and Lie 8-algebroids of I are a homotopy equivalence class of Lie 8-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular their common cohomologies, which is easily seen to be identified to Tor‚ OpA, Kq comes equipped with a graded Lie algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, Tor´1 O pA, Kq is a Lie algebra, Tor´2 O pA, Kq is a representation of this algebra, and the 3-ary bracket defines a class in the third Chevalley-Eilenberg cohomology of Tor´1 O pA, Kq valued in Tor´2 O pA, Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This class does not depend on any choice made in its construction by the previous corollaries, and trivially extends the class called NMRLA-class in [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If it is not zero, then there is no Lie algebroid of rank r equipped with a surjective Lie-Rinehart algebra morphism onto A, where r is the rank of A as a module over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' All these considerations can be obtained by repeating verbatim Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 in [LLS20] (where non-trivial examples are given).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A categorical approach of the results Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 means that acyclic Lie 8-algebroids that covers Lie-Rinehart algebra pA, r¨ , ¨sA , ρAq are terminal objects in the subcategory of Lie-8-alg-oids/O whose objects are Lie 8-algebroids that terminate in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Whence, acyclic Lie 8-algebroids over O that cover pA, r¨ , ¨sA , ρAq are deserved to be called "universal 8-algebroids of A".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From now on, we call them by this name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us re-state Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By associating to any Lie 8-algebroid its basic Lie-Rinehart algebra one obtains therefore a natural functor: from the category Lie-8-alg-oids/O, to the category Lie-Rhart-alg/O Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 gives a right inverse of this functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, this functor becomes an equivalence of categories when restricted to homotopy equivalence classes of acyclic Lie 8-algebroids over O, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is an equivalence of categories between: (i) Lie-Rinehart algebras over O, (ii) acyclic Lie 8-algebroids over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 56 This corollary justifies the title of the first part of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the language of categories, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12 means that there exists a functor from Lie-Rinehart ideals of a Lie-Rinehart algebra over O, to the category of Lie 8-algebroids, mapping a Lie-Rinehart ideal I to an equivalence class of Lie 8-algebroids over O{I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 Proof of main results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 A crucial bi-complex: PagepnqpE1, Eq Description of PagepnqpE1, Eq Let V be an O-module, and let pE, d, πq and pE1, d1, π1q be complexes of projective O-modules that terminates at V: ¨ ¨ ¨ dÝÑ E´2 dÝÑ E´1 πÝÑ V, ¨ ¨ ¨ d1 ÝÑ E1 ´2 d1 ÝÑ E1 ´1 π1 ÝÑ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) For every k ě 1, the pk ` 1q-th graded symmetric power Äk`1 E1 of E1 over O is a projective O-module, and comes with a natural grading induced by the grading on E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let k P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We call page number k of pE, d, πq and pE, d1, π1q the bicomplex of O-modules on the upper left quadrant Z´ ˆ N0 defined by: PagepkqpE1, Eqj,m :“ HomO ˜ k`1 ä E1 |´k´m´1 , Ej ¸ , for m ě 0 and j ď ´1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) PagepkqpE1, Eq0,m :“ HomO ˜ k`1 ä E1 |´k´m´1 , V ¸ , for m ě 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) together with the vertical differential Dv defined for any one of the two O-modules (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) by Dv : PagepkqpE1, Eqj,m ÝÑ PagepkqpE1, Eqj,m`1 Φ ÞÝÑ DvpΦq: Äk`1 E1 ÝÑ Ej x1 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' d xk`1 ÞÑ Φ ˝ d1 px1 d ¨ ¨ ¨ d xk`1q where d1 acts as an O-derivation on x1 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' d xk`1 P Äk E1 (and is 0 on E1 ´1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The horizontal differential, Dh : PagepkqpE1, Eqj,m ÝÑ PagepkqpE1, Eqj`1,m, is given by Φ ÞÑ d ˝ Φ or Φ ÞÑ π ˝ Φ depending on whether Φ is of type (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) with j ď ´2 or the type (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) with j “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is zero on elements of type (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by ´ Pagepkq ‚ pE1, Eq, D ¯ its associated total complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When E1 “ E we shall write Pagepkq ‚ pEq instead of Pagepkq ‚ pE, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 57 The following diagram recapitulates the whole picture of Pagepkq ‚ pE1, Eq: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ò Ò Ò ¨ ¨ ¨ Ñ HomO ´Äk`1 E1 |´k´3, E´2 ¯ Dh Ñ HomO ´Äk`1 E1 |´k´3, E´1 ¯ Dh Ñ HomO ´Äk`1 E1 |´k´3, V ¯ Ñ 0 Dv Ò Dv Ò Dv Ò ¨ ¨ ¨ Ñ HomO ´Äk`1 E1 |´k´2, E´2 ¯ Dh Ñ HomO ´Äk`1 E1 |´k´2, E´1 ¯ Dh Ñ HomO ´Äk`1 E1 |´k´2, V ¯ Ñ 0 Dv Ò Dv Ò Dv Ò ¨ ¨ ¨ Ñ HomO ´Äk`1 E1 |´k´1, E´2 ¯ Dh Ñ HomO ´Äk`1 E1 |´k´1, E´1 ¯ Dh Ñ HomO ´Äk`1 E1 |´k´1, V ¯ Ñ 0 Ò Ò Ò 0 0 0 "-2 column" "-1 column" "last column" (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) For later use, we spell out the meaning of being D-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' An element P P Pagepkq j pE1, Eq in ‘iě1HomO ´Äk`1 E1 |´j´i, E´i ¯ is D-closed if and only if: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the component P´1 : Äk`1 E1 |´j´1 Ñ E´1 is valued in the kernel of π : E´1 Ñ V, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the following diagram commutes: Äk`1 E1 |´j´i p´1qjd1 � P´i � Äk`1 E1 |´j´i`1 P´i`1 � E´i d � E´i`1 with P´i being the component of P in HomO ´Äk`1 E1 |´j´i, E´i ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For pE, dq “ pE1, d1q, the second condition above also reads rd, PsRN “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is an important technical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, d, πq be a resolution of V in the category of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, for every k ě 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the cohomology of the complex pPagepkq ‚ pEq, Dq for the total differential D‚ :“ Dh ´ p´1q‚Dv is zero in all degrees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, a D-closed element whose component on the “last column” of the diagram above is zero is the image through D of some element whose two last components are also zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More generally, for all n ě 1, for a D-closed element P P Pagepkq j pEq of the form à iěn HomO ˜ k`1 ä E1 |´j´i, E´i ¸ , one has P “ DpRq and R P Pagepkq j´1pEq can be chosen in À iěn`1 HomO ´Äk`1 E1 |´j´i`1, E´i ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since Äk E1|j`m´k is a projective O-module for all pj, mq P Z´ˆN0, and pE, d, πq is a resolution, all the lines of the above bicomplex are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The second and the third are obtained by diagram chasing (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 58 Interpretation of PagepnqpE1, Eq in our context We denote by Qp0q E and Qp0q E1 the differentials of polynomial-degree 0 on Ä‚ E and Ä‚ E1 induced by d and d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Whence, pHomOpÄ‚ E1, Ä‚ Eq, Bq with B: H ÞÑ Qp0q E ˝ H ´ p´1q|H|H ˝ Qp0q E1 is a complex of O-modules (see Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let φ: pE1, d1q Ñ pE, dq be a chain map, and let Φp0q : Ä‚ E1 Ñ Ä‚ E be its extension to a co-algebra morphism, namely: Φp0qpx1 ¨ ¨ ¨ ¨ ¨ xnq :“ φpx1q ¨ ¨ ¨ ¨ ¨ φpxnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have, Φp0q ˝ Qp0q E1 “ Qp0q E ˝ Φp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Φp0q-co-derivations form a sub-complex of pHomOpÄ‚ E1, Ä‚ Eq, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The first item can be easily checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The second item follows exactly the same pattern as Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following proposition is very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every k P N0, and φ: pE1, d1q Ñ pE, dq be a chain map as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The sub-complex of Φp0q-co-derivations of polynomial-degree k is isomorphic to the complex z Page pkqpE1, Eq obtained from PagepkqpE1, Eq by crossing its “last column”, see diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The chain isomorphism δ: HomOpÄ‚ E1, Ä‚ Eq Ñ HomOpÄ‚ E1, Eq consists in mapping a Φp0q-co-derivation H of polynomial-degree k and degree j to its unique Taylor coefficient Hk P Pagepkq j pE1, Eq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' δpHq “ pr ˝ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check that this map is indeed a chain map: for every x “ x1 d ¨ ¨ ¨ d xk`1 P Äk`1 E1 one has, δ ˝ pQp0q E ˝ H ´ p´1q|H|H ˝ Qp0q E1 qpxq “ ℓ1 ˝ Hkpxqp´1q|H|Hk ˜k`1 ÿ i“1 ´p´1qp|x1|`¨¨¨`|xi´1|q|xi|ℓ1 1pxiq d x1 d ¨ ¨ ¨ d xk`1 ¸ “ DhpHkqpxq ´ p´1q|Hk|DvpHkqpxq, by definition of Dh and Dv “ DpHkqpxq “ D ˝ δpHqpxq, by definition of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is another type of interpretation for z Page ‚pE1, Eq involving the Richardson-Nijenhuis bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [FN56] When E1 “ E, z Page ‚pE1, Eq with no 0-column is the bi-graded complex of exterior forms on E and the differential D of z Page ‚pE1, Eq is Dp¨q “ rd, ¨ sRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We finish the section with the following lemma that will be important to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It uses the consequence of the cone construction (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 59 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pR, dR, πRq be an arbitrary complex of projective O-module that terminates in a O-module V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There exists a projective resolution pE, dE, πEq of V, which contains pR, dR, πRq as a sub-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, we can assume that R admits a projective submodule in E in direct sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Resolutions of an O-modules V are universal objects in the category of complexes of projective O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, for every projective resolution pF, dF, πFq of V, there exist a (unique up to homotopy) chain map: φ: pR, dR, πRq Ñ pF, dF, πFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We apply the cone construction (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Cha14], Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) to: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the complex pR, dR, πRq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the direct sum of the complexes pR, dRq and pF, dFq namely, ` R ‘ F, dR ‘ dF, πR ‘ πF˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the chain map obtained by mapping any x P R to px, φpxqq P R ‘ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The differential is given by dEpx, y, zq “ p´dRx, dRy ´ x, dFz ´ ϕpxqq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) for all px, y, zq P E´i “ R´i`1‘R´i‘F´i, i ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the chain given in item 3 is a quasi-isomorphism, its cone is an exact complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We truncate the latter at degree ´1 without destroying its exactness by replacing the cone differential at degree ´1 as follows: πE : R´1 ‘ F´1 Ñ V, pr, eq ÞÑ πFpeq ´ πRprq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a visual description, see Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) below: the resolution of V described in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 is defined by: ¨ ¨ ¨ � F´3 dF � F´2 dF � F´1 πF � V ¨ ¨ ¨ � R´3 dR � R´2 dR � R´1 πR � ¨ ¨ ¨ � R´2 id � dR � φ � R´1 id � φ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) The proof of the exactness of this complex is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The henceforth defined complex pE, dE, πEq is a resolution of V, and obviously contains pR, dR, πRq as a sub-chain complex of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, dE, πEq be a free resolution of V and pR, dR, πRq a subcomplex of projective O-modules, as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that P P Pagepkq j pEq of the form ‘iěnHomO ´Äk`1 E |´j´i, E´i ¯ preserves R if Äk`1 R |´j´i is mapped by P to R´i for all possible indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In such case, it defines by restriction to Ä‚ R an element ι˚ RP in the graded O-module Pagepkq j pRq :“ ‘iěnHomO ´Äk`1 R |´j´i, R´i ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For the sake of clarity, let us denote by DE and DR the respective differentials of the bi-complexes Pagepkq j pEq and Pagepkq j pRq and by DE h, DR h and DR v , DR v the horizontal differential resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' vertical differential, of their associated bi-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, ι˚ RP stands for the restriction of P P Pagepkq j pEq to Ä‚ R (a priori it is not valued in R but in E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, dE, πEq be a free resolution of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let R Ă E be a subcomplex made of free sub-O-modules such that there exists a graded free O-module V such that E “ R ‘ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every k ě 0, a DE-cocycle P P Pagepkq j pEq which preserves R is the image through DE of some element Q P Pagepkq j´1pEq which preserves R if and only if its restriction ι˚ RP P Pagepkq j pRq is a DR-coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, if the restriction of dE and πE to R makes it a resolution of πEpR´1q Ă V, then any DE-cocycle P P Pagepkq j pEq which preserves R is the image through DE of some element Q P Pagepkq j´1pEq which preserves R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us decompose the element P P Pagepkq j pEq as P “ ř iě1 Pi with, for all i ě 1, Pi in HomO ´Äk`1 E |´j´i, E´i ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume P P Pagepkq j pEq is a DE-cocycle which preserves R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove one direction of item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If P is the image through DE of some element Q P Pagepkq j´1pEq which preserves R, then DR pι˚ RQq “ ι˚ RDEpQq “ ι˚ RP, with ι˚ RQ P Pagepkq j´1pRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, the restriction ι˚ RP P Pagepkq j pRq of P is a DR-coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conversely, let us assume that ι˚ RP P Pagepkq j pRq is a DR-coboundary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ι˚ RP “ DRQR for some QR P Pagepkq j´1pRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Take ˆQ P Pagepkq j´1pEq any extension of QR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' define ˆQ to be 0 as soon as one element in V is applied to it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then P ´ DEp ˆQq : Äk`1 E ÝÑ E is zero on Äk`1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have to check that it is a DE-coboundary of a map with the same property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Put κ “ P ´ DEp ˆQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3, item 1, there exists τ P Pagepkq j´1pEq such that DEpτq “ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The equation DEpτq “ κ is equivalent to the datum of a collection of equations DE v pτiq ` DE hpτi`1q “ κi`1, i ě 1, and DE hpτ1q “ κ1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) with, τi P HomO ´Äk`1 E |´j´i`1, E´i ¯ and κi P HomO ´Äk`1 E |´j´i, E´i ¯ for every i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ι˚ Rκ1 “ 0, we have that DE hpι˚ Rτ1q “ ι˚ R ` DE hpτ1q ˘ “ 0, (with the understanding that ι˚ Rτ1|V ” 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Using the exactness of the horizontal differential DE h, there exists C1 P PagepkqpEq such that DE hpC1q “ ι˚ Rτ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We now change τ1 to τ 1 1 and τ2 to τ 1 2 by putting τ 1 1 :“ τ1 ´ ι˚ Rτ1 and τ 1 2 :“ τ2 ` DE v pC1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can easily check that Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) still holds under these changes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', DE v pτ 1 1q ` DE hpτ 1 2q “ κ2 and DE hpτ 1 2q ` DE v pτ3q “ κ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can therefore choose τ such that ι˚ Rτ1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We then iterate this procedure, which allows us to choose τ P Pagepkq j´1pEq such that ι˚ Rτ “ 0 and DEpτq “ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, Q :“ τ ` ˆQ preserves R, while ι˚ RQ “ QR, and DEpQq “ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The second item follows from the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Proof on the existence In this section, we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider pE, d “ ℓ1, πq a resolution of A by free O-modules: such resolutions always exist, see Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To start, we define a binary bracket ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The pair pd, ℓ2q will obey the axioms of the object that we now introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Lav17] An almost differential graded Lie algebroid of a Lie-Rinehart algebra pA, ρA, r¨ , ¨sAq is a complex ¨ ¨ ¨ d ÝÑ E´3 d ÝÑ E´2 d ÝÑ E´1 π ÝÑ A of projective O-modules equipped a graded almost differential graded Lie algebroid pE‚, ℓ1, ℓ2, ρq over O such that CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρ “ ρA ˝ π: E´1 ÝÑ DerpOq, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' π is a morphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all x, y P E´1 πpℓ2px, yqq “ rπpxq, πpyqsA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We start by proving this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Every free resolution pE, d, πq of a Lie-Rinehart algebra pA, r¨ , ¨sA, ρAq comes equipped with a binary bracket ℓ2 that makes it an almost differential graded Lie algebroid of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all k ě 1, let us denote by pep´kq i qiPIk a family of generators of the free O-module E´k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction tai “ πpep´1q i q P A | i P I1u is a set of generators of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, there exists elements uk ij P O, such that for given indices i, j, the coefficient uk ij is zero except for finitely many indices k, and satisfying the skew-symmetry condition uk ij “ ´uk ji together with rai, ajsA “ ÿ kPI uk ijak @i, j P I1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) We now define: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' an anchor map by ρpep´1q i q “ ρApai) for all i P I, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a degree `1 graded symmetric operation ˜ℓ2 on E as follows: (a) ˜ℓ2 ´ ep´1q i , ep´1q j ¯ “ ř kPI uk ijep´1q k for all i, j P I´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) ˜ℓ2 ´ ep´kq i , ep´lq j ¯ “ 0 for all i P Ik, j P Il with k ě 2 or l ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (c) we extend ˜ℓ2 to E using O-bilinearity and Leibniz identity with respect to the anchor ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, ˜ℓ2 satisfies the Leibniz identity with respect to the anchor ρE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, ρ ˝ d “ 0 on E´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map defined for all homogeneous x, y P E by rd, ˜ℓ2sRNpx, yq “ d ˝ ˜ℓ2 px, yq ` ˜ℓ2 pdx, yq ` p´1q|x|˜ℓ2 px, dyq , is a graded symmetric degree `2 operation pE bEq‚ ÝÑ E‚`2, and rd, ˜ℓ2sRN|E´1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check that it is O-bilinear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all f P O, x, y P E: rd, ˜ℓ2sRNpx, fyq ´ frd, ˜ℓ2sRNpx, yq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if x P E´1, this quantity is zero in view of rd, ˜ℓ2sRNpx, fyq “ frd, ˜ℓ2spx, yq ` dρpxqrfs y ´ ρpxqrfs dy loooooooooooooomoooooooooooooon “0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if x P E´2, one has rd, ˜ℓ2sRNpx, fyq ´ frd, ˜ℓ2spx, yq “ ˜ℓ2pdx, fyq ´ f ˜ℓ2pdx, yq “ ρpdxqpfq y “ 0 since ρ ˝ d “ ρA ˝ π ˝ d “ 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if x P E´i with i ě 3, it is obvious by O-linearity of ˜ℓ2 on the involved spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 62 As a consequence, rd, ˜ℓ2sRN is a degree `2 element in the total complex Pagep1qpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction rd, ˜ℓ2sRN has no component on the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since πprd, ˜ℓ2sRN|E´1 q “ 0 and also rd, rd, ˜ℓ2sRNsRN|Eď´2 “ 0, the O-bilinear operator rd, ˜ℓ2sRN is D-closed in Pagep1qpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By virtue of the first item of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3, the operator rd, ˜ℓ2sRN is then a D-coboundary, so there exists τ2 P ‘jě2HomO ´Ä2 E|´j´1, E´j ¯ such as Dpτ2q “ ´rd, ˜ℓ2sRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Upon replacing ˜ℓ2 by ˜ℓ2 `τ2 we obtain a 2-ary bracket ℓ2 of degree +1 which satisfies all items of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof (of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 gives the existence of an almost differential graded Lie alge- broid with differential ℓ1 “ d and binary bracket ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have to construct now the higher brackets ℓk for k ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Step 1: Construction of the 3-ary bracket ℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (Its construction being different from the one of the higher brackets, we put it apart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We first notice that the graded Jacobiator defined for all x, y, z P E by Jacpx, y, zq :“ ℓ2pℓ2px, yq, zq ` p´1q|y||z|ℓ2pℓ2px, zq, yq ` p´1q|x||y|`|x||z|ℓ2pℓ2py, zq, xq is O-linear in each variable, hence is a degree `2 element in À jě1 HomOpÄ3 E |´j´2, E´jq Ă Pagep2qpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For degree reason, its component on the last column of diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' it belongs to z Page p1qpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check that it is D-closed: for this purpose we have to check that both conditions in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since π is a morphism from pE´1, ℓ2q to pA, r¨, ¨sA, ρAq, and since r¨, ¨sA satisfies the Jacobi identity, one has for all x, y, z P E´1: Jacpx, y, zq P ker π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Furthermore, a direct computation of rJac, dsRN gives in view item 2 of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3: dJacpx, y, zq “ Jacpdx, y, zq ` p´1q|x|Jacpx, dy, zq ` p´1q|x|`|y|Jacpx, y, dzq for all x, y, z P E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, DpJacq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3, item 2, Jac is a D-coboundary, and, more precisely, there exists an element ℓ3 “ ř jě2 ℓj 3 P z Page p2q 1 pEq with ℓj 3 P HompÄ3 E |´j´1, E´jq such that Dpℓ3q “ ´Jac i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' rd, ℓ3sRN “ ´Jac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) We choose the 3-ary bracket to be ℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Step 2: Recursive construction of the k-ary brackets ℓk for k ě 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recapitulate: ℓ1 “ d , ℓ2 and ℓ3 are constructed and the lowest polynomial-degree terms of rℓ1 ` ℓ2 ` ℓ3, ℓ1 ` ℓ2 ` ℓ3sRN satisfy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' rℓ1, ℓ1sRN “ 0 (since d2 “ 0), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' rℓ1, ℓ2sRN “ 0 (since d “ ℓ1 and ℓ2 define an almost Lie algebroid structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' rℓ2, ℓ2sRN `2rℓ3, ℓ1sRN “ 2pJac`rℓ3, ℓ1sRNq “ 0 by definition of ℓ3, and because rℓ2, ℓ2sRN “ 2Jac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 63 However, the following term of degree `2 and polynomial-degree 3 may not be equal to zero: rℓ3, ℓ2sRN P à HomO ˜ 4 ä Ej`1, E´j ¸ “ z Page p3q 1 pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) Let us check that this term is indeed a O-multilinear map: For x1 P E´1, x2, x3, x4 P Eď´2 and f P O, the only terms of pℓ3 ˝ ℓ2 ` ℓ2 ˝ ℓ3qpx1, fx2, x3, x4q where the anchor shows up are: $ ’ ’ ’ & ’ ’ ’ % ℓ3pℓ2px1, fx2q, x3, x4q “ ρpx1qrfsℓ3px2, x3, x4q ` fpℓ3pℓ2px1, x2q, x3, x4qq p´1q|x2|`|x3|`|x4|ℓ2pfℓ3px2, x3, x4q, x1q “ ´ρpx1qrfsℓ3px2, x3, x4q `fpp´1q|x2|`|x3|`|x4|ℓ2pfℓ3px2, x3, x4q, x1qq The terms containing the anchor map add up to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When there are more elements in E´1, the computation follows the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, by graded Jacobi identity of the Richardson-Nijenhuis bracket: rrℓ1 ` ℓ2 ` ℓ3, ℓ1 ` ℓ2 ` ℓ3sRN, ℓ1 ` ℓ2 ` ℓ3sRN “ 0 The term of polynomial-degree 4 in the previous expression gives rrℓ3, ℓ2sRN, ℓ1sRN “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6, rℓ3, ℓ2sRN is a D-cocycle in the complex Pagep3qpEq, whose components on the last column and the column ´1 are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is therefore a coboundary by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 item 3: we can continue a step further and define ℓ4 P ‘jě3Hom ´Ä4 E|´j´1, E´j ¯ such that: ´ rℓ2, ℓ3sRN “ rℓ1, ℓ4sRN “ rd, ℓ4sRN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) We choose the 4-ary bracket to be ℓ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We now proceed by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We assume that we have constructed all the k-ary brackets, ℓk such as : rd, ℓksRN “ ´ ÿ i`j“k`1 iďj rℓi, ℓjsRN “ ´1 2 ÿ i`j“k`1 i,jě1 rℓi, ℓjsRN (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) for every k “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n with n ě 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The (n`1)-ary bracket is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' First, the operator ř i`j“k`1 i,jě1 rℓi, ℓjsRN is checked to be O-linear as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now, we have ÿ i`j“n`2 i,jě1 “ d, rℓi, ℓjsRN ‰ RN “ ´2 ÿ i`j“n`2 i,jě1 “ ℓi, rd, ℓjsRN ‰ RN (by graded Jacobi identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ℓj satisfies Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) up to order n, we obtain ÿ i`j“n`2 i,jě1 “ d, rℓi, ℓjsRN ‰ RN “ ÿ i`j`k“n`3 i,j,kě1 “ ℓi, rℓj, ℓksRN ‰ RN “ 0, where we used the graded Jacobi identity of the Nijenhuis-Richardson bracket in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There- fore, ř i`j“n`2 i,jě1 rℓi, ℓjsRN, seen as an element in Pagepi`j´2qpEq by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6, is a cocycle and for degree reason it has no element on the last column, and the columns ´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , 3 ´ n in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The third item of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 gives the existence of an pn ` 1q-ary bracket ℓn`1 such as rd, ℓn`1sRN “ ´ ÿ i`j“n`2 iďj rℓi, ℓjsRN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 64 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 Proof (of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is a consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 and the third item of the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3: If the component of Jac on the column ´1 is zero, we can choose ℓ3 with no component on the last column and in column ´1 (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the restriction of ℓ3 to Ä3 E´1 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then ℓ3 has no component on the last column, the column ´1 and the column ´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' so rℓ2, ℓ3sRN has no component in the last column, ´1 and ´2 columns as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, ℓ4 can be chosen with no component on column ´1, ´2 and ´3 by the third item of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof continues by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We finish this section with a proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof (of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We prove this Proposition in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 guarantees the existence a free resolution pE, d, πq of the Lie-Rinehart algebra A such that E contains E1 and such that there exists a graded free module V with E1 ‘ V “ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let DE and DE1 be as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We construct the n-ary brackets on E by extending the ones of pE1, pℓ1 kqkě1, ρE1, π1q in the following way: (a) We first construct an almost Lie algebroid bracket ˜ℓ2 on E´1 that extends the 2-ary bracket of E1 ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the 2-ary bracket is determined by its value on a basis, the existence of a free module V´1 such that E1 ´1 ‘ V´1 “ E´1 allows to construct ˜ℓ2 on E such that its restriction to E1 is ℓ1 2 and such that it satisfies the Leibniz identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 (to be more precise: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10), we see that r˜ℓ2, dEsRN is O-linear, hence belongs to Pagep2q 2 pEq and is a DE-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since E1 is a Lie 8-algebroid, its restriction to Ä2 E1 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 allows to change ˜ℓ2 to an 2-ary bracket ℓ2 :“ ˜ℓ2 `τ2 with τ2 “ 0 on Ä2 E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, ℓ2 defines a graded almost Lie algebroid bracket, whose restriction to E1 is still ℓ1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) Since ℓ2 is an extension of ℓ1 2, its Jacobiator Jac P Pagep2q 2 pEq of the 2-ary bracket ℓ2 preserves E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, its restriction ι˚ E1Jac P Pagep2q 2 pE1q is the Jacobiator of ℓ1 2, and the latter is the DE1-coboundary of ℓ1 3 in view of the higher Jacobi identity of E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since Jac P Pagep2q 2 pEq is a DE-cocycle, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 assures that Jac is the image through DE of some element ℓ3 P Pagep2q 1 pEq which preserves E1 and whose restriction to Ä3 E1 is ℓ1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof continues by recursion: at the n-th step, we use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 to construct an n-ary bracket for E that extends the n-ary bracket of E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, the inclusion map ι: E1 ãÑ E is a morphism for the n-ary brackets for all n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 Proof of universality Before proving the universal character of the construction, we need to do some preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove the following lemma, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Ψ, Ξ : S‚ KpE1q Ñ S‚ KpEq be O-linear Lie 8-algebroid morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let n P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If Ξpiq “ Ψpiq for every 0 ď i ď n, there exists 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a Lie 8-morphism of algebroids J1 : SKpE1q Ñ SKpEq CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and a homotopy pJt, Htqr0,1s joining Ψ and J1, such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the components of polynomial-degree less or equal to n of Ht vanish, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Jpiq 1 “ Ξpiq for every 0 ď i ď n ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us consider pΞ ´ Ψqpn`1q : Ä‚ E1 ÝÑ Ä‚ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By assumption, pΞ ´ Ψqpiq “ 0 for all i ď n, so that in view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' pΞ ´ Ψqpn`1q is a Ψp0q-co-derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 means that the map the restriction of the map pΞ ´ Ψqpn`1q to Än`2 E1 corresponds to a closed element of degree 0 in Pagepn`1qpE1, Eq equipped with differentials ℓ1, ℓ1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 implies that there exists O-linear map Hn`1 : Än`2pE1q ÝÑ E, a degree ´1 and of polynomial-degree n ` 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' an element in Pagepn`1qpE1, Eq, such that, pΞ ´ Ψqpn`1q “ Qp0q E ˝ Hn`1 ` Hn`1 ˝ Qp0q E1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) We denote its extension to a Ψp0q-co-derivation of degree ´1 by Hpn`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We now consider the following differential equation for t P r0, 1s: dJt dt “ QE ˝ Ht ` Ht ˝ QE1, and J0 “ Ψ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) where Ht is the unique Jt-co-derivation of degree ´1 whose unique non-zero Taylor coefficient is Hn`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The existence of a solution for the differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) is granted by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By considering the component of polynomial-degree 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n, n ` 1 in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15), we find $ & % dJpiq t dt “ 0 for i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n , dJpn`1q t dt “ Qp0q E ˝ Hpn`1q ` Hpn`1q ˝ Qp0q E1 “ pΞ ´ Ψqpn`1q Hence: # Jpiq t “ Ψpiq for i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n , Jpn`1q t “ Φpn`1q ` tpΞ ´ Ψqpn`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, applying t “ 1 to the previous relation, one finds # Jpiq t “ Ψpiq for i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n , Jpn`1q 1 “ Ψpn`1q ` pΞ ´ Ψqpn`1q “ Ξpn`1q This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Construction of the Lie 8-morphism Proof (of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We construct the Taylor coefficients of the Lie 8- algebroid Φ by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Taylor coefficient of polynomial-degree 0 is obtained out of classical properties of projective resolutions of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given any complex pE1, ρ1, ℓ1 1, π1q which terminates in A through π, for CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 66 every free resolution pE, ρ, ℓ1, πq of A, there exists a chain map Φp0q : pE1, ℓ1 1q Ñ pE, ℓ1q as in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2), and any two such chain maps are homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We still denote by Φp0q its extension to an polynomial-degree 0 co-morphism Ä‚ E1 Ñ Ä‚ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To construct the second Taylor coefficient, let us consider the map: S2 KpE1q Ñ E px, yq ÞÑ Φp0q ˝ ℓ1 2px, yq ´ ℓ2pΦp0qpxq, Φp0qpyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) This map is in fact O-bililinear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' belongs to HomOpÄ2 E1, Eq, hence to Pagep1qpE1, Eq, see Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check that it is a D-cocycle: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If either one of the homogeneous elements x P E1 or y P E1 is not of degree ´1, a straightforward computation gives: ℓ1 ˝ ´ Φp0q ˝ ℓ1 2px, yq ´ ℓ2 ´ Φp0qpxq, Φp0qpyq ¯¯ “ Φp0q ˝ ℓ1 1 ˝ ℓ1 2px, yq ` ℓ2 ´ Φp0q ˝ ℓ1 1pxq, Φp0qpyq ¯ ` p´1q|x|ℓ2 ´ Φp0qpxq, Φp0q ˝ ℓ1 1pyq ¯ “ ´ Φp0q ˝ ℓ1 2 ´ ℓ2 ´ Φp0q, Φp0q¯¯ ˝ ℓ1 1px d yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If both x, y P E1 are of degree ´1: π ´ Φp0qℓ1 2px, yq ´ ℓ2pΦp0qx, Φp0qyq ¯ “ π1 ˝ ℓ1 2px, yq ´ π ˝ ℓ2 ´ Φp0qx, Φp0qy ¯ “ rπ1pxq, π1pyqs ´ ” π ´ Φp0qx ¯ , π ´ Φp0qx ¯ı “ rπ1pxq, π1pyqs ´ rπ1pxq, π1pyqs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 item 2), there exists Φp1q P HomO ´Ä2 E1, E ¯ , of degree 0, so that Φp0q ˝ ℓ1 2px, yq ` Φp1q ˝ ℓ1 1px d yq “ ℓ1 ˝ Φp1qpx, yq ` ℓ2pΦp0qpxq, Φp0qpyqq for all x, y P E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17) Φp0q is a chain map and Relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17) can be rewritten in terms of QE and QE1 as follows # Qp0q E ˝ Φp0q “ Φp0q ˝ Qp0q E1 Qp0q E ˝ Φp1q ´ Φp1q ˝ Qp0q E1 “ Φp0q ˝ Qp1q E1 ´ Qp1q E ˝ Φp0q (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18) The construction of the morphism Φ announced in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 is then done by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The recursion assumption is that we have already defined a O-multilinear co-morphism Φ : S‚ KpE1q Ñ S‚ KpEq with pΦ ˝ QE1 ´ QE ˝ Φqpkq “ 0 for all 0 ď k ď n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The co-morphism Φ : S‚ KpE1q Ñ S‚ KpEq with Taylor coefficients Φp0q and Φp1q satisfies the recursion assumption for n “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume now that we have a co-morphism Φ that satisfies this assumption for some n P N, and consider the map TΦ :“ Φ ˝ QE1 ´ QE ˝ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='26 implies that T pn`1q Φ is a O-multilinear Φp0q-co-derivation, and that it corresponds to a D-closed element1 in Pagepn`1qpE1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since it has no 1The following remark is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Under the assumptions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='26, pΦ˝QE1 ´QE ˝Φqpn`1q corresponds to a D- closed element of degree `1 in the bi-complex Pagepn`1qpE1, Eq through the chain isomorphism described in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, E, E1 are equipped with the differentials ℓ1, ℓ1 1 which are the restriction of the components Qp0q E , Qp0q E1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 67 component on the last column for degree reason, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 implies that T pn`1q Φ is a coboundary: That is to say that there is a Φp0q-co-derivation Θ P Pagepn`1qpE1, Eq (of polynomial-degree n ` 1 and degree 0) which can be seen as a map Θ : Än`2pE1q Ñ E such that: T pn`1q Φ “ Qp0q E ˝ Θ ´ Θ ˝ Qp0q E1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider now the co-morphism ˜Φ whose Taylor coefficients are those of Φ in polynomial-degree 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n and Φpn`1q ` Θ in polynomial-degree n ` 1: ˜Φpiq :“ $ & % Φpiq if 0 ď i ď n, Φpn`1q ` Θ if i “ n ` 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19) This is easily seen to satisfy the recursion relation for n`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This concludes the recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Taylor coefficients obtained by recursion define a Lie 8-algebroid Φ: S‚ KpE1q ÝÑ S‚ KpEq which is compatible by construction with the hooks π, π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By continuing this procedure, we construct a Lie 8-morphism from S‚ KpE1q to S‚ KpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves the first item of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Construction of a homotopy that joins two such morphisms Let us prove the second item in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that in the proof of the existence of the Lie 8-morphism between S‚ KpE1q and S‚ KpEq obtained in the first item, we made many choices, since we have chosen a coboundary at each step of the recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ, Ψ be two such Lie 8-morphisms between S‚ KpE1q and S‚ KpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The polynomial-degree 0 component of the co-morphisms Φ and Ψ restricted to E1 are chain maps: ¨ ¨ ¨ � E1 ´2 � h � Φp0q � Ψp0q � E1 ´1 Φp0q � Ψp0q � h � π1 � A ¨ ¨ ¨ � E´2 � E´1 π � � which are homotopy equivalent in the usual sense because pE, ℓ1q is a projective resolution of A: said differently, there exists a degree ´1 O-linear map h: E1 Ñ E such that Ψp0q ´ Φp0q “ ℓ1 ˝ h ` h ˝ ℓ1 1 on E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20) Let us consider the following differential equation: $ & % dJt dt “ QE ˝ HtpJtq ` HtpJtq ˝ QE1, for t P r0, 1s J0 “ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21) with HtpJtq being a Jt-co-derivation of degree ´1 whose Taylor coefficient of polynomial-degree 0 is h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This equation does admit solutions in view of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By looking at the component polynomial-degree 0 of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21) on E1, one has: dJp0q t dt “ ℓ1 ˝ h ` h ˝ ℓ1 1 “ Ψp0q ´ Φp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 68 Hence, Jp0q t “ Φp0q ` t ` Ψp0q ´ Φp0q˘ is a solution such that Jp0q 1 “ Ψp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, J1 is homotopic to Φ via the pair pJt, Htq over r0, 1s, and its polynomial-degree 0 Taylor coefficient coincides with the Taylor coefficient of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From there, the construction goes by recursion using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, this lemma allows constructing recursively a sequence of Lie 8-algebroids morphism pΨnqně0 and homotopies pJn,t, Hn,tq (with t P rn, n ` 1s) between Ψn and Ψn`1 such that: Hpiq n,t is zero for t ě n and i ‰ n ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11, all these homotopies are compatible with the hooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These homotopies are glued in a homotopy pJt, Htqr0,`8r such that for every n P N0, the components of polynomial-degree n of the Lie 8-algebroids morphism Jpnq t are constant and equal to Ψpnq for t ě n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22, these homotopies can be glued to a homotopy on r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Explicitly, since t ÞÑ t 1´t maps r0, 1r to r0, `8r and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22, the pair ´ J t 1´t , 1 p1´tq2 Hk, t 1´t ¯ is a homotopy between Φ and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves the second item of the Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 Examples of universal Lie 8-algebroids of Lie-Rinehart algebras 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 New constructions from old ones In this section, we explain how to construct universal Lie 8-algebroids of some Lie-Rinehart algebra which is derived from a second one through one of natural constructions as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 (localization, germification, restriction), when a universal Lie 8-algebroid of the latter is already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Localization Localization is an useful algebraic tool, specially in algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When O is an algebra of functions, it corresponds to study local properties of a space, or germs of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let S Ă O be a multiplicative-closed subset containing no zero divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recall from item 2, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 that the localization S´1A – A bO S´1O of A at S comes equipped with a natural structure of Lie-Rinehart algebra over the localization algebra S´1O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Recall that for ϕ: E ÝÑ T a homomorphism of O-modules, there is a well-defined homomorphism of O-modules, ϕ b id: E bO S´1O ÝÑ T bO S´1O, ϕ b id px b f s q :“ ϕpxq b f s that can be considered as a S´1O-module homomorphism S´1ϕ: S´1E ÝÑ S´1T with S´1ϕ ´x s ¯ :“ ϕpxq s , x P E, pf, sq P O ˆ S, called the localization of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a Lie 8-algebroid structure pE‚, ℓ‚, ρq that covers A through π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The triplet pE1 ‚, ℓ1 ‚, ρ1q is a Lie 8-algebroid structure that covers S´1A through the hook π1 where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' E1 “ S´1E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The anchor map ρ1 is defined by ρ1 : S´1E´1 ÝÑ DerpS´1Oq x s ÞÝÑ ρ1 ` x s ˘ : S´1O ÝÑ S´1O f u ÞÝÑ 1 s ¨ ´ ρpxqrfsu´fρpxqrus u2 ¯ for x P E, f P O, ps, uq P S ˆ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ℓ1 k “ S´1ℓk, for all k P Nzt2u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The binary bracket is more complicated because of the anchor map: we set ℓ1 2 ˆ1 s x, 1 uy ˙ “ 1 suℓ2px, yq ´ ρpxqrus su2 y ` ρpyqrss s2u x for x, y P E, ps, uq P S2 (with the understanding that ρ ” 0 on E´i with i ě 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' π1 “ S´1π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can check that these operations above are well-defined and for all z P S´1E´1 the map ρ1pzq is indeed a derivation on S´1O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The previously defined structure is also a Lie 8-algebroid that we call localization of the Lie 8-algebroid pE‚, ℓ‚, ρq with respect to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let S Ă O be a multiplicative subset containing no zero divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The localization of a universal Lie 8-algebroid of a Lie-Rinehart algebra A is a universal Lie 8-algebroid of S´1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The object ppE1 ‚, ℓ1 ‚, ρ1qq described above is also a Lie 8-algebroid terminating in S´1A through π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is universal because localization preserves exact sequences [And].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Restriction When OY is the ring of functions of an affine variety Y (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1), to every subvariety X Ă Y corresponds its zero locus, which is an ideal IX Ă OY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie 8-algebroid or a Lie-Rinehart algebra over OY may not restrict to a Lie-Rinehart algebra over OX: it only does so when one can quotient all brackets by IX, which geometrically means that the anchor map takes values in vector fields tangent to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can then “restrict”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' replace OY by OY {IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This operation has already been defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2, and here is an immediate consequence of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let I Ă O be a Lie-Rinehart ideal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' an ideal such that ρApAqrIs Ă I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The quotient of a universal Lie 8-algebroid of A with respect to an ideal I is a Lie 8-algebroid that terminates in A{IA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is universal if and only if pp E´i IE´i qiě1, ¯ℓ1, πq is exact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if Tor‚ OpA, O{Iq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that the anchor map ρ: E´1 Ñ DerpOq goes to quotient to E´1 IE´1 Ñ DerpOq as an O-linear map, but needs the extra condition ρpE´1qrIs Ă I to induce an O{I- linear map E´1 IE´1 Ñ DerpO{Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Germification Let W Ď CN be an affine variety and OW its coordinates ring (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a P W, consider OW,a the local ring at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that OW,a » pOW qma [Har77], where ma “ tf P OW | fpaq “ 0u and pOW qma is the localization w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t the complement of ma, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 implies the following statement: CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 70 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W be an affine variety with functions OW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every point a P W and any Lie-Rinehart A over OW , the germ at a of the universal Lie 8-algebroid of A is the universal Lie 8-algebroid of the germ of A at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, the germ at a of a Lie-Rinehart algebra or a Lie 8-algebroid is simply its localization w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t the complement of ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Sections vanishing on a codimension 1 subvariety Let pA, r¨ , ¨s , ρAq be an arbitrary Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any ideal I Ă O, IA is also a Lie-Rinehart algebra (see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When O are functions on a variety M, I are functions vanishing on a subvariety X and A is a O-module of sections over M, IA corresponds geometrically to sections vanishing along X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is not an easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In codimension 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' when I is generated by one element, the construction can be done by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρAq be a Lie-Rinehart algebra over a commutative algebra O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, ℓk “ t¨ ¨ ¨ ukě1, ρq be a Lie 8-algebroid that terminates in A through a hook π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any element χ P O, the O-module A1 “ χA Ď A is closed under the Lie bracket, so the triple pχA, r¨, ¨sA, ρAq is a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie 8-algebroid pE1 “ E, ℓ1 k “ t¨ ¨ ¨ u1 kě1, ρ1q hooked in χA through π1 can be defined as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The brackets are given by (a) t¨u1 1 “ t¨u1, (b) the 2-ary bracket: tx, yu1 2 :“ χtx, yu2 ` ρpxqrχs y ` p´1q|x||y|ρpyqrχs x, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22) for all x, y P E‚, with the understanding that ρ “ 0 on Eď´2, (c) t¨ ¨ ¨ u1 k “ χk´1t¨ ¨ ¨ uk for all k ě 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρE1 “ χρ, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' π1 “ χπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We leave it to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If χ is not a zero-divisor in O, and pE, t¨ ¨ ¨ ukě1, ρ, πq is a universal Lie 8- algebroid of A, then the Lie 8-structure described in the four items of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 is the universal Lie 8-algebroid of χA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If χ is not a zero-divisor in A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if a ÞÑ χa is an injective endomorphism of A), then the kernel of π1 coincides with the kernel of π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' with the image of t¨u1 “ t¨u1, so that pE, ℓ1, χπq is a resolution of χA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 Algebra extension Recall that for O a unital algebra with no zero divisor, derivations of O induce derivations of its field of fractions O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let O be an unital algebra with no zero divisor, O its field of fractions, and ˜O an algebra with O Ă ˜O Ă O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every Lie-Rinehart algebra A over O whose anchor map takes values in derivations of O preserving ˜O, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' any Lie 8-algebroid structure pE, pℓkqkě1, ρ, πq that terminates at A extends for all i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n to a Lie 8-algebroid structure on ˜O bO E, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and this extension ˜O bO E is a Lie 8-algebroid that terminates at the Lie-Rinehart algebra ˜O bO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since they are O-linear, the hook π, the anchor ρ, and the brackets ℓk for k ‰ 2 are extended to ˜O-linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the image of ρ is the image of ρA, it is made of derivations preserving O, which is easily seen to allow an extension of ℓ2 to ˜O bO E using the Leibniz identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Of course, the Lie 8-algebroid structure obtained on ˜O bO E is not in general the universal Lie 8-algebroid of ˜O bO A, because the complex p ˜O bO E, ℓ1, πq may not be a resolution of ˜O bO A (see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since any module over a field is projective, any Lie-Rinehart algebra over a field is a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If we choose ˜O “ O therefore, the Lie-Rinehart algebra O bO A is a Lie algebroid, so is homotopy equivalent to any of its universal Lie 8-algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Unless A is a Lie algebroid itself, the Lie 8-algebroid in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 will not be homotopy equivalent to a Lie 8-algebroid whose underlying complex is of length one, and is therefore not a universal Lie 8-algebroid of O bO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 Blow-up Let us investigate the behavior of the universal Lie 8-algebroids under blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recall that the blowup of CN`1 at the origin is given by, B0pCN`1q “ tpx, ℓq P CN`1 ˆ PN | x P ℓu together with the map π: B0pCN`1q ÝÑ CN`1 px, ℓq ÞÑ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For O “ Crz0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , zNs the coordinate ring of CN`1, the blow-up of CN`1 at the origin is covered by affine charts: in the i-th affine chart Ui, the coordinate ring is OUi “ Crz0{zi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , zi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , zN{zis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 for any Lie-Rinehart algebra A whose anchor map takes values in vector fields vanishing at 0 P CN`1, we obtain a Lie 8-algebroid of OUi bO A that we call blow-up of at 0 in the chart Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 then says that the blow-up at 0 of the universal Lie 8-algebroid of A, in each chart, is a Lie 8-algebroid that terminates in the blow-up of A (as defined in remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It may not be the universal one, see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 72 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 (Universal Lie-8-algebroids and blow-up: a counter example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the poly- nomial function in N ` 1 variables ϕ “ řN i“0 z3 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us consider the singular foliation Fϕ Ă XpCNq as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Its generators are ∆ij :“ z2 i B Bzj ´ z2 j B Bzi , for 0 ď i ă j ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us consider its blow-up in the chart UN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Geometrically speaking, Ă Fϕ “ OUN bO Fϕ is the OUN -module generated by the blown-up vector fields r∆ij “ zN ´ z2 i B Bzj ´ z2 j B Bzi ¯ , j ‰ N, and r∆iN “ zN ´ zNz2 i B BzN ´ B Bzi ´ z2 i řN´1 j“0 zj B Bzj ¯ of the vector fields ∆ij, for i, j P t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Nu, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t affine chart UN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The vector fields r∆ij, j ‰ N, belong to the OUN -module generated by the vector fields r∆iN, (explicitly r∆ij “ z2 j r∆iN ´z2 i r∆jN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Said differently, the vector fields r∆iN, i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' N´1 are generators of Ă Fϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since they are independent, the singular foliation Ă Fϕ is a free OUN -module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A universal Lie 8-algebroid for it is therefore concentrated in degree ´1 and is given by the OUN -module generated by some set pei, i “ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' N ´ 1q, equipped with the Lie bracket: rei, ejs :“ 2zN ` z2 i ej ´ z2 j ei ˘ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23) together with the anchor map ρ which assigns ei to r∆iN, for i “ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' N ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On the other hand, the blow-up of the universal Lie 8-algebroid of Fϕ is not homotopy equivalent to a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this case, the Lie 8-algebroid is given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6: notice that E´i ‰ 0 for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , N and that ℓ1|0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The pull-back Lie 8-algebroid p ˜E, p˜ℓkqkě1, ˜ρ, ˜πq verifies by construction that ˜E´i ‰ 0 for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , N and that ˜ℓ1|x “ 0 for every x in the inverse image of zero, and such a complex can not be homotopy equivalent to a complex of length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In other words, OUi bO ¨ is not an exact functor in this case (which is a classical fact in algebraic geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This example tells us that the blow-up of the universal Lie 8-algebroid of an affine variety W may not be the universal Lie 8-algebroid of its blow-up, (even locally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 Universal Lie 8-algebroids of some singular foliations Singular foliation is defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The coming example uses the notion "multi-derivation", it is useful to recall the definition and write down some basic operations on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 of [LGPV13] for more details on this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A skew symmetric k-multilinear map P P HomKpOk, Oq is a said to a k-multi- derivation of O if P is a derivation in each of its argument, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', for every i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ku and for all f, g P O we have Prf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fg loomoon i´th slot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fks “ Prf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , f loomoon i´th slot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fks g ` f Prf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , g loomoon i´th slot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fks (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24) When O happen to be the coordinate ring of some affine variety W, k-multi-derivations of O are called k-multi-vector fields on W, and denote by XkpWq the module of k-multi-vector fields on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In general, it is denoted by XkpOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By skew-symmetry argument, it is enough that the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24) holds for the first slot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for i “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 73 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The graded module X‚pOq :“ À kě0 XkpOq comes equipped with graded algebra struc- ture whose product ^ is defined as follows, pP ^ Rqrf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fk`ls :“ ÿ σPSpk,lq ϵpσqPrfσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fσpkqsRrfσpk`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fσpk`lqs, for P P XkpOq, R P XlpOq and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fk`l P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By convention X0pOq “ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, f ^ X :“ fX for all f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every f P O, the contraction by f, ιf : XkpOq Ñ Xk´1pOq, @ k ě 1, which is defined as follows2 P ÞÑ ιfpPqrf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fk´1s :“ Prf, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fk´1s, for all f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fk´1 P O (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25) satisfies ιf ˝ ιf “ 0, for all P P XkpOq and R P XlpOq, ιfpP ^ Rq “ ιfpPq ^ R ` p´1qkP ^ ιfpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The first item is true by skew-symmetry of multi-derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now let us prove the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fk`l´1 P O, we have ιfpP ^ Rqrf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fk`l´1s “ pP ^ Rqrf, f1, ¨ ¨ ¨ , fk`l´1s “ ÿ σPSpk´1,lq ϵpσqPrf, fσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fσpk´1qsRrfσpkq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fσpk`l´1qs` p´1qk ÿ σPSpk,l´1q ϵpσq Prfσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fσpkqsRrf, fσpk`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fσpk`l´1qq looooooooooooooooooooooooooooomooooooooooooooooooooooooooooon here, the inversion number with f is k “ ιfpPq ^ R ` p´1qkP ^ ιfpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have used the fact that for every σ P Spk, lq, one has that either σp1q “ 1 or σpk ` 1q “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice the following: For all f P O and X P DerpOq, one has ιfpXq “ Xrfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any two derivation X, Y P DerpOq we have pX ^ Y qrf, gs “ XrfsY rgs ´ Y rgsXrfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a family of derivations X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Xk P DerpOq, the k-multi-derivation X1 ^ ¨ ¨ ¨ ^ Xk applied to f1, ¨ ¨ ¨ , fk P O is equal to the determinant �������� X1rf1s ¨ ¨ ¨ X1rfks .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Xkrf1s ¨ ¨ ¨ Xkrfks �������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When O is the algebra of polynomial functions in d variables Krx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds, a k-multi-derivation P admits the coordinate expression P “ ÿ 1ďi1㨨¨ăikďd Prxi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xiks B Bxi1 ^ ¨ ¨ ¨ ^ B Bxik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2It should be understood that ιfpOq :“ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 74 Lie derivative For P P XkpOq and X P DerpOq, the Lie derivative LXP P XkpOq of P along X is defined as pLXPqrf1, ¨ ¨ ¨ , fks :“ X rPrf1, ¨ ¨ ¨ , fkss ´ kÿ j“1 Prf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xrfjs, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='27) There is another important operation on multi-derivations, the so-called "Schouten bracket" which is a generalization of the commutator of derivations, also Lie derivative of multi-derivation along a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We will not recall how it is defined here, since we do not really make use of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g [LGPV13] for this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 Vector fields annihilating a Koszul function ϕ This universal Lie 8-algebroid was already described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 of [LLS20], where the brackets were simply checked to satisfy the higher Jacobi identities - with many computations left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, we give a theoretical explanation of the construction presented in [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let O be the algebra of all polynomials on V :“ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A function ϕ P O is said to be a Koszul polynomial, if the Koszul complex .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ιϕ ÝÑ X3pV q ιϕ ÝÑ X2pV q ιϕ ÝÑ XpV q ιϕ ÝÑ O ÝÑ 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='28) is exact in all degree, except in degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By virtue of a theorem of Koszul [Eis95], see [Hid89] Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 piq, ϕ is Koszul if ´ Bϕ Bx1 , ¨ ¨ ¨ , Bϕ Bxd ¯ is a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From now on, we choose ϕ a Koszul function, and consider the Lie-Rinehart algebra (which is a singular foliation) Fϕ :“ tX P XpV q : Xrϕs “ 0u “ Kerpιϕq: XpV q ιϕ ÝÑ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='29) The Koszul complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='28) truncated of its degree 0 term gives a free resolution pE, d, ρq of Fϕ, with E´i :“ Xi`1pV q, d :“ ιϕ, and ρ :“ ´ιϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Exactness of the Koszul complex implies in particular that Fϕ is generated by the vector fields: " Bϕ Bxi B Bxj ´ Bϕ Bxj Bϕ Bxi , | 1 ď i ă j ď d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='30) In [LLS20], this resolution is equipped with a Lie 8-algebroid structure, whose brackets we now recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A universal Lie 8-algebroid of Fϕ Ă XpV q is given on the free resolution ` E´‚ “ X‚`1pV q, d “ ιϕ, ρ “ ´ιϕ ˘ by defining the following n-ary brackets: tBI1, ¨ ¨ ¨ , BInun :“ ÿ i1PI1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',inPIn ϵpi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , inqϕi1¨¨¨inBIi1 1 ‚¨¨¨‚Iin n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='31) and the anchor map given for all i, j P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , nu by ρ ˆ B Bxi ^ B Bxj ˙ :“ Bϕ Bxj B Bxi ´ Bϕ Bxi B Bxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='32) Above, for every multi-index J “ tj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , jnu Ď t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , du of length n, BJ stands for the n-vector field B Bxj1 ^ ¨ ¨ ¨ ^ B Bxjn and ϕj1¨¨¨jn :“ Bnϕ Bxj1¨¨¨Bxjn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, I1 ‚ ¨ ¨ ¨ ‚ In is a multi-index obtained by CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 75 concatenation of n multi-indices I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every i1 P I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , in P In, ϵpi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , inq is the signature of the permutation which brings i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , in to the first n slots of I1 ‚ ¨ ¨ ¨ ‚ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Last, for is P Is, we define Iis s :“ Iszis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To understand this structure, let us first define a sequence of degree `1 graded symmetric poly- derivations on X‚pV q (by convention, i-vector fields are of degree ´i ` 1) by: tBi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Biku1 k :“ Bkϕ Bxi1 ¨ ¨ ¨ Bxik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='33) We extend them to a graded poly-derivation of X‚pV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The poly-derivations pt¨ ¨ ¨ u1 kqkě1 are O-multilinear and equip X‚pV q with a (graded symmetric) Poisson 8-algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, t¨u1 1 “ ιϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For degree reason, tF, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xk´1u1 k “ 0 for all X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xk´1 P X‚pV q and all F P X0pV q “ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies the required O-multilinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is clear that the higher Jacobi identities hold since brackets of generators tδi1, ¨ ¨ ¨ , δinu1 are elements in O, and all brackets are zero when applied an element in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof (of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The brackets introduced in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17 are modifications of the Poisson 8-algebra described in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, t¨ ¨ ¨ u 1 n “ t¨ ¨ ¨ un when all arguments are generators of the form BI for some I Ă t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , nu of cardinal ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By O-multilinearity, this implies t¨ ¨ ¨ u 1 n “ t¨ ¨ ¨ un when n ě 3, or when n “ 2 and no argument is a bivector-field, or when n “ 1 and the argument is not a bivector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As a consequence, all higher Jacobi identities hold when applied to n-vector fields with n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us see what happens when one of the arguments is a bivector field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' in the case where we deal with at least an element of degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us assume that there is one such element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Q1 “ Bi ^ Bj, Q2 “ BI2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qn “ BIn with |Ij| ě 3, j “ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, in view of the higher Jacobi identity for the Poisson 8-brackets pt¨ ¨ ¨ u1 kqkě1 gives: 0 “ ÿ 2ďkďn´2 ÿ σPSk,n´k ϵpσq !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='␣ Qσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qσpkq (1 k , Qσpk`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qσpnq )1 n´k`1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='34) ` ÿ σPSn´1,1,σpnq‰1 ϵpσq !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='␣ Qσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qσpn´1q (1 n´1 , Qσpnq )1 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='35) ` ÿ σPS1,n´1,σp1q‰1 ϵpσq !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='␣ Qσp1q (1 1 , Qσp2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qσpnq )1 n (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='36) ` p´1q řn k“2|BIk| ␣ tQ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qnu1 n´1 , Q1 (1 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='37) ` ␣ tQ1u1 1 , Qσp2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Qσpnq (1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='38) In lines (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='34)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='35)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='36) above, we have t¨ ¨ ¨ u1 “ t¨ ¨ ¨ u for all the terms involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is not the case for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='37)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' tBI2, ¨ ¨ ¨ , BInu 1 n´1 , Bi ^ Bj )1 2 “ ␣ tBI2, ¨ ¨ ¨ , BInun´1 , Bi ^ Bj ( 2 ´ ÿ i2PI2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',inPIn ϵpi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , inqρpBi ^ Bjqrϕi2¨¨¨ins BIi2 2 ‚¨¨¨‚Iin n CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 76 and ␣ tBi ^ Bju1 1 , BI2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , BIn (1 n “ p´1q řn k“2|BIk|`1 ` ϕi tBj, BI2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , BInu1 n ´ ϕj tBi, BI2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , BInu1 n ˘ “ ´p´1q řn k“2|BIk| ÿ i2PI2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',inPIn ϵpi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , inqιϕpBi ^ Bjqrϕi2¨¨¨ins BIi2 2 ‚¨¨¨‚Iin n “ p´1q řn k“2|BIk| ÿ i2PI2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',inPIn ϵpi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , inqρpBi ^ Bjqrϕi2¨¨¨ins BIi2 2 ‚¨¨¨‚Iin n since ρ “ ´ιϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, the quantities in lines (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='38) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='37) add up, when we re-write them in terms of the new brackets t¨ ¨ ¨ uk, to yield precisely the higher Jacobi identity for this new bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is then not difficult to see this is still the case if there is more than one bivector field, by using many times the same computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 Restriction to ϕ “ 0 of vector fields annihilating ϕ We keep the convention and notations of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us consider the restriction i˚ W Fϕ of the Lie-Rinehart algebra Fϕ to the zero-locus W of a Koszul polynomial ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since all vector fields in Fϕ are tangent to W, this restriction is now a Lie-Rinehart algebra over OW “ O Oϕ, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ϕ be a Koszul Polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The restriction of the universal Lie 8-algebroid of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17 to the zero-locus W of ϕ is a universal Lie 8-algebroid of the Lie-Rinehart algebra i˚ W Fϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the image of its anchor map are vector fields tangent to W, it is clear that the universal Lie 8-algebroid of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17 restricts to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19, it suffices to check that the restriction i˚ W XpV q to W of the Koszul complex is still exact, except in degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The restriction to the zero locus W of ϕ of the Koszul complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='28), namely the complex, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ιϕ ÝÑ i˚ W X3pV q ιϕ ÝÑ i˚ W X2pV q ιϕ ÝÑ i˚ W F is a free resolution of i˚ W F in the category of OW -modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let k ě 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a given P P i˚ W XkpV q, the relation ιϕP “ 0 means that for any ˜P P XkpV q extending P, there exists U P Xk´1pV q such that ιϕ ˜P “ ϕU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By exactness of the Koszul complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='28), one has, U “ ιϕ ˜Q for some ˜Q P XkpV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, ˜P ´ ϕ ˜Q is an extension of the bivector field P such that ιϕp ˜P ´ ϕ ˜Qq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Using one more time exactness of the Koszul complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='28), we construct ˜R P Xk`1pV q such that ˜P “ ϕQ ` ιϕ ˜R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, P “ i˚ W ˜P “ ιϕi˚ W ˜R “ ιϕR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 Vector fields vanishing on subsets of a vector space Let O be the algebra of smooth or holomorphic or polynomial or formal functions on Kd, and I Ă O be an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then IDerpOq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' vector fields of the form: řd i“1 fi B Bxi , with f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fd P I, is a Lie-Rinehart algebra (It is also a singular foliation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Geometrically, when I corresponds to functions vanishing on a sub-variety N Ă Kn, IDerpOq must be interpreted as vector fields vanishing along N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 77 Let us describe a Lie 8-algebroid that terminates at IDerpOq, then discuss when it is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pϕiqiPI be generators of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the free graded algebra K “ OrpµiqiPIs generated by variables pµiqiPI of degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The degree ´1 derivation B :“ ř iPI ϕi B Bµi squares to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-module K´j of elements degree j in K‚ is made of all sums ř i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ijPI fi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='ijµi1 ¨ ¨ ¨ µij with fi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='ij P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the complex of free O-modules ¨ ¨ ¨ BbOid ÝÑ K´2 bO DerpOq BbOid ÝÑ K´1 bO DerpOq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='39) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='39) comes equipped with a Lie 8-algebroid structure that ter- minates in IDerpOq through the anchor map given by µi B Bxj ÞÑ ϕi B Bxj for all i P I, and j P 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' First, one defines a O-linear Poisson-8-algebra structure on the free algebra generated by pµiqiPI (in degree ´1) and ´ B Bxj ¯d j“1 (in degree 0) and 1 by: " µi, B Bxj1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , B Bxjr 1 r`1 :“ Brϕi Bxi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Bxir (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='40) all other brackets of generators being equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the brackets of generators take values in O, and since an n-ary bracket where an element of O appears is zero, this is easily seen to be a Poisson 8-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The general formula is ␣ µI1 bO Bxa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , µIn bO Bxan (1 n :“ ÿ j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n ij P Ij ϵ Bn´1ϕij Bxa1 ¨ ¨ ¨ x Bxaj ¨ ¨ ¨ Bxan µI1 ¨ ¨ ¨ µij Ij ¨ ¨ ¨ µInbOBxaj , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='41) where µJ “ µj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' µjs for every list J “ tj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , jsu, where ϵ is the Koszul sign, and where for a list J containing j, Jj stands for the list J from which the element j is crossed out, as in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-module generated by µi1 ¨ ¨ ¨ µik bO Bxa , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='39) is easily seen to be stable under the brackets t¨ ¨ ¨ u1 k for all k ě 1, so that we can define on K bO DerpOq a sequence of brackets pℓk “ t¨ ¨ ¨ ukqkě1 by letting them coincide with the previous brackets on the generators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' t¨ ¨ ¨ un is given by Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='41) for all n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The brackets are then extended by derivation, O-linearity or Leibniz identity with respect to the given anchor map, depending on the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, t¨ ¨ ¨ u1 k “ t¨ ¨ ¨ uk for k ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For k “ 1, t¨ ¨ ¨ u1 1 “ t¨ ¨ ¨ u1 on ‘iě2K´i bO DerpOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For k “ 2, we still have t¨ ¨ ¨ u1 2 “ t¨ ¨ ¨ u2 on ‘i,jě2K´i d K´j bO DerpOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us verify that all required axioms are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For n “ 2, Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='41) specializes to: ℓ2pµi bO Bxa, µj bO Bxbq “ Bϕj Bxa µi bO Bxb ´ Bϕi Bxb µj bO Bxa which proves that the anchor map is a morphism when compared with the relation: rϕiBxa , ϕjBxbs “ Bϕj Bxa ϕi Bxb ´ Bϕi Bxb ϕj Bxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The higher Jacobi identities are checked on generators as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When there are no degree ´1 generators, it follows from the higher Jacobi identities of the Poisson 8-structure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='40) and the O-multilinearity of all Lie 8-algebroid brackets involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When generators of degree ´1 are involved, the higher Jacobi identities are obtained by doing the same procedure as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18, that is, we first consider the higher Jacobi identities for the Poisson 8-structure (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='40), and we put aside the terms where t¨u1 is applied to these degree ´1 generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We then check that the latter terms are exactly the terms coming from an anchor map when the 2-ary bracket is applied to generators of degree ´1 and the pn ´ 1q-ary brackets of the remaining generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9 Vector fields vanishing on a complete intersection Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W Ă Cn be an affine variety defined by a regular sequence ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕk P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, the Lie 8-algebroid described in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22 is the universal Lie 8-algebroid of the singular foliation of vector fields vanishing along W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the notation of the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22, K‚ equipped with the derivation B “ řk i“1 ϕi B Bµk is a free O-resolution of the ideal IW of functions vanishing along W, since ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕk is a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since XpCdq is a flat O-module, the sequence ¨ ¨ ¨ BbOid � K´2 bO XpCdq BbOid � K´1 bO XpCdq BbOid � IW XpCdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='42) is a free O-resolution of the singular foliation IW XpCdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Lie 8-algebroid structure of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22 is therefore universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As a special case of the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23, let us consider a complete intersection defined by one function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' an affine variety W whose ideal xϕy is generated by a regular polynomial ϕ P CrX1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has a free resolution of the space of vector fields vanishing on W given as follows: 0 � Oµ bO XpCdq ϕ B Bµ bOid � IW XpCd q, where µ is a degree ´1 variable, so that µ2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The universal Lie 8-algebroid structure over that resolution is given on the set of generators by : tµ bO Bxa, µ bO Bxbu2 :“ Bϕ Bxa µ bO Bxb ´ Bϕ Bxb µ bO Bxa and t¨ ¨ ¨ uk :“ 0 for every k ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is a Lie algebroid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that this construction could be also be recovered using Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' MAIN RESULTS OF PART I 79 Conclusion: This chapter described the 1-1 correspondence "Lie-Rinehart algebras ÐÑ Lie 8-algebroids on acyclic complexes".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It extends greatly [LLS20] for singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The functor "ÐÝ" consists in the 1-truncation of the Lie 8-agebroid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The converse functor consists in taking any free resolution, and constructing the brackets by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We prove that it is unique by proving it is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that we need the "complicated" notion of homotopy given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Last, some examples of [LLS20] are conceptually understood, and new examples are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Some algebraic constructions (blow-up, localization, germs, quotient) are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Obstruction classes to the existence of a Lie algebroid with a surjective morphism onto the Lie-Rinehart algebra are also described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We never assume finite rank here!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Part II Geometric Applications 80 CHAPTER 5 Universal Lie 8-algebroids of affine varieties In this chapter, we apply the results of Chapter 4 to answer some elementary but open questions that have to do with algebraic geometry, such as the interaction between the singularities of an affine variety and its Lie algebra of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 of Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 only says that it is possible to associate a Lie 8- algebroid structure to an affine variety by considering the Lie-Rinehart algebra made of its vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But the construction can be extremely complicated, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We only have an existence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This leads to the natural question: Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' How is the geometry of an affine variety related to its universal Lie 8-algebroid?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For instance, in view of the construction of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is also relevant to ask about the effect of blow-ups on this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We will see in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 that blow-ups may change a universal Lie 8-algebroid to a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But this example does not tell us really how the higher brackets disappear under the effect of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One of the important question we may ask is about the description of “the big theorem” of Hironaka [Já07] in terms of the universal Lie 8-algebroids obtained at each step while resolving singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This question remains open, but there are several other problems about which we are able to make some progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Those are quite modest, but, at least, we want to have the question clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Application to "blowup-up" will be discusses in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Background on affine varieties and some constructions We recall definitions and some main properties of the notion of affine variety in order to fix notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Our main references for this chapter are [Har77, Eis95, LB15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this chapter we will sometimes see Cd as the d-dimensional affine space which is commonly denoted by Ad C, forget about its vector space structure, but here we will not make any notational distinctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter is equipped with the Zariski topology that is, the topology whose closed subsets are the zero set of some ideal I Ď Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds “: O, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' which are of the form ta “ 81 CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 82 pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , adq P Cd | fpaq “ 0, @f P Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can check that these subsets indeed define a topology on Cd [KES00, Har77, LB15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' An affine variety W Ď Cd is a the zero locus of an ideal I Ď O, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', W :“ ZpIq :“ ta “ pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , adq P Cd | fpaq “ 0, @f P Iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It admits a topology, induced by the Zariski topology in Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, we do not exclude irreducible varieties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' W “ tpx, yq P C2 | xy “ 0u is an affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following facts and remarks are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For W Ď Cd an affine variety in the notation of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' we denote by IW the vanishing ideal of W, namely, IW “ tf P O | fpxq “ 0, @ x P Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In general, we have IW ‰ I because the vanishing ideal of the affine variety W “ tx2 “ 0u is the ideal IW “ xxy ‰ xx2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that IW can be defined for any arbitrary subset W Ă Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easy to check that (a) for S Ď T Ď O, one has ZpSq Ě ZpTq, (b) for U Ď V Ď Cd, one has IU Ě IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The ideal IW is larger than I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hilbert’s Nullstellensatz theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 of [Eis95]) claims that IW “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' I :“ tf P O | fN P I, for some N P Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have, W “ ZpIW q: if x P W, then by definition of IW , one has fpxq “ 0 for all f P IW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Whence, W Ď ZpIW q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conversely, it is clear that I Ď IW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This fact proves the other inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, by Noetheriality of the polynomial ring O, the ideal IW Ă O is generated by a finite number of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the sequel, we shall define an affine variety W as the zero locus of an ideal generated by a finite set of polynomials ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕr P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Zariski closure V of a subset V Ď Cd is equal to ZpIV q: by definition of IV , one has V Ď ZpIV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conversely, V Ď V “ ZpIq for some ideal I Ď O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b), IV Ď IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, I Ď IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (a), this implies ZpIV q Ď ZpIq “ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W Ď Cd be an affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A function F : W Ñ C is said to be a polynomial if Fpxq “ fpxq, @x P W, for some element f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The set CrWs of polynomial functions on W is, under the restriction map f P O ÞÑ f|W , isomorphic the quotient O{IW “: OW , called the coordinate ring of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Elements of the Lie algebra of C-linear derivations, DerpOW q “: XpWq, of OW are called vector fields on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the coordinate ring OW of an affine variety besides being a ring is also a vector space over C, hence it is a C-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This algebra is generated by the images ¯x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ¯xd in OW through the projection map of the coordinate functions x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xd P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For each a P W, the kernel of the evaluation map eva : OW Ñ C, F ÞÑ Fpaq, is the maximal ideal ma :“ kerpevaq made of all polynomial functions on W that vanish at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' W “ Cd is an affine variety with IW “ t0u, and OW “ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' W “ tau Ď Cd is an affine variety with IW “ px1 ´ x1paq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xd ´ xdpaqq the maximal ideal of a, and OW “ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For an affine variety W Ă Cd with corresponding ideal IW , we have DerpOW q » tX P XpCdq | XrIW s Ă IW u IW XpCdq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every affine variety W Ď Cd, the ring OW is Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let I be an ideal of OW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Denote by p: O Ñ O{IW be the quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then p´1pIq is also an ideal of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Noetheriality of O, p´1pIq is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, I “ ppp´1pIqq is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This shows that OW is Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Germs Here we mention the notion of local rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to [Cha14, Hid89, Eis95] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W Ď CN be an affine variety and OW its coordinates ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recall for U Ď W an open subset, a function f : U ÝÑ C is said to be regular at a P U if there exists g, h P OW with hpaq ‰ 0 such that f “ g h in a neighborhood of a, namely there exists an open set V Ă U that contains a such that f|V “ g h|V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A function germ at a point a P W is an equivalence class pfqa of pairs pU, fq with a P U Ă W an open subset containing a, and f : U ÝÑ C is regular at a, under the relation equivalence: pU, fq „ pV, gq if f|W “ g|W on an open subset W Ď U XV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The set of equivalence classes of the above equivalence relation inherits naturally an associative C-algebra, that is called germs of regular functions at a and is denoted by OW,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, a function germ pfqa at a P W has a well-defined value at a, given by the image of any representative pU, fq at a, namely pfqapaq :“ fpaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the map OW,a ÝÑ pOW qmW,a, pU, fq ÞÑ f|U “ g h with g, h P OW and h does not vanish on U, is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has, OW,a » pOW qmW,a [Har77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here mW,a “ tf P OW | fpaq “ 0u and pOW qmW,a is the localization w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t the complement of mW,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is important to notice that OW,a is a local ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote the unique maximal ideal of OW,a again by mW,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, It is worth it to notice that OW,a isomorphic to the quotient Oa{Ia of the local ring Oa of Cd at a by the ideal Ia which is spanned by the ideal IW in Oa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Zariski tangent space Let a P W Ď Cd a point of an affine variety W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are several equivalent descriptions of the tangent space of W variety at the point a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here we define it as pointwise derivations of the local ring CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 84 OW,a, see [Har77, vIR94] or Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 of [LGPV13], for more details on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A pointwise derivation of OW at a is a C-linear map δa : OW,a Ñ C satisfying the following Leibniz identity, δapFGq “ δapFqGpaq ` FpaqδapGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, if a regular function f is constant in a neighborhood of a then its germ pfqa at a satisfies δappfqaq “ 0, since δap1 ¨ 1q “ δap1q ¨ 1 ` 1 ¨ δap1q “ 2δap1q δap1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is not hard to check that the set of all pointwise derivations of OW at a is a C-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume now that W “ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Denote by pe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , edq the canonical basis of Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , du peiqa : OW,a Ñ C, pfqa ÞÑ lim tÑ0 fpa ` teiq ´ fpaq t “: Bf Bxi paq, is a well-defined pointwise derivation of O at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can show that (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g [LGPV13], Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) any pointwise derivation δa of O at a P Cd has the form δa “ dÿ i“1 δappxiqaq peiqa, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', δapfqa “ dÿ i“1 Bf Bxi paq δapxiqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) Hence, pointwise derivations peiqa, i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d form a basis for the vector space of pointwise derivation of O at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Zariski tangent space TaW of W Ď Cd at a P W is the vector space of all pointwise derivations of OW at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [vIR94] For a P W, one has TaW » ´ mW,a{m2 W,a ¯˚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The vector space mW,a{m2 W,a is called the cotangent space to W at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6, we have TaCd » Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the tangent space TaW of W Ď Cd at a P W can be seen as pointwise derivations δa of O at a of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) such that δappfqaq “ 0 for all f P IW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From this point of view, one has TaW » # pv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , vdq P Cd ˇˇˇˇˇ dÿ i“1 vi Bf Bxi paq “ 0, @f P IW + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 85 Singularities of an affine variety W In this section we recall some definitions and some facts on singularities on affine varieties and fix some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to [Har77, vIR94] for the full theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are various equivalent ways to define the dimension of an affine variety W, we refer the reader to Page 4 of [Har77] also to the Chapter 11 of [Joe92] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The dimension dim W of W is defined to be the maximal length d of the chains W0 Ă W1 Ă ¨ ¨ ¨ Ă Wd of distinct nonempty irreducible sub-varieties of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that a chain of irreducible sub-varieties corresponds to a chain of prime ideals in OW , by Noetheriality it must be of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A point a P W is said to be regular if dim TaW “ dim W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Otherwise, we say that a is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The set of regular points of W is denoted by Wreg, and the singular ones by Wsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that W is regular if Wsing “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Har77, vIR94]We have the following 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every a P W, dim TaW ě dim W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is an open dense open subset of W such that the map a ÞÑ dim TaW is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, regular points of W form an open dense subset of W, and singular points a (closed) proper sub-variety of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, a P W is a singular point of W if only if dim TaW ą dim W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' That is, Wsing “ ta P W | dim TaW ą dim Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Local coordinates at a point Let a P W Ď Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recall that (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g [vIR94]) that a family of elements t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , tr P OW,a are called local coordinates of W at a, if they vanish at a (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ti P mW,a for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , r), and if the classes of t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , tr P mW,a{m2 W,a form a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If a “ pa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , adq P Cd, then x1 ´ a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xd ´ ad are local coordinates of Cd at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that: Local coordinates at a P Cd generate the maximal ideal ma of Oa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, let t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , td P Oa be local coordinates at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By applying Nakayama Lemma (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) to R “ Oa Ě ma and V “ ma: the basis t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , td P ma{m2 a lifts to a (minimal) generating set t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , td for ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following proposition explains how an affine variety looks around a regular point (see [Hau14] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5, also [dJP00]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W Ď Cd be affine variety of codimension k, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', k “ d ´ dim W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, W is regular at a point a P W if and only if there exist local coordinates y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , yd of Cd at a such that W is locally of the form y1 “ ¨ ¨ ¨ “ yk “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 86 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Three main constructions Consider an affine variety W which is given by an ideal IW Ă O, with O the algebra of polynomials in d-variables, and OW “ O{IW the algebra of functions on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are three natural Lie-Rinehart algebras associated to W: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The OW -module XpWq of vector fields on W (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' derivations of OW ) is a Lie-Rinehart algebra over OW ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' its anchor map is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-module XW pCdq of vector fields on Cd tangent to W (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' derivations of O preserving IW ) is a Lie-Rinehart algebra with respect to the O-module structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' its anchor map is the inclusion XW pCdq ãÑ XpCdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-module IW XpCdq of vector fields on Cd vanishing at every point of W (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' IW -valued derivations of O) is a Lie-Rinehart algebra with respect to the O-module structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' its anchor map is again the inclusion IW XpCdq ãÑ XpCdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These three Lie-Rinehart algebras are related: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is an inclusion IW XpCdq Ă XW pCdq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the restriction of XW pCdq to W coincides with XpWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us justifies this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Every vector field on W extends to Cd: to see that, let δ P XpWq, we have δpxi ` IW q “ fi ` IW for some fi P Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds, i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define the vector field rδ :“ dÿ i“1 fi B Bxi on Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The vector field rδ restricts to δ on W, since for every f P Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xns, rδpfq ` IW “ δpf ` IW q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, rδpIW q Ă IW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that the Lie-Rinehart algebras XW pCdq, IW XpCdq Ă XpCdq are finitely generated as O-modules, since O is Noetherian (Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Whence, these Lie-Rinehart algebras are singular foliations on the complex manifold Cd in the sense of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' What happens if we take a look at the evaluation map at some point a P Cd?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any vector field X “ dÿ i“1 Xrxis B Bxi P XW pCdq tangent to W induces a pointwise derivation of OW at a P W as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We extend X by localization at the maximal ideal ma to a derivation pXq P DerpOW,aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define X|appfqaq :“ dÿ i“1 Xrxispaqpeiqappfqaq P TaW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' X|a is well-defined, since XrIW s Ă IW for all f P IW , in particular X|appfqaq “ 0 for all f P IW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In fact, we do not need to localize X to define X|a :“ pXrx1spaq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xrxdspaqq P TaW ãÑ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 87 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The image of the map Eva : XW pCdq Ñ Cd, X ÞÑ X|a (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) is denoted by TaXW pCdq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if a P W, then TaXW pCdq Ď TaW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If a R W, then TaXW pCdq “ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If W is a complete intersection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' IW “ pϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕrq and dim W “ d ´ r), then TaXW pCdq “ TaW for all a P Wreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More generally, for an arbitrary affine variety W, TaXW pCdq “ TaW for all a P Wreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is given by the construction in item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show item 2: let pv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , vdq P Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a R W, there exists ϕ P IW such that ϕpaq ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The vector field X “ ϕ ϕpaq dÿ i“1 vi B Bxi belongs to IW XpCdq Ă XW pCdq “ Cd and Xpaq “ pv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , vdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now we prove item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , vdq P TaW “ kerpJpaqq, with J :“ ´ Bϕi Bxj ¯ i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By assumption, we have rkpJpaqq “ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, J admits pr, rq-minor µ such that µpaq ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can assume that µ is the determinant of the first r-columns of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the vector fields Hj :“ ������������ B Bx1 ¨ ¨ ¨ B Bxr B Bxj Bϕ1 Bx1 ¨ ¨ ¨ Bϕ1 Bxr Bϕ1 Bxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Bϕr Bx1 ¨ ¨ ¨ Bϕr Bxr Bϕr Bxj ������������ , for j P tr ` 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , du, understood as the cofactor expansion along the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since for each i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ru, Hjrϕis has two repetitive lines, therefore it vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore Hj’s are tangent to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We claim that the following vector fields does the job, namely X “ p´1qr dÿ j“r`1 vj µpaqHj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) Indeed, if we denote by µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , µr the minors associated to the partials B Bx1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , B Bxr , respectively, then for every j the decomposition of Hj reads Hj “ rÿ i“1 p´1qi`1µi B Bxi ` p´1qrµ B Bxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, X “ dÿ j“r`1 vj µpaqµ B Bxj ` p´1qr µpaq ˜ rÿ i“1 dÿ j“1 p´1qi`1vjµi B Bxi ¸ CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 88 But, by developing the minors µipaq’s along their last columns, it takes the form, µipaq “ ˘Bϕ1 Bxj paqC1 ˘ ¨ ¨ ¨ ˘ Bϕr´1 Bxj paqCr´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that the determinants C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Cr´1 are the same for each Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, dÿ j“1 vjµipaq “ r´1 ÿ s“1 ˘ ˜ dÿ j“1 vj Bϕs Bxj paq ¸ looooooooomooooooooon “0 Cs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is obtained as follows: the local ring at a is by definition the localization Oa of Crx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds with respect to the multiplicative set of all polynomials that do not vanish at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15 a P W is a regular point if and only if there exists "local coordinates" y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , yd P Oa such that W is locally of the form y1 “ ¨ ¨ ¨ “ yk “ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the localization of IW is generated by these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, the tangent space at m is the vector space, spant B Byi |m, i ě k ` 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, for v P TaW the local vector field X “ dim W ÿ i“1 vi B Byk`i maps Oa to Oa, in particular it maps O to Oa and we have XrIW s Ă pIW qma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, for every, i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , du there exists a polynomial function gi that does not vanish at a such that giY rxis P Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, the vector field ˆX “ g1¨¨¨gr g1paq¨¨¨grpaqX is tangent to W satisfies ˆXpaq “ v and ˆXrIW s Ă IW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We may not have equality in item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To see this, consider the cups, W “ tpx, yq | C2 | x3 ´ y2 “ 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is clear that the tangent space T0W of W at 0 P W is the whole space C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But the vector fields in XW pC2q vanish at zero, since it is spanned as a Crx, ys-module by the Hamiltonian 2y B Bx ` 3x2 B By and the weighted Euler vector field 2x B Bx ` 3y B By (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following lemma shows that the vector fields that are tangent to W are also tangent to every strata of the stratification that consists of by taking the singular locus Wsing of the singular locus of W then the singular locus pWsingqsing of the singular locus Wsing and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='. We obtain a sequence of inclusions of the form W Ą pWsingqsing looooomooooon “:W1 Ą ppWsingqsingqsing looooooooomooooooooon “:W2 Ą ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have the following inclusions XW pCdq Ď XW1pCdq Ď ¨ ¨ ¨ Ď XWipCdq Ď ¨ ¨ ¨ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 89 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove that if X P XpCdq is such that XrIW s Ă IW then XrIWsings Ă IWsing, where IWsing is the ideal of functions on the singular part of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since IWsing is obtained by considering the minors of order k “ d ´ dim W of k elements chosen into the generators ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' That is, Wsing is given the ideal A ϕ1, ¨ ¨ ¨ ϕr, Prϕi1, ¨ ¨ ¨ , ϕiks, P P XkpCdq, for integers 1 ď i1 ă ¨ ¨ ¨ ă ik ď r E (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) Let us explain why the vector fields that tangent to W are also tangent to its singular locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a vector field X P XW pCdq one has by Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='27) that, X rPrϕi1, ¨ ¨ ¨ , ϕikss “ pLXPqrϕi1, ¨ ¨ ¨ , ϕiks ` kÿ j“1 Prϕi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xrϕijs, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕiks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) Notice that pLXPqrϕi1, ¨ ¨ ¨ , ϕiks P Ising since pLXPq P XkpCdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On the other hand, for every j there exists polynomial functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fr such that Xrϕijs “ řr i“1 flϕl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since P is a multi-derivation, one has, Prϕi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xrϕijs, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕiks “ rÿ l“1 ϕlPrϕi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fl, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕiks` rÿ i“1 flPrϕi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕl, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕiks It is now clear that the RHS of the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) is in the ideal Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof goes by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is a direct consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Every vector field X P XpWq is tangent to the stratification (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' X P XpWiq for each i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The coming example shows that the inclusions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) may be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This Example can also be found in the problem list [LLG22] of the lecture on singular foliations, Poisson 2022 [LGLR22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W “ tpx, y, zq P C3 | xypx ` yqpx ` yzq “ 0u Ă C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Staatq Stnoto Lin 9 Saotas ARCHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 90 The straight line x “ y “ 0 is a strata of the previous affine variety W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any vector field tangent to W is tangent to this straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show that it has to vanish at every point of this straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If not, its flow at time t would map a point p0, 0, z0q to a point p0, 0, z1q with z1 ‰ z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Its differential then induce a linear automorphism of the normal bundle of that straight line that has to preserve the straight lines x “ 0, y “ 0, x ` y “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since a linear endomorphism of C2 preserving three straight lines has to be a multiple of the identity map, this differential cannot map the straight line x ` z0y to the straight line x ` z1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On their universal Lie 8-algebroids Let W be an affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following is a direct consequence of the results of Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W be an affine variety as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' XpWq admits a universal Lie 8-algebroid made of free OW -modules of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' IW XpCdq and XW pCdq admit universal Lie 8-algebroids made of finitely many free O-modules of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Classical theorems of commutative algebras allows equipping the three Lie-Rinehart algebras above with resolutions of a certain type: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since O is a Noetherian regular ring, XW pCdq and IW XpCdq admit free resolutions by finitely generated O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Hilbert Syzygy Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8, those can be chosen to be of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since OW is a Noetherian ring, XpWq admits a free resolution by finitely generated OW -modules (by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In view of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, these three resolutions do admit Lie 8-algebroid structures over their respec- tive algebras, and those are universal Lie 8-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by UOW pXpWqq, UOpXW pCdqq and UOpIW XpCdqq the universal Lie 8-algebroids associated to the three Lie-Rinehart algebras above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There exist natural Lie 8-algebroid morphisms between these structures: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since IW XpCdq Ă XW pCdq, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 implies the existence of a unique up to homotopy Lie 8-algebroid morphism Ψ: UOpIW XpCdqq ÝÑ UOpXW pCdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In view of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6, the morphism Ψ may be represented by the inclusion map for well-chosen representation of UOpIW XpCdqq and UOpXW pCdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The anchor map of UOpXW pCdqq is tangent to W, hence the restriction i˚ W UOpXW pCdqq to W of UOpXW pCdqq exists, and is a Lie 8-algebroid over XpWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In general, it does not need to be a universal one, but Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 implies the existence of a unique up to homotopy Lie 8-algebroid morphism: Φ: i˚ W UOpXW pCdqq ÝÑ UOW pXpWqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that this morphism is in general not a Lie 8-quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Universal Lie 8-algebroid of an affine variety We give the following definition, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie 8-algebroid of an affine variety W is the homotopy class of Lie-8-algebroid associated to the Lie-Rinehart algebra XpWq over OW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 (Vector fields on hyperelliptic curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We follow the notations of [BF18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us consider on C2 the hyperelliptic curve H given by the equation y2 “ 2hpxq, where h is a monic polynomial of odd degree 2ν ` 1 ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let OH “ Crx,ys xy2´2hpxqy be the coordinate ring of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let XpHq “ DerpOHq be the Lie-Rinehart algebra of derivations of OH, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of vector fields on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As an OH-module, XpHq is the submodule " f B Bx ` g B By | f, g P OH, yg ´ h1pxqf “ 0 Ă OH B Bx ‘ OH B By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The curve H is non-singular if and only if gcdphpxq, h1pxqq = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, the Lie algebra of vector fields XpHq is a free OH-module of rank 1 generated by the vector field X “ y B Bx `h1pxq B By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A universal Lie 8-algebroid is given by the Lie-Rinehart algebra E´1 “ OHX Ă OH B Bx ‘ OH B By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the singular case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', gcdphpxq, h1pxqq “ dpxq ‰ 1, the OH -module XpHq is not free and has two generators, X “ y B Bx ` h1pxq B By and Y “ 2hpxq dpxq B Bx ` y h1pxq dpxq B By with a relation yX “ dpxqY (see [BF18] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A free resolution is described as follows: E´1 is the OH-module, generated by two elements that we denote by τ, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The anchor is defined then by ρpτq “ X, ρpµq “ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then we choose E´2 to be the OH-module given by the generator η, and we set E´i “ 0 for i ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The differential map ℓ1 “ d is chosen to be zero, except on degree ´2 where it is the OH-linear map d: E´2 ÝÑ E´1 given by dpηq “ yτ ´ dpxqµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us now describe a universal Lie 8-algebroid structure: The 2-ary bracket is defined on generators of E´1 by tτ, µu2 “ h1pxq dpxq τ ´ yd1pxq dpxq µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then we extend this bracket to the whole space E´1 by OH-linearity, skew-symmetric and Leibniz identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that tdη, τu2 “ tdη, µu2 “ 0, thus, one can define the 2-bracket by tη, τu2 “ tη, µu2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We extend all brackets using Leibniz identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' All k-ary brackets are zero for k ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie 8-algebra of an affine variety at a point: Let UOW pXpWqq “ pE, ℓ‚, ρq be a universal Lie algebroid of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us choose a P W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As stated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, the Lie 8-algebroid structure of W restricts at a (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' goes to the quotient with respect to the maximal ideal ma) if and only if ρrmas Ď ma, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ma is a Lie-Rinehart ideal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Define the OW -submodule Skerapρq :“ te P E´1 | ρpeqrOs Ď mau Ď E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 92 The k-ary bracket ℓk, k ě 1 and ρ restrict to the exact complex ¨ ¨ ¨ � E´3 ℓ1 � E´2 ℓ1 � Skerapρq ρ � DerpOW q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) Let us check that ℓ2 is well-defined: for all e, e1 P Skerapρq, ρpℓ2pe, e1qqrOs “ rρpeq, ρpe1qspOq, (since ρ is morphism of brackets) Ă ma, (by definition of the commutator r¨ , ¨s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter Lie 8-algebroid goes to quotient to a Lie 8-algebroid ˆ p‘iě2 E´i maE´i q ‘ kerapρq, ¯ℓ‚, ρ ˙ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) over OW {ma, where kerapρq :“ Skerapρq maSkerapρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since this quotient is the base field C, we obtain in fact a Lie 8-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, if a P W is an isolated singular point, then E´1 “ Skerapρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, UOW pXpWqq is a universal of the Lie-Rinehart algebra Aa “ tδ P XpWq | δrmas Ă mau “ XpWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 again, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) is a Lie 8-algebra on TorOW pXpWq, Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie 8-algebroids on minimal resolutions The germ at a of the Lie-Rinehart algebra of vector fields on W is easily checked to coincide with the Lie-Rinehart algebra of derivations of OW,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is an immediate consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W be an affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every a P W, the germ at a of the universal Lie 8-algebroid of W is the universal Lie 8-algebroid of DerpOW,aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To describe this structure, let us start with the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let a be a point of an affine variety W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The universal Lie 8-algebroid of DerpOW,aq can be constructed on a resolution ppEa ´iqiě1, ℓ1, πq, with Ea ´i free OW,a-modules of finite rank for all i ě 1, which is minimal in the sense that ℓ1pE´i´1q Ă maE´i for all i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since Noetherian property is stable by localization, the ring OW,a is a Noetherian local ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 in [May] assures that OW,a bOW DerpOW q admits a free minimal resolution by free finitely generated OW,a-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since OW,a is a local ring with maximal idea ma, we can assume that this resolution is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In view of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, there exists a Lie 8-algebroid structure over this resolution, and the latter is an universal of DerpOW,aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, a resolution of DerpOW,aq as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 comes equipped with a universal Lie 8-algebroid structure for DerpOW,aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The quotient with respect to ma is a Lie 8-algebra of the isolated singular point a with trivial 1-ary bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13 and its subsequent discussion, we can prove the next statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 93 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any universal Lie 8-algebroid structure on a resolution of DerpOW,aq as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5, the quotient with respect to the ideal ma is a representative of the Lie 8-algebra of the isolated singular point a, with trivial 1-ary bracket, on a graded vector space canonically isomorphic to TorOW pXW , Cq (C being a OW -module through evaluation at a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, its 2-ary bracket is a graded Lie bracket on TorOW pXW , Cq which does not depend on any choice made in the construction, and its 3-ary bracket is a Chevalley-Eilenberg cocycle whose class is also canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 Some examples of universal Lie-algebroids over an affine variety 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Vector fields tangent to W: a codimension one example The zero-set in W Ă V “ Cd of a weight homogeneous polynomial function ϕ P CrX1, ¨ ¨ ¨ , Xds admitting only an isolated singularity at the origin is one of the simplest possible example of a non- smooth affine variety W Ă V “ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We give in this section a description of UOpXpWqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Although our description is not complete, it will show how complex the universal Lie 8-algebroids may be, even for simple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' First, let us describe XpWq “ DerpOW q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ωd the weights of the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xd and by |ϕ| the weighted degree of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Recall that, by definition, OW :“ CrX1,¨¨¨ ,Xds xϕy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As a OW -module, XpWq :“ DerpOW q is generated by the restrictions to W of the following vector fields in Cd: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the weighted Euler vector field ÝÑ E :“ řd i“1 ωixiBxi 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the dpd ´ 1q{2 vector fields given by: Xij :“ Bϕ Bxi Bxj ´ Bϕ Bxj Bxi with 1 ď i ă j ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We start with a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Recall that a homogenous function ϕ with an isolated singularity at 0 is a Koszul function (see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6), so that the Koszul complex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the complex of poly-vector fields on V “ Cd, equipped with ιϕ: ¨ ¨ ¨ ιϕ ÝÑ X2pCdq ιϕ ÝÑ X1pCdq ιϕ ÝÑ O has no cohomology except in degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote it by pX‚, ιϕq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If P P Xi`1pCdq, Q P XipCdq satisfy ιϕpPq “ ϕQ, then there exists R P Xi`2pCdq such that P “ 1 |ϕ| ÝÑ E ^ Q ` ιϕpRq Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This follows from the easily checked fact that ιϕpQq “ 0, so that ιϕ ˆ 1 |ϕ| ÝÑ E ^ Q ˙ “ ϕQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that ιϕ ˆ P ´ 1 |ϕ| ÝÑ E ^ Q ˙ “ 0 and the existence of R now follows from the exactness of the Koszul complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 94 Proof (of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any X P XpWq is the restriction to W of a vector field ˜X in V tangent to W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' that satisfies ˜Xrϕs “ fϕ for some f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 the vector field ˜X can be written as ˜X “ f |ϕ| ÝÑ E ` ιϕpPq, for some bivector field P P X2pCdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The restriction to W of the first (resp the second) term is tangent to W and is of the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' second) type described in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We now intend to construct a free resolution of XpWq in the category of OW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us denote by ^ÝÑ E : XipV q Ñ Xi`1pV q the map ω ÞÑ ω ^ ÝÑ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map ^ÝÑ E is a chain map from the restriction i˚ W X‚ to W of the Koszul complex, shifted by one to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, ^ÝÑ E in the diagram below is a chain map: ¨ ¨ ¨ ιϕ� i˚ W X2pV q ιϕ � ^ÝÑ E � i˚ W XpV q iϕ � ^ÝÑ E � OW ^ÝÑ E � ¨ ¨ ¨ ιϕ� i˚ W X3pV q ιϕ � i˚ W X2pV q ιϕ � XpWq Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The chain map ¨ ^ ÝÑ E is a quasi-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is quite clear for i ě 1 since the complex pE1r1s, ιϕq has no cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check bijectivity in the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Surjectivity follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To check injectivity consider a function f P O such that fÝÑ E “ ιϕpπq ` ϕQ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) for some bivector field π P X2pV q and vector field Q P XpV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Upon taking ιϕ to both sides, Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) implies, f|ϕ|ϕ “ ϕιϕpQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, we obtain f “ 1 |ϕ|ιϕpQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 implies that the mapping cone construction provides a free resolution of the OW -module of vector fields of XpWq on W namely: ˜ E´i “ i˚ W Xi´1pV q ‘ i˚ W Xi`1pV q, i ě 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' D “ ˜ ´ιϕ 0 ´ ^ ÝÑ E ιϕ ¸ , π ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) In degree ´1 we read E´1 “ OW ‘i˚ W X2pV q and π is defined on the generators of E´1 as πp1‘0q :“ ÝÑ E and πp0 ‘ Bxi ^ Bxjq :“ Bϕ Bxi Bxj ´ Bϕ Bxj Bxi, for all, 1 ď i ă j ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us now describe some of the k-ary brackets: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Lie-Rinehart algebra XpWq of vector fields on W admits a universal Lie 8-algebroid whose 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' underlying complex is (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) (which is a free resolution of XpWq), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1-ary bracket is given by the resolution (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' anchor map ρ :“ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL LIE 8-ALGEBROIDS OF AFFINE VARIETIES 95 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the case where the generator 1‘0 appears together with a bivector field on W, one can define the 2-ary bracket on elements of degree ´1 which makes the anchor map a morphism as follows, t1 ‘ 0, 0 ‘ Bxi ^ Bxju2 :“ p|ϕ| ´ 2q0 ‘ Bxi ^ Bxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The k-ary brackets can be chosen to be the same as in the structure given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17 on i˚ W Xi`1pV q for i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the 3-ary bracket can be chosen to be zero when evaluated at 1 ‘ 0 and with two other elements of i˚ W X2pV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' XpWq does not come from a Lie algebroid of rank 1 ` dpd`1q 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In general, it is hard to compute the generators of XpWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But, there is a case where we know all generators: it is when W is a complete intersection with isolated singularity at zero such that IW is generated by weight homogeneous polynomials ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, XW pCdq is generated by IW XpCdq, the Euler vector field, and the Hamiltonian vector fields (see [HM93, Sie96]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The computation of the Lie 8-algebroid remains complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conclusion: As a particular case of Section 4, we notice that a Lie 8-algebroid can be associated to any affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Explicit computations are difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Some applications to Mohsen’s resolution of singularities will be given in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 There are still many open questions on the geometric meaning of this Lie 8-algebroid struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Although, we have not answered many questions , we state concepts, lemmas, and counter- examples that we hope to be able to use in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6 Universal Q-manifolds of a singular foliation The aim of this chapter is to lay the ground for the subsequent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, to explain how Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 extends the results on singular foliations of Sylvain Lavau’s PhD [Lav17] followed by a referred version by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Laurent-Gengoux, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lavau and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Strobl in [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The results of Section Chapter 4 extend the latter for arbitrary Lie-Rinehart algebras and also to the infinite case, and it still holds even when we do not have a geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We introduce the notion of longitudinal vector fields on a NQ-manifold and prove a new result on their cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This chapter is taken from the textbook [LGLR22] in which I am co-author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to [Vor10, BP13, LMP20, LLS20] for more details on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Throughout of this chapter M is a smooth, real analytic or complex manifold and K P tR, Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by O the sheaf of functions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Q-manifolds Let us first define N-graded manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Graded manifolds In words, as the name suggests graded manifolds are for graded vector spaces what manifolds are for vector spaces, in the sense that roughly speaking manifolds look locally like Rn and graded manifolds are locally like Rn ˆ V for some graded vector space V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this chapter we introduce the notion of graded manifolds, their vector fields, their morphisms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A (positively) graded manifold over the base manifold M is a sheaf E : U ÞÑ EpUq 96 CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 97 of graded commutative algebras over K such that every m P M admits an open neighborhood U Ă M on which the sheaf structure takes the form EpUq “ OU bK SpE˚ ´‚q for some graded vector space E “ ‘8 i“1E´i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Sections of the sheaf E are called functions on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is convenient to denote a graded manifold as a pair pM, Eq where E is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A function ξ P Ej is a formal sum ξ “ `8 ÿ i“0 ξpiq (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) with ξpiq P E an element of polynomial-degree i and degree j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For degree reasons, the sum must be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Local coordinates of a graded manifold Recall that for U Ă M an open set, one has pE˚ i qU „ ÝÑ U ˆ KrkpE˚ i q for every i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, the graded coordinates on the graded manifold pM, Eq is the data made of: In degree 0: a system of coordinates px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xnq of M on U In degree i ě 1: a local trivialization pξ1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξ rkpE˚ i q i q of E˚ i on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' That is, a system of graded coordinates of pM, Eq on U is px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn, ξ1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξ rkpE˚ 1 q 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξ1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξ rkpE˚ i q i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Elements of EpUq are "polynomials" in tpξj i qj“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',rkpE˚ ´iq, i ě 1u with coefficients in OpUq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The sheaf of differential forms pM, E “ ΩpMqq on a manifold M is a graded manifold since for every point m P M, it takes the form OU bK ^‚T ˚ mM where U is an open neighborhood of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Exterior forms can be seen as sections on the graded vector bundle E´1 “ TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A finite dimensional vector space E and its dual E˚ can be seen as graded vector bundles of respective degree ´k and k over a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' E is a graded manifold over M “ tptu, with functions isomorphic to ^E˚ for k odd and SpE˚q for k even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A morphism of graded manifolds between the two graded manifolds pM, Eq and pM1, E1q with respective base manifolds M and M1 is a pair made of a smooth or real analytic or holomorphic map φ: M ÝÑ M1 called the base map and a sheaf morphism over it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a family of graded algebra morphisms: E1pU1q Ñ Epφ´1pU1qq, compatible with the restriction maps, such that Φpfαq “ φ˚pfqΦpαq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 98 for all f P O1 U1 and α P E1pU1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A homotopy between two morphisms of graded manifolds Φ, Ψ: pM, Eq ÝÑ pN, E1q is a morphism of graded manifold pM, Eq ˆ pr0, 1s, Ωpr0, 1sqq ÝÑ pM1, E1q whose restrictions to t “ 0 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' t “ 1 coincide with Φ and Ψ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Vector fields on graded manifolds Vector fields on manifolds are derivations of its algebra of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a graded manifold, the analog of functions are the sheaf of sections ΓpS‚pE˚qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since it is not commutative but graded commutative, one has to consider graded derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A graded derivation of degree k of E is the data, for every U Ă M of a linear map Q: E‚pUq ÝÑ E‚`kpUq, compatible with all restriction maps, that increases the degree by `k and satisfies: QrFGs “ QrFsG ` p´1qkiFQrGs for every F P EipUq, G P EpUq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since we think geometrically, we say "vector fields of degree k" instead of graded derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, Eq be a graded manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For U Ă M and k P Z let XkpEqpUq :“ DerkpEpUqq be the EpUq-module of derivation of degree k on EpUq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The correspondence U ÞÝÑ X‚pEqpUq is a sheaf of E-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Its sections are called vector fields on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us list some important facts on vector fields on E: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the E-module X‚pEq :“ ‘kPZXkpEq of vector fields on E is naturally graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The E-module X‚pEq of vector fields on E is a graded Lie subalgebra of the graded Lie algebra HomKpE, Eq whose graded Lie bracket is the graded commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Precisely, the graded Lie bracket rP, Qs “ P ˝ Q ´ p´1qklQ ˝ P (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) of two vector fields P, Q of degree k, l respectively is a vector field of degree k ` l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easily checked that the bracket (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) fulfills (a) rP, Qs “ ´p´1qjkrQ, Ps, (graded skew-symmetry) (b) p´1qjlrP, rQ, Rss ` p´1qjkrQ, rR, Pss ` p´1qklrR, rP, Qss “ 0, (graded Jacobi identity) (c) rP, fQs “ PrfsQ ` frP, Qs, (Leibniz identity) for f P O and P, Q, R are vector fields of degree j, k and l respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Their description in local coordinates: notice that any homogeneous element e P ΓpE´kq cor- responds to a vertical1 vector field ιe P X´kpEq (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' it is O-linear) of degree ´k defined by contraction with e ιepξq :“ xξ, ey, ξ P ΓpE˚q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) and we extend by O-linear derivation, where x¨ , ¨y is the dual pairing between E˚ and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that ιe is by construction of polynomial-degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pU, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xnq be a coordinate chart of M and pξj i qj“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',rkpE˚ i q with i ě 1 be a homogeneous local trivialization of E˚ ´i, it should be understood that ξj i is the j-th elements of the local frame in ΓpE˚ ´iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pej iqj“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',rkpE´iq, i ě 1 be the dual basis of pξj i qj“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',rkpE˚ ´iq, i ě 1, then for every pair i, j, ιej i “ B Bξj i is the partial derivative with respect to ξj i P ΓpE˚ ´iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By choosing a TM-connection on E, it is easy to check that for any k P Z the family ˆ ξj1 i1 d ¨ ¨ ¨ d ξjl il B Bxj ˙ l ě 0 i1 ¨ ¨ ¨ il “ k j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , jl j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n Y ˜ ξj1 i1 d ¨ ¨ ¨ d ξjl il B Bξj i ¸ l ě 0 i1 ¨ ¨ ¨ il ´ i “ k j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , jl i ě 1, j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , rkpE´iq form a basis for XkpEqpUq up to permutations of the ξj1 i1 d ¨ ¨ ¨ d ξjl il ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here we adopt the convention i0 “ j0 “ 0 and ξ0 “ 1 P ΓpS0pE˚qq » O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Whence, any vector field Q P XkpEqpUq admits coordinates decomposition as follows Q “ ÿ l ě 0 i1 ¨ ¨ ¨ il “ k j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , jl j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' jQj1¨¨¨jl i1¨¨¨il ξj1 i1 d ¨ ¨ ¨ d ξjl il B Bxj ` ÿ i ě 1, l ě 0 i1 ¨ ¨ ¨ il ´ i “ k j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , jl j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , rkpE´iq 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ijQj1¨¨¨jl i1¨¨¨il ξj1 i1 d ¨ ¨ ¨ d ξjl il B Bξj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for some functions Qj1¨¨¨jl i1¨¨¨il P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These functions can be chosen in a unique manner to satisfy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ijQ jσp1q¨¨¨jσplq iσp1q¨¨¨iσplq “ ϵpσqQj1¨¨¨jl i1¨¨¨il for any permutation σ of t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For example, if Q is of degree `1, then it can be written in these notations as Q “ ÿ 1 ď u ď rkpE´1q j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n jQu 1 ξu 1 B Bxj ` ÿ i ě 1, “ l ě 0 i1 ¨ ¨ ¨ il ´ i “ 1 j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , jl j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ijQj1¨¨¨jl i1¨¨¨il ξj1 i1 d ¨ ¨ ¨ d ξjl il B Bξj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This following lemma says that vector fields of polynomial-degree ´1 are all the types (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, Eq be a graded manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For i ě 1, a vector field P P X´ipEq of polynomial- degree ´1 and of degree ´i, is of the form ιe for some section e P ΓpE´iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that a vector field P P X´ipEq of polynomial-degree ´1 is vertical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ppfq “ 0, since functions of M are of polynomial-degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, in local coordinates pU, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xnq M and pξj i qj“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',rkpE˚ ´iq with be a homogeneous local trivialization of E˚ ´i and pej iqj“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',rkpE´iq be the dual 1A vector field P P XpEq is said to be vertical if it is linear with respect to functions on M, in other words if Prfs “ 0 for all f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 100 basis of pξj i qj“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',rkpE˚ ´iq: a polynomial-degree ´1 vector field P P X´ipEq of degree ´i is forced to be of the form P|U “ rkpE´iq ÿ j“1 fjpxq B Bξj i (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) with fj P C8pUq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now, choose a local section eU P ΓUpE´iq of the form, eU :“ ÿ jě1 fjej i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) It is then clear that ιeU “ P|U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since local sections of E´i given on different coordinates chart domains Ua and Ub coincide on Ua X Ub and then lift to a global section of E´i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, ιe “ P for some e P ΓpE´iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 NQ-manifolds Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A dg-manifold or NQ-manifold is a positively graded manifold pM, Eq endowed with a degree `1 homological vector field Q on E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', Q P X1pEq is such that Q2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' They shall be denoted as a triple pM, E, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a finite dimension Lie algebra pg, r¨ , ¨ sq of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We assume that g is concentrated in degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is clear that pM “ tptu, E “ ^‚g˚q is a graded manifold over M “ tptu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This graded manifold carries a dg-manifold structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Precisely, we define the corresponding homological vector field as follows: fix a basis peiqi“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',n of g and let these global coordinate functions pξiqi“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',n on g be its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have rei, ejs “ nÿ l“1 λl ijel for some coefficients λl ij P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can check that the degree `1 vector field Q “ 1 2 nÿ i,j,l“1 λl ijξi ^ ξj B Bξl corresponds to the Chevalley-Eilenberg differential pdCE, ^‚g˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, Q2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that Q2 “ 0 is equivalent to the Jacobi identity for r¨ , ¨s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a differential graded vector bundle ppE´iqiě1, dq over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is a natural dg manifold given by its sheaf of sections pM, E “ ΓpSpE˚qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, the deferential map d: E ÝÑ E is dualized as a degree `1 map S1pE˚q ÝÑ S1pE˚q that we extend to a C8pMq-linear derivation on E squared to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let E “ Tr1sM be the shifted bundle of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It induces a graded manifold structure pM, E “ ΩpMqq over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This graded manifold carries a dg-manifold structure Q that corresponds to the de Rham differential on ΩpMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In terms of coordinates, the homological vector field Q reads nÿ i“1 dxi B Bxi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us introduce some vocabularies that will need to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 101 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, E1, Q1q and pM, E, Qq be two NQ-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A linear map Φ: E ÝÑ E1 is said to be of polynomial-degree/degree j P Z provided that, for all function α P E of polynomial-degree/degree i, Φpαq is of polynomial-degree/degree i ` j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any map Φ: E ÝÑ E1 of degree i decomposes w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t the polynomial-degree as follows: Φ “ ÿ rPZ Φprq with Φprq : E ÝÑ E1 a map of polynomial-degree r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When Φ: E ÝÑ E1 is a graded morphism of algebras, necessarily one has Φprq “ 0 for all r ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Furthermore, for all n, r P N and all ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξk P ΓpV q one has: Φprqpξ1 d ¨ ¨ ¨ d ξnq “ ÿ i1`¨¨¨`in“r Φpi1qpξ1q d ¨ ¨ ¨ d Φpinqpξnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) Obviously, in this case Φ is determined uniquely by the image of ΓpV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, E, Qq and pM, E1, Q1q be two NQ-manifolds over M with sheaves of functions E and E1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A morphism of NQ-manifold over M from pM, E1, Q1q to pM, E, Qq is a morphism of graded manifolds Φ: E ÝÑ E1 (of degree 0) over the identity map which intertwines Q and Q1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', Φ ˝ Q “ Q1 ˝ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is important to notice that morphisms of NQ-manifolds over M are by definition O-linear, since they are defined over the identity map the component Φprq of polynomial-degree r ě 0 of any O-linear map Φ: E ÝÑ E1 maps ΓpE˚q to ΓpSr`1pE1˚qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By O-linearity, it gives rise to a section φr P ΓpSr`1pE1˚q b Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, one has Φprqpξq “ xφr, ξy (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) for all ξ P ΓpE˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows that Φ is entirely determined by the collection ` φr P ΓpSk`1pE1˚q b Eq ˘ rě0 when Φ is an algebra morphism or a Ξ-derivation for some map Ξ: E ÝÑ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In such case, for r ě 0, φr P ΓpSr`1pE1˚q b Eq is then called the r-th Taylor coefficient of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We also call the 0-th Taylor coefficient φ0 : E1 Ñ E the linear part of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter is a chain map ¨ ¨ ¨ � E1 ´3 φ0 � d1p3q � E1 ´2 φ0 � d1p2q � E1 ´1 φ0 � ρ1 � TM id � ¨ ¨ ¨ � E´3 dp3q � E´2 dp2q � E´1 ρ � TM (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 102 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 NQ-manifolds - Lie 8-algebroids We have seen in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 that Lie 8-algebroids (possibly infinite dimension) over O are one-to-one with co-differentials of the graded symmetric algebra, which are compatible with the action of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This correspondence provides a simple characterization of Lie 8-algebroids over an arbitrary commutative unital algebra O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the finite dimensional case, we can work without co-differentials by using T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Voronov’s higher derived brackets construction [Vor10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Roughly speaking, it is shown in [Vor10] that (finite dimensional) Lie 8-algebroids as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13 are the same as NQ-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, it is important to note the latter correspondence fails in infinite dimension case, since the identification Γ pS‚pE˚qq » Γ pS‚pEq˚q fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We will not explain entirely the construction here, but we refer the reader to [Vor04, Vor05, Vor10] also to [BP13, LMP20] for more details on the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following statement is similar to our Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 the difference lies in the fact that ours remains valid in infinite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16 (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Voronov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, Eq be a graded manifold of finite dimension in each degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is a one-to- one correspondence between: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (finitely generated) negatively graded Lie 8-algebroids pE, pℓkqkě1, ρq over M, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' homological vector fields Q P X`1pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The relation stated in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16 can be enlightened in terms of the k-ary brackets pℓkq as follows: r¨ ¨ ¨ rrQ, ιe1s , ιe2s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ιeksp´1q “ ιℓkpe1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ekq, for all e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ek P ΓpEq (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) ρpeqrfs “ rQ, ιesp0qpfq, for all f P O, e P ΓpE´1q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) In particular2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all f P O, e P ΓpE´1q xQp1qrfs, ey “ ρpeqrfs, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all α P ΓpE˚q and e P ΓpEq: A Qp0qrαs, e E “ xα, ℓ1peqy , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all homogeneous elements e1, e2 P ΓpEq and α P ΓpE˚q A Qp1qrαs, e1 d e2 E “ ρpe1qrxα, e2ys ´ ρpe2qrxα, e1ys ´ xα, ℓ2pe1, e2qy, with the understanding that the anchor ρ vanishes on E´i when i ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2Our sign’s convention are those of [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 103 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for every k ě 3, the k-ary brackets ℓk : ΓpSk KpEqq ÝÑ ΓpEq and the polynomial-degree k ´ 1 component Qpk´1q : ΓpE˚q Ñ ΓpSk KpE˚qq of Q are dual to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Qpk´1qrαs, e1 d ¨ ¨ ¨ d ek E “ xα, ℓkpe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ekqy , for α P ΓpE˚q and e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ek P ΓpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Convention 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From now on, unless otherwise mentioned, we shall simply say "Lie 8-algebroids over M" for "finitely generated 8-algebroids over M".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, Lie 8-algebroids pE, pℓkqkě1, ρq over M shall be denoted as pE, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Whether we see Lie 8-algebroids as NQ-manifolds or as co-differentials, these two approches have in common to give the same notion of Lie 8-morphism of 8-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This can be seen directly by writing the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) in terms of the k-ary brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Universal Q-manifolds This section can be understood as a consequence of the main results of the first part of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recover the result on universal Q-manifolds of a singular foliation [LLS20, Lav17], whose existence was proved under the condition that geometric resolutions exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recall that the existence of such a resolution is not guaranteed, unlike the case of resolution by free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to Appendix B and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 for more details on resolutions of modules and geometric resolutions of singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any resolution of F by free O-modules (which may not be a geometric resolution) ¨ ¨ ¨ d ÝÑ P´3 d ÝÑ P´2 d ÝÑ P´1 ρ ÝÑ F ÝÑ 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) carries a Lie 8-algebroid structure over F whose unary bracket is ℓ1 :“ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, when F admits a geometric resolution pE, d, ρq, there exists a Lie 8-algebroid pE, Qq over F whose linear part is pE, d, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 to F seen as a Lie-Rinehart algebra over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given, a) a Lie 8-algebroid pM, E1, Q1q that terminates in F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e, ρ1pΓpE´1qq Ď F, b) a universal Lie 8-algebroid pM, E, Qq of F, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' there exists a Lie 8-morphism from pM, E1, Q1q to pM, E, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and any two such morphisms are homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Two universal Lie 8-algebroid of a singular foliation are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, the homotopy equivalence between them is unique up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Apply Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 The complex defined by adQp0q Let F be a singular foliation on M that admits a universal algebroid pE, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a fixed k P N0 Yt´1u consider the bicomplex defined on Hom‚ O ´ ΓpE˚q, Γ ´ Sk`1 K pE˚q ¯¯ where the horizontal differential Bh (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' vertical differential Bv) are given by left composition (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' right composition with Qp0q), namely, for all Ψpkq P Hom‚ O ´ ΓpE˚q, Γ ´ Sk`1 K pE˚q ¯¯ one has, Bh ´ Ψpkq¯ :“ $ & % Ψpkq ˝ Qp0q when Ψpkq restricts to ΓpE˚ ´iq, with i ‰ 1, Ψpkq ˝ ρ˚ when Ψpkq restricts to ΓpE˚ ´1q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) and Bv ´ Ψpkq¯ :“ $ & % ρ˚ ˝ Ψpkq when Ψpkq is of degree i and restricts to ΓpE˚ ´i´1q, Qp0q ˝ Ψpkq otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) With the understanding that Γ ` S0 KpE˚q ˘ » O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The total differential is given by the formula, B ´ Ψpkq¯ “ Bh ´ Ψpkq¯ ´ p´1q|Ψpkq|Bv ´ Ψpkq¯ for all Ψpkq P Hom‚ O ´ ΓpE˚q, Γ ´ Sk`1 K pE˚q ¯¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following diagram pictures the idea of the bicomplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The total degree is given by the anti- diagonals lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ò Ò Ò Ñ HomO ˆ ΓpE˚ ´2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚q ¯ k`3 ˙ Bh Ñ HomO ˆ ΓpE˚ ´1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚q ¯ k`3 ˙ ¨ ˝ρ˚ Ñ HomO ´ F˚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚qk`3 ¯¯ Ñ 0 Bv Ò Bv Ò Dv Ò Ñ HomO ˆ ΓpE˚ ´2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚q ¯ k`2 ˙ Bh Ñ HomO ˆ ΓpE˚ ´1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚q ¯ k`2 ˙ ¨˝ ρ˚ Ñ HomO ´ F˚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚qk`2 ¯¯ Ñ 0 Bv Ò Bv Ò Bv Ò Ñ HomO ˆ ΓpE˚ ´2q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚q ¯ k`1 ˙ Bh Ñ HomO ˆ ΓpE˚ ´1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚q ¯ k`1 ˙ ¨ ˝ρ˚ Ñ HomO ˆ F˚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γ ´ Sk`1 K pE˚q ¯ k`1 ˙ Ñ 0 Ò Ò Ò 0 0 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) There is no cohomology for the total complex which is governed by B ” r ¨ , Qp0qs since the lines are exact (see Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When k “ ´1 the bicomplex (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) is just the line complex: ¨ ¨ ¨ Bh ÝÑ HomO ` ΓpE˚ ´2q, O ˘ Bh ÝÑ HomO ` ΓpE˚ ´1q, O ˘ Bh ÝÑ HomO pF˚, Oq ÝÑ 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17) which is also exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 A result on longitudinal vector fields and examples In this section, we study the cohomology of longitudinal vector fields, which will help in proving the main results stated in the beginning of Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let E be a splitted graded manifold over M with sheaf of function E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A vector field L P XpEq is said to be a longitudinal vector field for F if there exists vector fields X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xk P F and functions Θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Θk P E such that Lpfq “ kÿ i“1 XirfsΘi, @f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18) CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 105 We also need to define the following Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A vector field X P XpMq is said to be an infinitesimal symmetry of F, if rX, Fs Ă F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Lie algebra of infinitesimal symmetries of F is denoted by, spFq Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here are some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Vertical vector fields on a graded manifold are longitudinal for any singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any Q-manifold pE, Qq over a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The homological vector field Q P XpEq is a longitudinal vector field for F :“ ρpΓpE´1qq: in local coordinates pU, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xnq on M and a local trivialization ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of graded sections in ΓpE˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The vector fields Q are of the form: Q “ ÿ j ÿ k, |ξk|“1 Qj kpxqξk B Bxj ` ÿ j ÿ k,ι1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ιk 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='Qj ι1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ιkpxqξ1 d ¨ ¨ ¨ d ξk B Bξj (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19) “ ÿ k, |ξk|“1 ξk ˜ÿ j Qj kpxq B Bxj ¸ ` ¨ ¨ ¨ Take Xk :“ ř j Qj kpxq B Bxj P FpUq for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has for all f P C8pUq, Qpfq “ ř k ξkXkrfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For pE, Qq a Q-manifold and F :“ ρpΓpE´1qq its basic singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any extension of a symmetry X P spFq of F to a degree zero vector field p X P XpEq, the degree `1 vector field rQ, p Xs is longitudinal for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show this last point using local coordinates px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xnq on M and a local trivialization ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of graded sections in ΓpE˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The vector fields Q and p X take the form: Q “ ÿ j ÿ k, |ξk|“1 Qj kpxqξk B Bxj ` ÿ j ÿ k,ι1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ιk 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='Qj ι1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ιkpxqξ1 d ¨ ¨ ¨ d ξk B Bξj p X “ X ` ÿ j ÿ k,ι1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ιk 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='Xj ι1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ιkpxqξ1 d ¨ ¨ ¨ d ξk B Bξj (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20) where X “ nÿ i“1 Xipxq B Bxi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By using Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20) we note that all the terms of rQ, p Xs are vertical except maybe for the ones where the vector field X appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For k ě 1, the vector field rQj ι1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ιkξ1 d ¨ ¨ ¨ d ξk B Bξj , Xs is vertical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and for every fix k, one has « nÿ j“1 Qj kξk B Bxj , X ff “ ξk « nÿ j“1 Qj k B Bxj , X ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, « nÿ j“1 Qj k B Bxj , X ff P F, since X is a symmetry for F and nÿ j“1 Qj k B Bxj P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Longitudinal vector fields are stable under the graded Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by LpEq the graded Lie algebra of longitudinal vector fields for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us study vector fields on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Sections of E are identified with derivations under the isomorphism mapping e P ΓpEq ÞÝÑ ιe P XpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This allows us to identify a vertical vector field with (maybe infinite) sums of tensor products of the form Θ b e with Θ P E, e P ΓpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any connection on ΓpE˚q induces a vector field of degree zero r∇X P XpEq by setting for f P O, r∇Xpfq :“ Xrfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Once a connection is chosen, we have for all k P Z XkpEq » à jě1 Ek`jbOΓpE´jq‘EkbOXpMq » ‘jě1ΓpSpE˚qk`jbE´jq‘ΓpSpE˚qkbTMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21) Thus, one can realize a vector field P P XkpEq as a sequence P “ pp0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q, where p0 P ΓpSpE˚qk b TMq and pi P ΓpSpE˚qk`i b E´iq for i ě 1 are called components of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the diagram (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23), P “ pp0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q is represented as an element of the anti-diagonal and pi is on column i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that P is of depth n P N if pi “ 0 for all i ă n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, vector fields of depth greater or equal to 1 are vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Under the decomposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21), the differential map adQ takes the form D “ Dh ` ÿ sě0 Dvs (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22) with D2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here Dh “ idbd or idbρ, and Dvs : ΓpSpE˚qkbE´iq Ñ ΓpSpE˚qk`s`1b E´i´sq for i ě 0, s ě 0, we shall denote E0 :“ TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote the latter complex by pX, Dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' They can be represented as anti-diagonal lines in the following commutative diagram, whose lines are complexes of O-modules .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ¨ ¨ ¨ � ΓpSpE˚qk`2 b E´2q � idbd � ΓpSpE˚qk`2 b E´1q � idbρ � ΓpSpE˚qk`2 b TMq � ¨ ¨ ¨ � ΓpSpE˚qk`1 b E´2q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Qbid ` ¨¨¨ � idbd � ΓpSpE˚qk`1 b E´1q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Qbid ` ¨¨¨ � idbρ � ΓpSpE˚qk`1 b TMq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Qbid ` ¨¨¨ � ¨ ¨ ¨ � ΓpSpE˚qk b E´2q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Qbid ` ¨¨¨ � idbd � ΓpSpE˚qk b E´1q � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Qbid ` ¨¨¨ � idbρ � ΓpSpE˚qk b TMq � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Qbid ` ¨¨¨ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' � column 2 column 1 column 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23) Under this correspondence, we understand longitudinal vector fields as the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A graded vector field P “ pp0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q P X is longitudinal if p0 P E bO F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following theorem is crucial for Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, Qq be a universal Q-manifold of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Longitudinal vector fields form an acyclic complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, any longitudinal vector field on E which is a adQ-cocycle is the image through adQ of some vertical vector field on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More generally, if a vector field on E of depth n is a adQ-cocycle, then it is the image through adQ of some vector field on E of depth n ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since pE, Qq is an universal Q-manifold of F, lines in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23) are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is now a diagram chasing phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let P “ pp0, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , q P X be a longitudinal element which is a D-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By longitudinality there exists an element b1 P ΓpSpE˚q b E´1q such that pid b ρqpb1q “ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Set P1 “ p0, b1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q, that is we extend b1 by zero on ΓpSpE˚q b Eď´2q and ΓpSpE˚q b TMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is clear that P ´ DpP1q “ p0, p1 1, p1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q is also a D-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, we have Dhpp1 1q “ 0 by exactness there exists b2 P ΓpSpE˚qbE´2q such that Dhpb2q “ p1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As before put P2 “ p0, 0, b2, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Similarly, P ´ DpP1q ´ DpP2q “ p0, 0, p2 2, p2 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q is a D-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By recursion, we end up to construct P1, P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' that satisfy P ´ DpP1q ´ DpP2q ` ¨ ¨ ¨ “ 0, that is, there exists an element B “ p0, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q P X such that DpBq “ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To prove item 2 it suffices to cross out in the diagram (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23) the columns number 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , n ´ 1, which does not break exactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof now follows as for item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, we deduce from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 the following exact subcomplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, Qq be a universal Q-manifold of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The subcomplex VQ of pXpEq, adQq made of vertical vector fields P P XpEq that satisfy P ˝ Qpfq “ 0 for all f P O is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let P P XpEq be a vertical vector field which is an adQ-cocycle (note that we have automatically P ˝ Qpfq “ 0 for all f P O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 there exists a vertical vector field rP P XpEq such that rQ, rPs “ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, rP P VQ, since for all f P O, 0 “ rQ, rPspfq “ p´1q| rP| rP ˝ Qpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following remark will be used especially for the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 of Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a cocycle P P VQ of degree 0 one has P p´1q “ 0 (for degree reason).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11, P is the image by adQ of an element rP P VQ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' such that rQ, rPs “ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, one can choose rP p´1q “ 0: we have rQ, rPsp´1q “ rQp0q, rP p´1qs “ P p´1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By exactness of adQp0q (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1), we have P p´1q “ rQp0q, ϑs for some O-linear map ϑ P HompΓpE˚q, ΓpS0pE˚qqq of degree ´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Put sP :“ rP ´ rQ, ϑs, where ϑ is extended to a derivation of polynomial-degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Clearly, rQ, sPs “ P and sP p´1q “ rP p´1q ´ rQ, ϑsp´1q “ rP p´1q ´ rQp0q, ϑs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, P “ adQp sPq with sP p´1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is direct consequence of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F a singular foliation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, Qq be a universal Q-manifold of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, H‚ pXpEq, adQq » H‚ ˆXpEq LpEq, adQ ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24) where adQ is induced by adQ on the quotient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 108 Since all vertical vector fields are longitudinal, there is an isomorphism of O-modules XpEq LpEq » ΓpS‚pE˚qq bO ˆXpMq F ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25) Let us describe the differential map adQ using this isomorphism: let pU, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xnq be local coordi- nates on M and take local trivialisations ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of graded sections in ΓpE˚q, also let ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the corresponding dual sections in ΓpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that locally, any P P XpEq LpEq is represented by a vector field of the form ÿ j, i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ik fj i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ikpxqξi1 d ¨ ¨ ¨ d ξik b B Bxj , fj i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ik P C8pUq since the vector fields in B Bθi are vertical vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' That is to say, P can be represented by an element of the form P “ Θ b X with Θ P ΓUpS‚pE˚qq and X P XpUq, using the decomposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since rQ, Ps “ QrΘsX ` p´1q|Θ|Θ ¨ rQ, Xs “ QrΘsX ` p´1q|Θ|Θ d ÿ k, |ξk|“1 ξk «ÿ j Qj kpxq B Bxj , X ff ` vertical vector fields “ QrΘsX ` p´1q|Θ| ÿ k, |ξk|“1 Θ d ξk rρpξkq, Xs ` vertical vector fields we have adQpPq “ QrΘs b X ` p´1q|Θ| ÿ k, |ξk|“1 Θ d ξk b rρpξkq, Xs, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='26) where X stands for the class of a vector field X P XpMq in XpMq F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, locally adQ “ Q b id ` ÿ k, |ξk|“1 id d ξk b ∇Bott ξk where ∇Bott : ΓpE´1qˆ XpMq F ÝÑ XpMq F , is the Bott connection associated to F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ∇Bott e X :“ rρpeq, Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume that M “ Rd and F is a singular foliation that admits two leaves: t0u and Rdzt0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then XpEq LpEq » ΓpS‚pE˚qq|0 bR ˆ Rd ‘ I0XpMq F ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The differential may be complicated to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F “ I0XpRdq and pE, Qq one of its universal Lie algebroid such that E´1 “» glpRdq is the transformation Lie algebroid for the action of glpRdq on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For this singular foliation, XpRdq F » Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, we obtain XpEq LpEq » ΓpS‚pE˚qq|0 bC8pRdq Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and the Bott connection is simply the Chevalley-Eilenberg differential for the action of glpRdq on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, it is easy to see that H1pXpEqq “ H1 ˆXpEq LpEq ˙ “ H1 CEpglpRdq, Rdq “ 0 In other words, the universal Lie 8-algebroid is, in that case, rigid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' any formal deformation Q ` ϵQ1 ` ϵ2Q2 ` ¨ ¨ ¨ of its derivation Q is formally trivial [LGL22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' UNIVERSAL Q-MANIFOLDS OF A SINGULAR FOLIATION 109 Here is a second example where the universal Lie 8-algebroid is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be the singular foliation given by the action of Lie algebra g “ slp2, Rq on R2 through the vector fields: e “ x B By, f “ y B Bx, h “ x B Bx ´ y B By, where e, f, h are the standard basis elements of slp2, Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The singular foliation F admits a universal Lie 8-algebroid pE, Qq built on a geometric resolution ([LLS20], Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='31) of length 2 0ÝÑE´2 d ÝÑ E´1 ρ ÝÑ TM, where E´1 » slp2, Rq ˆ R2 is a trivial vector bundle of rank 3, and E´2 a trivial vector bundle of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, the bracket between two constant sections of E´1 is defined as being their bracket in slp2, Rq and all other brackets between generators is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, Q takes the form Q “ xe˚ B By `yf˚ B Bx`h˚px B Bx´y B Byq`e˚df˚ B Bh˚ ´2e˚dh˚ B Be˚ `2f˚dh˚ B Bf˚ `px2f˚`y2e˚´xyh˚q B Bµ˚ where µ is the generator of E´2, and pe˚, f˚, h˚, µ˚q is the dual basis of ppe, f, h, µqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this case, one has XpR2q F » R2 ‘ R is generated by the classes of B B Bx, B By, x B Bx ` y B By F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, as a graded vector space: XpEq LpEq » p^‚sl˚ 2 ‘ Rr2sq bR pR2 ‘ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The differential induced by Q is easily checked to be the Chevalley-Eilenberg differential for the natural slp2, Rq-action on these modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, H1pXpEqq “ H1 ˆXpEq LpEq ˙ “ H1 ˆ slp2, Rq, XpR2q F ˙ “ 0, since the Lie algebra slp2, Rq is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that the universal Lie 8-algebroid pE, Qq is rigid in the sense of [LGL22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conclusion: We recalled the duality "Lie 8-algebroid ÐÑ Q-manifolds of finite rank in all degree" so that the universal Lie 8-algebroid becomes a universal Q-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recover [LLS20] the notion of universal Q-manifold pE, Qq of a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We prove that adQ hasa no longitudinal cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' aWhich squares to zero on vector fields on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 7 Isotropy Lie algebras of a singular foliation In this chapter, we look at an extension of the Androulidakis and Skandalis isotropy Lie algebra [AZ13] of a singular foliation at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recall some definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Im “ tf P C8pMq | fpmq “ 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The tangent space of F at a point m P M is the vector space TmF :“ tX|m | X P Fu whose elements are evaluations of the vector fields of F at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In other words, TmF is the image of the evaluation map evm : F Ñ TmM, X ÞÑ X|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The set !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' m P M ˇˇˇ F ImF ÝÑ TmF is bijective ) is open and dense in M [AS09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is the set of regular points of pM, Fq and is denoted by Mreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any point m P M, the O-module Fpmq :“ tX P F | Xpmq “ 0u “ kerpevmq is a Lie subalgebra of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The evaluation map goes to quotient and induces the following exact sequence, 0 � Fpmq ImF � F ImF � TmF � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ImF Ď Fpmq is a Lie ideal, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the quotient space gm “ Fpmq ImF is a Lie algebra, which is called the isotropy Lie algebra of F at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The isotropy Lie algebras detect singular and regular points of pM, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It measures how far F is of being regular in neighborhood of a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This can be stated as follows (see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 of [AZ13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any point m P L of a leaf L Ă M, gm “ t0u if and only if, L is a regular leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, the gm’s are all isomorphic as Lie algebras while m P L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 110 CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 111 For A a Lie-Rinehart algebra over O, the same construction can be done for any maximal ideal m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let A|m :“ ta P A | ρpaqrOs Ď mu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The quotient A|m mA is a Lie algebra over O{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For O “ C8pMq and m “ Im for a point m P M, the following sequence 0 � A|Im ImA � � � A ImA � � TmρpAq � 0 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) is exact, where TmρpAq Ă Hompm{m2, O{mq is the image of A Ñ Hompm{m2, O{mq, a ÞÑ ρpaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 is however subtle to extend to this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a finitely generated Lie-Rinehart A over C8pMq, there exists a Lie algebroid pA, ρA, r¨ , ¨sAq such that A » ΓpAq for some vector bundle A Ñ M if and only if A ImA, as a vector space, has constant rank at all points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case Apmq ImA is kerpρA|mq for all m P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, it is the case in a neighborhood of every point where the dimension of A ImA is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is a consequence of Nakayama Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The germ Om of functions on a neighbor- hood of U Ď M of m is a local ring [RS94] with maximal ideal still denoted by Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The tensor product OmbO A is a module over Om.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let r “ dimp A ImAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Nakayama Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 any basis pe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , erq of A ImA lifts to minimal generators on Om bO A in a neighborhood of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since r “ dimp A Im1Aq for all points m1 P U, again Nakayama Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 implies pe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , erq are minimal generators on a neigh- borhood of m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any representative pe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , erq of pe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , erq are local sections of A in a neighborhood U of m that span A on U so that A|U is a free module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This applies that A is a C8pUq-module, which is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Serre-Swan theorem, A “ ΓpAq for some vector bundle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Specialization of a Lie 8-algebroid at a point Let pM, E, Qq “ pE‚, ℓ‚, ρq be a Lie 8-algebroid over a manifold M with anchor ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every point m P M, the k-ary brackets restrict to the graded vector space evpE, mq :“ ˜ à iě2 E´i|m ¸ ‘ kerpρmq and equipped the latter with a Lie 8-algebra structure that we denote by IstropympE, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every k ě 1, the restriction goes as follows: tx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xkuk :“ ℓkps1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , skq|m (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xk P evpE, mq and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , sk P ΓpEq sections of E such that sipmq “ xi with i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These brackets are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is clear that for k ‰ 2, since ℓk is linear over functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But it is not immediate that the 2-ary bracket is well-defined as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On one hand, the new brackets t¨ ¨ ¨ uk have values in evpE, mq for degree reason, except may be for the 2-ary bracket when applied to elements of degree ´1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' elements of the kernel of ρm) CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 112 but in that case it is in the kernel of ρm since ρmptx1, x2u2q “ ρmpℓ2ps1, s2q|mq “ ρpℓ2ps1, s2qq|m “ rρps1q, ρps2qs|m “ 0 In the last line we have used the fact that the Lie bracket of two vector fields that vanish at m is a vector field that vanishes again at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On the other hand, the 2-ary bracket t¨ , ¨ u2 is also well-defined when applied to elements of degree less or equal to ´2, we need to verify when we take the bracket with at least an element of degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pei 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ei rkpE´iqq be a local trivialization of E´i on a neighborhood U of the point m P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x1 P kerpρmq and x2 P E´i|m write x1 “ rkpE´1q ÿ k“1 λke1 kpmq, x2 “ rkpE´iq ÿ k“1 µkei kpmq for some scalars pλiq in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The scalars pλkq, pµkq extend to functions pfkq, pgkq on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, we have tx1, x2u2 “ ℓ2ps1, s2q|m with s1 “ rkpE´1q ÿ k“1 fke1 k, s2 “ rkpE´iq ÿ k“1 gkei k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If rs2 is another extension of x2, then ps2 ´ rs2qpmq “ 0 and this is equivalent to pgk ´ rgkqpmq “ 0 for k “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , rkpE´iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows that ℓ2ps1, s2 ´ rs2q|m “ rkpE´iq ÿ k“1 ℓ2 ` s1, pfk ´ rgkqei k ˘ |m “ rkpE´iq ÿ k“1 ((((((( pfk ´ rgkqpmqℓ2 ` s1, ei k ˘ |m ` (((((((( ρps1q|mrfk ´ rgksei k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, evpE, mq : ¨ ¨ ¨ t¨u1“ℓ1|m ÝÑ E´3|m t¨u1“ℓ1|m ÝÑ E´2|m t¨u1“ℓ1|m ÝÑ kerpρmq comes equipped with a Lie 8-algebra whose brackets are pt¨ ¨ ¨ ukqkě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any Lie 8-morphism of algebroids Φ: pM, E1, Q1q Ñ pM, E, Qq induces a Lie 8-algebra morphism Φ|m : S‚pV 1 |mq Ñ S‚pV|mq since it is O-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define the graded vector space à iě1 H´ipE‚, mq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) as the cohomology group of the complex ¨ ¨ ¨ ℓ1|m� E´3|m ℓ1|m � E´2|m ℓ1|m � kerpρmq � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 113 One can check that when pM, E, Qq is universal for a singular foliation F, the graded space (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) does not depend on the underlying geometric resolution of F and is denoted HpF, mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This construction can be extended to any Lie 8-algebroid over O for any maximal ideal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, ℓ‚, ρq be a Lie 8-alegrboid over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Define E´i|I “ $ ’ & ’ % te P E´1 | ρpeqrOs Ď Iu for i “ 1 E´i IE´i for i ě 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) The construction is purely formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, the cohomology of the complex ¨ ¨ ¨ ℓ1|I � E´3|I ℓ1|I � E´2|I ℓ1|I � E´1|I denoted by H‚pE, Iq does not depend on the choice of a free resolution of the Lie-Rinehart algebra A, we denote it by H‚pE, Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The isotropy Lie 8-algebra of a singular foliation We assume that pM, E, Qq is universal for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that the Lie 8-algberoid obtained by specialising at some point m P M does not induce directly a Lie 8-algberoid on the graded space HpF, mq but the 2-ary bracket t¨ , ¨ u2 goes to quotient directly on elements of degree ´1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to H´1pF, mq, because tdp2q m px1q, x2u2 “ dp2q m ptx1, x2u2q for all x1 P E´2|m and x2 P kerpρmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' That endows H´1pF, mq with Lie algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Androulidakis and Skandalis isotropy Lie algebra gm “ Fpmq ImF of the singular foliation F at a point m P M, is isomorphic to H´1pF, mq equipped with the induced Lie algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For m P M, we construct a Lie algebra isomorphism ζ : kerpρmq impdp2q m q Ñ gm as follows: For an element u P kerpρmq, let ru be an extension of u to a local section on E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, one has ρpruq P Fpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let rρm be the surjective linear map defined by rρm : kerpρmq ÝÑ gm, u ÞÝÑ rρpruqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since any other extension ru for u differs from the first one by a section in ImΓpE´1q, the map rρm is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Surjectivity is due to the fact that every vector field of F vanishing at m P M is of the form ρpeq with e a (local) section of E´1 whose value at m belongs to kerpρmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In addition, it is not hard to see that rρm is a morphism of brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It remains to show that kerprρmq “ impdp2q m q: let u P kerprρmq Ă kerpρmq and ru be a local section of E´1 that extends u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By definition of u, the class of ρpruq is zero in gm, therefore, there exists some functions fi P Im and Xi P F, i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , k, local generators such that ρpruq “ kÿ i“1 fiXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that, ρpru ´ kÿ i“1 fieiq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 114 where for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , k, ei is a (local) section of E´1 whose image through ρ is Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since pE‚, d‚, ρq is a geometric resolution, there exists a (local) section q P ΓpE´2q such that ru “ kÿ i“1 fiei ` dp2qq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) By evaluating Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) at m, we find out that u P impdp2q m q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conversely, for v P E´2|m, choose a (local) section q of E´2 through v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, dp2qq P ker ρ, is a (local) extension of dp2q m v P impdp2q m q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The image of dp2q m v through rρm is obviously zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves that kerprρmq “ impdp2q m q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, if the underlying complex of pM, E, Qq is minimal at m then, for every i ě 2, the vector space H´ipF, mq is canonically isomorphic to E´i|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, H´1pF, mq is canonically isomorphic to kerpρmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, E, Qq be a universal Lie 8-algebroid of a singular foliation F whose underly- ing complex is minimal at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, HpF, mq carries a Lie 8-algebra structure given by IstropympE, Qq called the isotropy Lie 8-algebra of the singular foliation F at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can show that this definition is independent of any choices made in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, the isotropy Lie algebra of the singular foliation F at a point m P M in the sense of Androulidakis and Skandalis, is isomorphic to the degree minus one component H´1pF, mq » kerpρmq of the isotropy Lie 8-algebra of F at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For an arbitrary Lie-Rinehart algebra A and a maximal ideal I, it is still true that pH‚pA, Iq, d “ 0q is quasi-isomorphic to pE|I, ℓ1|Iq and it is still true that the bracket of the universal Lie 8-algrbroid over O constructed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 restricts to a 8-algebra structure on pE|I, ℓ1|Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Homotopy transfer, this Lie 8-algebra can be transferred to pH‚pA, Iq, d “ 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is not clear whether Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 holds true or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, Fq be a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, Qq be a universal Lie 8-algebroid of F and let pE, d, ρq : ¨ ¨ ¨ dp4q ÝÑ E´3 dp3q ÝÑ E´2 dp2q ÝÑ E´1 ρ ÝÑ TM (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) be its linear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all m P M, we have kerpρmq impdp2q m q » gm as Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The subset of regular points of F in M satisfies Mreg “ tm P M | rkpdp2q m q “ dimpker ρmqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is open and dense in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The restriction of the foliation F to Mreg is the set of sections of a subbundle of TM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', is a regular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If pE, d, ρq is of finite length, then the regular leaves have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is proved in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 (see also Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14 [LLS20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For items 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' we refer the reader to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 of [AS09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 115 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 A blow-up procedure for a singular foliation We present in this section an interpretation of a blow-up procedure invented by Mohsen in [Moh21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Grassmann bundle For E a finite dimensional vector space, we denote by Gr´ℓpEq the set of all codimension ℓ P N vector subspaces in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recall a few facts on Gr´ℓpEq see [KES00, Joe92, LB15], our main reference is [vIR94]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is a metric space for the distance dpV, V 1q “ ∥PV ´ PV 1∥, where PV stands for the orthogonal projection of E onto V Ă E with respect to an arbitrary metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The topology does not depend on the metric and makes Gr´ℓpEq a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is moreover a compact manifold and an projective variety1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Affine coordinates charts: One of the ways of defining the standard affine coordinates on the Grassmanian Gr´ℓpEq is as follows (Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24 of [vIR94]): Fix a basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ed“dim E for E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any element V P Gr´ℓpEq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a vector subspace V Ă E of codimension ℓ, can be viewed as a d ˆ pd ´ ℓq matrix MV “ pC1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Cd´ℓq whose columns are formed by linearly independent column vectors obtained by choosing a basis for V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The homogeneous coordinates of V in Gr´ℓpEq are the components of the n ˆ pd ´ ℓq matrix MV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any other choice of basis for V gives another maximal rank matrix M1 V and an invertible pd ´ ℓq ˆ pd ´ ℓq-matrix P P GLpd ´ ℓ, Kq such that MV “ M1 V ˝ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, since MV has full rank, there exists a family of integers 1 ď i1 ă ¨ ¨ ¨ ă id´ℓ ď d, such that MV is equivalent to a matrix M1 V whose submatrix made of the rows i1, ¨ ¨ ¨ , id´ℓ is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For example, if the first d ´ ℓ rows of MV are linearly independent, then the matrix is equivalent to the matrix ˜ Id´ℓ A ¸ where A “ paijq is a ℓ ˆ pd ´ ℓq-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, V admits a basis of the form vj :“ ej ` ℓÿ k“1 akjek, j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d ´ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) V is completely determined by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One define an atlas on Gr´lpEq as follows: Consider the map ψ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',d : Ml,d´lpKq ÝÑ Md,d´lpKq A ÞÝÑ ˜ Id´ℓ A ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1For the notion of projective variety and notations, we refer the reader e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to the book [vIR94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 116 For a permutation σ P Sd, we define the map ψσp1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',σpdq that associates every A P Ml,d´lpKq the matrix in Md,d´lpKq that permutes the lines of ˜ Id´ℓ A ¸ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t σ, that is to say ψσp1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',σpdqpAq :“ Ppσq ˝ ψ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',dpAq where Ppσq is the permutation matrix associated to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Statement: For every ordered integers 1 ď i1 ă ¨ ¨ ¨ ă id´ℓ ď d, the coordinates chart on Gr´ℓpEq is the open set Ui1,¨¨¨ ,id´ℓ Ă Gr´ℓpEq consisting of all sub-vector space of E such that for every basis the submatrix which is made of the rows i1, ¨ ¨ ¨ , id´ℓ is invertible, and the coordinate map is ψσp1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',σpdq with σ is a permutation sending 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d ´ ℓ on i1, ¨ ¨ ¨ , id´ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Grassmann bundle: For E´1 Ñ M a vector bundle of rank d, the disjoint union: Gr´ℓpE´1q :“ ž mPM Gr´ℓpE´1|mq comes equipped with a natural manifold structure and Π: Gr´ℓpE´1q ÝÑ M is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is called the Grassmann pd ´ ℓq-plane bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To fix some notations, the fiber at m P M is Π´1pmq “ tpV, mq | V P Gr´ℓpE´1|mqu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every open subset U Ă M on which E´1 is trivial, Π´1pUq » Gr´ℓpRdq ˆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 A blow-up procedure Settings: In what follows, we are given a foliated manifold pM, Fq with M connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We assume that a geometric resolution of finite length exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Under these assumptions, all the regular leaves have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Mreg the set of the regular points of pM, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Denote by ℓ the common dimension of the regular leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For most of the present discussion, the whole geometric resolution is not needed: is it sufficient to assume that there exists vector bundles E´1, E´2 and that all regular leaves have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE‚, ℓ‚, ρq be a universal Lie 8-algebroid of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the underlying geometric resolution pE, d, ρq : ¨ ¨ ¨ ℓ1“dp4q ÝÑ E´3 ℓ1“dp3q ÝÑ E´2 ℓ1“dp2q ÝÑ E´1 ρ ÝÑ TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) Notice that for every point m P Mreg, dimpdp2q m q “ dim kerpρmq “ rkpE´1q ´ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, there exists a natural section of Π on Mreg defined by: σ: Mreg ÝÑ Gr´ℓpE´1q, m ÞÝÑ impdp2q m q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 117 We consider Ă M :“ σpMregq the closure of the graph of σ in Gr´ℓpE´1q, and denote by π the restriction of Π to Ă M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Recall from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 that evpE, mq : ¨ ¨ ¨ ÝÑ E´3|m dp3q m ÝÑ E´2|m dp2q m ÝÑ kerpρmq comes equipped with a Lie 8-algebra pt¨ ¨ ¨ ukqkě1 whose unary bracket is dp‚q m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The projection map π: Ă M Ă Gr´ℓpE´1q ÝÑ M, pV, mq ÝÑ m (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) fulfills the following properties 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For each point m P M, the set π´1pmq “ " V Ă E´1|m ˇˇˇˇ D pmnq P MN reg, such that, impdp2q mnq ÝÑ nÑ`8 V as mn ÝÑ nÑ`8 m is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all m P M, and V P π´1pmq one has, (a) impdp2q m q Ď V Ď kerpρmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) The 2-ary bracket t¨ , ¨ u2 on ker ρm restricts to V and the image of V in kerpρmq impdp2q m q » gm, is a Lie subalgebra of codimension ℓ ´ dimpLmq, where Lm is the leaf through m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all m P Mreg, π´1pmq “ kerpρmq “ impdp2q m q is reduced to a point in Gr´ℓpRrkpE´1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ă M does not depend on the choice of a geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The projection π: Ă M Ñ M is proper and onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove item 1 for m P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By compactness of the Grassmanian manifold Gr´ℓpE´1|mq » Gr´ℓpRrkpE´1qq, out of any sequence in MN reg that converges to m, we can extract a sequence n ÞÑ κ ´ impdp2q mnq ¯ P Gr´ℓpE´1|mq that has a limit, where κ is a local trivialization of the vector bundle E´1 which identifies the fibers E´1|m and E´1|mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='paq: Let V P π´1pmq and pmnq P MN reg such that mn ÝÑ nÑ`8 m and impdp2q mnq ÝÑ nÑ`8 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let v P impdp2q m q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have v “ dp2q m u for some u P E´2|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Choose a (local) section ru of E´2 through u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It implies that dp2q mnrupmnq ÝÑ nÑ`8 dp2q m u, hence dp2q m u P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, impdp2q m q Ď V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any element v P V , there exists a sequence vn P kerpρmnq “ impdp2q mnq, n P N that converges to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, ρmnpvnq “ 0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, by continuity, one has v P kerpρmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To prove item pbq, choose a local frame e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , er, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' eℓ`r of E´1 such that e1pmq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , erpmq is an orthogonal basis of V for an arbitrary Hermitian structure on E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For i, j P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , l ` ru, let pck ijq P OpUq be a family of functions over U such that ℓ2pei, ejq “ l`r ÿ k“1 ck ijek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 118 In particular, for every ti, ju Ă t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , l ` ru, teipmq, ejpmqu2 “ l`r ÿ k“1 ck ijpmqekpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let v1, v2 P V with v1 “ rÿ i“1 αieipmq and v2 “ rÿ i“j βieipmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There exists sequences vn 1 “ r`l ÿ i“1 αi neipmq ÝÑ nÑ`8 v1 and vn 2 “ r`l ÿ i“1 βi neipmq ÝÑ nÑ`8 v2 with vn 1 , vn 2 P ker ρxn, for all n P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, the sequences pαi nq, pβi nq P KN with i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , r ` lu satisfy αi n ÝÑ nÑ`8 αi, βi n ÝÑ nÑ`8 βi (αi “ βi “ 0 for i ě r ` 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every n P N we have, r`l ÿ i,j,k“1 αi nβj nck ijpmnqekpmnq “ tvn 1 , vn 2 u2 P ker ρxn “ impdp2q mnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) Since r`l ÿ i,j,k“1 αi nβj nck ijpmnqekpmnq ÝÑ nÑ`8 r`l ÿ i,j,k“1 αiβjck ijpmqekpmq “ tv1, v2u2, one has, tv1, v2u2 P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is a consequence of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='paq, since m P Mreg if and only if kerpρmq “ impdp2q m q (by item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' follows from the existence of homotopy equivalence between any two geometric resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' follows from the fact that the projection Π is with compact fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The corollary below is a direct consequence of item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is a natural inclusion Ă M ãÑ ž mPM Grℓ´dimpLmqpgmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let m P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since elements V P π´1pmq satisfy impdp2q m q Ď V Ď kerpρmq, they correspond injectively to a (unique) sub-vector space of codimension ℓ ´ dim Lm in gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, this implies π´1pmq ãÑ Grℓ´dimpLmqpgmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every m P M, the set Km “ tV P Gr´ℓpE´1|mq | V Ď ker ρmu Ă Gr´ℓpE´1|mq is locally an affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ed be a local frame of E´1 on an open subset U Ă M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has, ρpeiq “ dim M ÿ k“1 fk i B Bxk P F, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 119 for some local functions fk i P C8pUq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Without any lost of generality, consider for example the standard coordinates chart U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',d´ℓ for the grassmanian Gr´ℓpE´1|mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let V P U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',d´ℓ and let paijq be the homogeneous coordinates of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Define the sections rvj :“ ej ` ℓÿ k“1 akjek, j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d ´ ℓ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) By construction, the rvj’s, evaluated at m, form a basis for V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have, ρprvjq “ dim M ÿ k“1 ˜ fk j ` ℓÿ s“1 asjfk s ¸ B Bxk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' V Ď ker ρm if and only if fk j pmq ` ℓÿ s“1 asjfk s pmq “ 0, j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d ´ ℓ k “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , dim M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) Therefore, Km is defined by the polynomial Equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If F is a polynomial singular foliation on M P ␣ CN, RN( , then Ă M is a locally affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can choose a polynomial geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Denote by x “ px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xNq the local coordinates on W Ď M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We are using notations of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13), the functions fk i are polynomial in px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By item 1, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2, every element V of Ă M “ Ť xPM π´1pxq is obtained as a limit impdp2q xn q ÝÑ nÑ`8 V P π´1pxq with pxnq P MN reg, such that, xn ÝÑ nÑ`8 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Fix n P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the notations of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4, take V “ impdp2q xn q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, in the coordinate chart U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',d´ℓ (see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1), the coordinates pan ijq of impdp2q xn q satisfy the polynomial equations fk j pxnq ` ℓÿ s“1 an sjfk s pxnq “ 0, j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d ´ ℓ k “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) One has, fk j pxnq ` ℓÿ s“1 an sjfk s pxnq “ 0 ÝÑ nÑ`8 fk j pxq ` ℓÿ s“1 asjfk s pxq “ 0, where for all s, j, an sj ÝÑ nÑ`8 asj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, Ă M is given in local coordinates WˆU1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',d´ℓ by elements that satisfy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and that are limit of elements of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) in nearby regular points, elements which are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, it is, on this affine variety, the irreducible components of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) that projects onto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 120 The pull-back of F to Ă M: Let X P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There exists a section υ of the vector bundle p: E´1 Ñ M such that ρpυq “ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the linear vector field r X P XpE´1q defined as follows r Xrp˚fs :“ p˚pρpυqrfsq, @ f P C8pMq, x r Xrαs, ey :“ ρpυqrxα, eys ´ xα, ℓ2pυ, eqy, @ α P ΓpE˚q, e P ΓpE´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that r X depends on the choice of the almost Lie algebroid bracket ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following items hold (see [LGLR22], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The flow φ r X t : E´1 Ñ E´1 of r X when it is defined, is a vector bundle isomorphism over φX t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The diagram below commutes, E´1 φĂ X � ρ � E´1 ρ � TM dφX t � TM (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17) where φX t is the flow of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, φ r X t preserve the grassmanian Gr´ℓpE´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every X P F, choose υ P ΓpE´1q such that ρpυq “ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The vector field r X induces a vector field Ă Ă X on Gr´ℓpE´1q such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ă Ă X is tangent to Ă M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ă Ă X projects onto X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since for every x P M, φ r X t |x is an isomorphism of E´1|x, therefore φ r X t |x preserves Π´1pxq “ Gr´ℓpE´1|xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define Ă Ă X for pV, xq P Gr´ℓpE´1q by Ă Ă XpV q :“ d dt|t“0 ´ φ r X t |x ¯ |V P TV Gr´ℓpE´1|xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18) This shows item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show item 1, φ r X t preserves Ă M: to see this take pV, xq P Ă M, let xn ÝÑ nÑ`8 x be such that imdp2q xn ÝÑ nÑ`8 V with pxnq Ă Mreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every n P N0, one has φ r X t |xn ´ imdp2q xn ¯ “ imdp2q φX t pxnq, since ρ ˝ φ r X t “ dφX t ˝ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, φ r X t |xpV q “ lim nÑ`8 φ r X t |xn ´ imdp2q xn ¯ “ lim nÑ`8 ´ imdp2q φX t pxnq ¯ P π´1 ` φX t pxq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By consequence, for V P Ă M, we have Ă Ă XpV q P TV Ă M, by Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 121 Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ă Ă X does not depend on the choice of the almost bracket ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is an obvious consequence of item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6, since π: Ă M ÝÑ M is invertible on an open (dense) subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Denote by rF the module generated by the Ă Ă X’s, with X P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For F polynomial, rF is a singular foliation on the locally affine variety Ă M, because r X is still polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If F is a polynomial singular foliation, then it is lifted to a projective singular foliation rF on the locally affine variety Ă M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The same construction would apply if, instead of considering impdp2qq Ď E´1, we consider impdpi`1qq Ď E´i or impρq Ď TM for i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Denote by Ă Mi the transformation that we would obtain by such a construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A lift rFi of F can be defined exactly in the same manner by considering a lift r X of X P F on E´i (or TM) associated to ℓ2 : ΓpE´1qˆΓpE´iq Ñ ΓpE´iq or to rX, ¨ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, consider for i ě 0, the Grassmann bundle Gr´ℓipE´iq, with E0 :“ TM, where ℓi :“ rkpdi`1 m q (with the understanding that dp1q “ ρ for i “ 0) at regular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These bundles are manifolds that project to M through a proper map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Over Mreg, there exists a unique V P Gr´ℓipE´iq such that dpiq m pV q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theses define maps σi : Mreg ãÝÑ Gr´ℓipE´iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define Ă Mi :“ σipMregq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is not a manifold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, it is a locally affine variety if F is a polynomial singular foliation on RN or, CN as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' independent of the choice of a geometric resolution, as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the projection Ă Mi Ñ M is proper and onto, as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' F lifts to a singular foliation on Ă Mi, and it is one-to-one to Mreg, as in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For i “ 0 the same conclusion holds for σ0 : M ÝÑ Grℓ0pTMq m ÞÝÑ TmF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us give some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that Gr´ℓpCℓq “ tptu, so that if dpiq is into on an open dense subset, the construction degenerates, in this case Gr´ℓpCℓq ˆ M » M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If F is a projective singular foliation, then Ă M1 » M: because ΓpE´1q » F and E´1 ρÑ TM is injective on the open dense subset Mreg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ℓ0 “ rkpE´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence Gr´ℓ0pE´1q » M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If F admits open dense orbits, Ă M0 » M, since Gr´ dim MpTMq » M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W Ď M “ Cd be an affine variety generated by (independent functions) ϕ, ψ P Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' IW “ xϕ, ψy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let FW be the singular foliation of (polynomial) vector fields vanishing on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' At regular points x P Mreg, TxFW “ d, hence CHAPTER 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ISOTROPY LIE ALGEBRAS OF A SINGULAR FOLIATION 122 (a) Ă M0 » M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) Also, E´1 “ pCd ‘ Cdq ˆ M, ρ: ei ‘ 0 ÞÑ ϕ B Bxi and 0 ‘ ei ÞÑ ´ψ B Bxi , for all i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' E´2 “ Cd ˆ M, and dp2q : ei ÞÑ ψ K ei ‘ ϕ K ei with, K “ gcdpϕ, ψq impdp2q x q “ ker ρx “ B ψpxq Kpxqu ‘ ϕpxq Kpxqu, with u P Cd F For a convergent sequence pynq in MzW, pker ρynq converges if and only if r ψpynq Kpynq : ϕpynq Kpynqs converges in P1pCq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, Ă M1 is the closure of the graph tpy, rψpyq: ϕpyqsq, y P MzWu, which is the blow up of Cd along W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ă M2 » M since dp2q x is injective at regular points, so that Gr´dpE´2q » M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W be the affine variety defined by ϕ P Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the singular foliation Fϕ “ tX P XpCdq | Xrϕs “ 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every y P Cd, pTyFϕqK “ x∇yϕy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For convergent sequence yn ÝÑ nÑ`8 y P W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The sequence impρynq converges if and only if ∇ynϕ converges in Gr´pd´1qpCdq, that is ” Bϕ Bx1 pynq: ¨ ¨ ¨ : Bϕ Bxd pynq ı converges in the projective space Pd´1pCq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, Ă M0 is the closure of the graph of the map, y ÞÑ py, ” Bϕ Bx1 pyq: ¨ ¨ ¨ : Bϕ Bxd pyq ı q which is the blow up of Cd along the singular locus of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us conclude this chapter by an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Can we desingularize a singular affine variety W Ď Cd by applying the constructions above to the singular foliation F “ XpWq of vector fields tangent to W?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We should then understand Ă W0 as the Nash modification of W [LU81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The meaning of Ă W1 is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We would like to relate the Ă Wi’s and the rFi together and go further in the universal Lie 8-algebroid, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to understand the role of the 3-ary bracket in this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conclusion: In this chapter, we recall the notion of isotropy Lie (8)-algebra of a singular foliation (and of a Lie-Rinehart algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, we use the geometric resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the resolution on which the universal Lie 8- algebroid is built) to recover several notions of resolution of singularities: one being due to Nash and a second one to Mohsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8 Symmetries of singular foliations through Lie 8-algebroids This chapter is one of the main application of results in Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 and Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These results are taken from my article [Lou22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We introduce the notion of weak symmetry actions of a lie algebra g on a singular foliation F and study the interaction of those on the universal Lie 8-algebroids of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, in the subsequent chapter, we apply these results to the problem of extending a strict Lie algebra action on a sub-affine variety on the ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Convention 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Throughout this chapter, M stands for a smooth or complex manifold, or an affine variety over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote the sheaf of smooth or complex, or regular functions on M by O and the sheaf of vector fields on M by XpMq, and Xrfs stands for a vector field X P XpMq applied to f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, K stands for R or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Definitions and examples Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F Ă XpMq be a singular foliation over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A diffeomorphism φ: M ÝÑ M is said to be a symmetry of F, if φ˚pFq “ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A vector field X P XpMq is said to be an infinitesimal symmetry of F, if rX, Fs Ă F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Lie algebra of infinitesimal symmetries of F is denoted by spFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, F Ă spFq, since rF, Fs Ă F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [AS09, GY18] Let M be a smooth or complex manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The flow of an infinites- imal symmetry of F, if it exists, is a symmetry of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As we will see later, one of the consequences of our future results is that any symmetry X P spFq of a singular foliation F admits a lift to a degree zero vector field on any universal NQ-manifold over F that commutes with the homological vector field Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This allows us to have an alternative proof and 123 CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS124 interpretation of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 (see Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pg, r¨ , ¨sgq be a Lie algebra over K “ R or C, depending on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From now on and in the sequel g is concentrated in degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A weak symmetry action of the Lie algebra g on a singular foliation F on M is a K-linear map ϱ: g ÝÑ XpMq that satisfies: @ x P g, rϱpxq, Fs Ď F, @ x, y P g, ϱprx, ysgq ´ rϱpxq, ϱpyqs P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When x ÞÝÑ ϱpxq is a Lie algebra morphism, we speak of strict symmetry action of g on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is an equivalence relation on the set of weak symmetry actions which is defined as follows: two weak symmetry actions, ϱ, ϱ1 : g ÝÑ XpMq are said to be equivalent if there exists a linear map ϕ: g ÝÑ F such that ϱ ´ ϱ1 “ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is important to notice that when F is a regular foliation and M{F is a manifold, any weak symmetry action of a Lie algebra g on F induces a strict action of g over M{F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 is a way of extending this idea to all singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is a list of some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let π: M ÝÑ N be a submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since any vector field on N comes from a π- projectable vector field on M, therefore any Lie algebra morphism g ÝÑ XpNq can be lifted to a weak symmetry action g ÝÑ XpMq on the regular foliation Γpker dπq, and any two such lifts are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Furthermore, any weak action of a Lie algebra g on a singular foliation F on N can be lifted to a class of weak symmetry actions on the pull-back foliation π´1pFq, (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9 in [AS09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any point m P M, consider gm “ Fpmq ImF the isotropy Lie algebra of F at m (see Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us denote its Lie bracket by r¨ , ¨sgm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider ϱ: gm Ñ Fpmq Ă XpMq a section of the projection map, ImF� � � Fpmq � � gm ϱ � (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) Then, rϱpxq, ImFs Ă ImF and ϱprx, ysgmq ´ rϱpxq, ϱpyqs P ImF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, the map ϱ: gm Ñ XpMq is a weak symmetry action of the singular foliation ImF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A different section ϱ1 of the projection map yields an equivalent weak symmetry action of gm on ImF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' An obstruction class for having a strict symmetry action equivalent to ϱ will be given later in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, for k ě 1, let us denote by gk m the isotropy Lie algebra of the singular foliation Ik mF at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any section ϱk : gk m ÝÑ XpMq of the projection map Ik`1 m F� � � IkF � � gk m ϱk � (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) is a weak symmetry action of the Lie algebra gk m on the singular foliation Ik`1 m F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS125 Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pA, r¨ , ¨sA , ρq be an almost Lie algebroid on a smooth manifold M, and let F “ ρpΓpAqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume there exists p P M such that A|p is a Lie algebra and ρ|p “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The question of finding a map A|p ÝÑ ΓpAq such that the composition A|p ÝÑ ΓpAq ρ ÝÑ XpMq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) be a Lie algebra morphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', to ask whether the singular foliation F comes from the transformation Lie algebroid of A|p, can be formulated as a weak symmetry action of A|p on the singular foliation IpF as follows: Consider a linear map A|p ÝÑ ΓpAq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) a ÞÝÑ ra such that for all a P A|p and ra a section of A such that ra|p “ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easily checked that the map in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) is not a Lie algebra morphism, but it satisfies Ć ra, bsA|p ´ ” ra,rb ı A P IpΓpAq and ” Ą A|p, IpΓpAq ı Ă IpΓpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, the map a ÞÝÑ ρpraq is a weak symmetry action of A|p on IpF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that the isotropy Lie algebra g1 p “ IpF I2pF of the singular foliation IpF, is Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In Chapter 9, we show that in this case, the obstruction of having a Lie algebra morphism in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) is a cocycle of Chevalley-Eilenberg of A|p valued in g1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The next example comes from [LGR22], and follows the same patterns as in Examples 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is based on the notion of Ehresmann connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us first recall quickly this concept for the sake of completeness and clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are several equivalent manners of viewing Ehresmann connections (see [Iva93] Section 9, Page 76), the most relevant approach in this context is the following: An Ehresmann connection on a smooth fiber bundle1 π: E Ñ M is a vector subbundle H of TE, such that TE “ H ‘V , where V :“ tξ P TE | π˚pξq “ 0u is called the "vertical bundle" whose fiber at e P E is Ve “ TepEπpeqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The subbundle H is called the "horizontal bundle".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a connection as defined above, for every e P E, the linear map π˚ : TeE Ñ TπpeqM restricts to an isomorphism, He Ñ TπpeqM, whose inverse TπpeqM Ñ He is called the horizontal lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, Fq be a singular foliation on a smooth manifold M and L Ă M a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let rL, Ms be a neighborhood of L in M equipped with some projection2 π: rL, Ms Ñ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' According to [LGR22], upon replacing rL, Ms be a smaller neighborhood of L if necessary, there exists an Ehresmann connection (that is a vector sub-bundle H Ă TrL, Ms with H ‘ kerpπ˚q “ TrL, Ms) which satisfies that ΓpHq Ă F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Such an Ehresmann connection is called an Ehresmann F-connection and induces a C8pLq-linear section ϱH : XpLq Ñ Fproj of the surjection Fproj Ñ XpLq, where Fproj stands for vector fields of F π-projectable on elements of XpLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The section ϱH is a weak symmetry action of XpLq on the transverse foliation T :“ Γpker π˚q X F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When the Ehresmann connection H is flat, ϱH is bracket-preserving, and defines a strict symmetry of XpLq on the transverse foliation T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1this generalizes connections to arbitrary fiber bundle π : E Ñ M, here the total space E is itself a smooth manifold, and has its own tangent bundle TE Ñ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2such projection comes with the Tubular neighborhood theorem [dS01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS126 Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider, for a fixed k, the singular foliation Fk :“ Ik 0 XpRdq generated by all vector fields vanishing to order k at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The action of the Lie algebra glpRdq on Rd which is given by, glpRq ÝÑ XpRdq, paijq1ďi,jďd ÞÝÑ ÿ 1ďi,jďd aijxi B Bxj is a strict symmetry of Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ϕ :“ pϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ϕrq be a r-tuple of homogeneous polynomial functions in d variables over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the singular foliation Fϕ (see [LGL22b] Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) which is generated by all polynomial vector fields X P XpKdq that satisfy Xrϕis “ 0 for all i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The action K Ñ XpKdq, λ ÞÑ λÝÑ E , is a strict symmetry of Fϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here ÝÑ E stands for the Euler vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W Ă Cd be an affine variety and IW Ă Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds its corresponding ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us denote by XpWq :“ DerpCrx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds{IW q the Lie algebra of vector field of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any vector field on W can be extended (not unique) to a vector field on Cd (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let FW :“ IW XpCdq the singular foliation made of vector fields vanishing on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since every vector field on W can be extended to a vector field on Cd tangent to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any Lie algebra morphism ϱ: g ÝÑ XpWq extends to a linear map rϱ: g ÝÑ XpCdq that makes this diagram commutes XpCdq �� g rϱ � ϱ � XpWq This extension rϱ is a weak symmetry action of g on FW over the ambient space Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Two different extensions yield equivalent symmetry actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 A Lie 8-morphism lifting a weak symmetry of a foliation We recall that O is the sheaf of smooth/complex functions on a smooth/complex manifold M, or the algebra of regular functions on an affine variety over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader to the Chapters 6 and 4 for the notion of (universal) Lie 8-algebroid of a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote them by pE, Qq and their functions by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For the sake of clarity, let put this chapter in context, and fix some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pg, r¨ , ¨sgq a Lie algebra and pE, Qq a Lie 8-algebroid over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the sequel, the Lie algebra g is concentrated in degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The differential graded Lie algebra pXpEq, r¨ , ¨s , adQq of vector fields on E is shifted by 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a derivation of degree k in XkpEq is of degree k ´ 1 as an element of the shifted space XkpEqr1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The graded symmetric Lie bracket on XpEqr1s is of degree `1 and given on homogeneous elements u, v P XpEqr1s as tu, vu :“ p´1q|v|ru, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the sequel, we write pXpEqr1s, r¨ , ¨s , adQq instead of pXpEqr1s, t¨ , ¨u, adQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS127 Let pS‚ Kg, Qgq respectively pS‚ KpXpEqr1sq, ¯Qq be the corresponding formulations in terms of co- derivations of the differential graded Lie algebras pg, r¨ , ¨sgq and pXpEqr1s, r¨ , ¨s , adQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Precisely, Qg is the co-derivation defined by putting for every homogeneous monomial x1 ^ ¨ ¨ ¨ ^ xk P Sk Kg, Qgpx1 ^ ¨ ¨ ¨ ^ xkq :“ ÿ 1ďiăjďk p´1qi`j´1rxi, xjsg ^ x1 ^ ¨ ¨ ¨ pxi ¨ ¨ ¨ pxj ¨ ¨ ¨ ^ xk, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) and ¯Q “ ¯Qp0q ` ¯Qp1q is the co-derivation of degree `1 where the only non-zero Taylor coefficients are, ¯Qp0q : S1 KpXpEqr1sq adQ ÝÑ XpEqr1s and ¯Qp1q : S2 KpXpEqr1sq t¨ ,¨u ÝÑ XpEqr1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following is a particular case of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5, see also [LM94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Lie 8-morphism3 Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq is a graded coalgebra morphism ¯Φ: pS‚ Kg, Qgq ÝÑ pS‚ K pXpEqr1sq , ¯Qq of degree zero which satisfies, ¯Φ ˝ Qg “ ¯Q ˝ ¯Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) In order words, it is the datum of degree zero linear maps ´ ¯Φk : Sk`1 K g ÝÑ X´kpEqr1s ¯ kě0 that satisfies ÿ 1ďiăjďn`2 p´1qi`j´1 ¯Φnprxi, xjsg, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , pxij, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn`2q “ rQ, ¯Φn`1px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xn`2qs ` ÿ i ` j “ n i ď j σ P Si`1,j`1 ϵpσqr¯Φipxσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xσpi`1qq, ¯Φjpxσpi`2q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xσpn`2qqs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) where pxij means that we take xi, xj out of the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When there is no risk of confusion, we write Φ for ¯Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is important to notice that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 and Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14 are compatible when M “ tptu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, morphisms in both sense match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In [MZ12], Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 corresponds to the definition of actions of a Lie 8-algebras of finite dimension on Lie 8-algebroids of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here we only have a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In contrast to theirs, we do not assume that g is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows from the axioms (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) that for all x, y P g, rQ, Φ0pxqs “ 0 and Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqs “ rQ, Φ1px, yqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) In particular, if the homological vector field Q vanishes at some point m P M, then the map x ÞÝÑ pP P XpEq, P|m ÞÑ rΦ0pxq, Ps|mq endows the vector space XpEq|m » pSpE˚q b Eq|m with a g-module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, the restriction of the map Φ1 : ^2 g ÝÑ X´1pEq|m at m is a 2-cocycle of Chevalley-Eilenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, Qq be a Lie 8-algebroid and F its basic singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any Lie 8- morphism Φ: pg, r¨ , ¨sgq ÝÑ pX‚pEqr1s, r¨ , ¨s , adQq gives a weak symmetry action of g on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If Q|m “ 0 for some point m P M, the g-action defined in Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3, is independent of the equivalence class of the weak symmetry action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3Here, we use "ù" to emphasize that Φ is not a DGLA morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS128 The following lemma explains what the 0-Taylor coefficient of a Lie 8-morphism as in Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 induces on the linear part of pE, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More details will be given in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 and Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The 0-th Taylor coefficient Φ0 : g ÝÑ X0pEq induces 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a linear map ϱ: g ÝÑ XpMq, x ÞÝÑ pϱpxqrfs :“ Φ0pxqrfs, f P Oq and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a linear map x P g ÞÝÑ ∇x P Der0pEq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for each x P g, ∇x : E ÝÑ E is a degree zero map that satisfies ∇xpfeq “ f∇xpeq ` ϱpxqrfse, for f P O, e P ΓpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' such that xΦ0pxqp0qpαq, ey “ ϱpxqrxα, eys ´ xα, ∇xpeqy, for all α P ΓpE˚q, e P ΓpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) Φ0pxqp0q stands for the polynomial-degree zero of Φ0pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7, we have for every x P g, and e P ΓpEq, rΦ0pxq, ιesp´1q “ ι∇xe, for some K-bilinear map ∇x : ΓpE´‚q ÝÑ ΓpE´‚q that depends linearly on x P g and that satisfies ∇xpfeq “ f∇xpeq ` ϱpxqrfse, for f P O, e P ΓpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) To see (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10), compute rΦ0pxq, ιfesp´1q: ι∇xpfeq “ rΦ0pxq, ιpfeqsp´1q “ Φ0pxqrfsιe ` frΦ0pxq, ιesp´1q “ ιϱpxqrfse`∇xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, one has for all α P ΓpE˚q, e P ΓpEq, xΦ0pxqp0qpαq, ey “ Φ0pxqp0qrxα, eys ´ rΦ0pxqp0q, ιesp´1qpαq “ ϱpxqrxα, eys ´ xα, ∇xpeqy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Homotopies The following is a particular case of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14 of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We rewrite it in this special context for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ¯Φ, ¯Ψ: pS‚ Kg, Qgq ù ` S‚ KpXpEqr1sq, ¯Q ˘ be Lie 8-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say ¯Φ, ¯Ψ are homotopic over the identity of M if the following conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' there a piecewise rational continuous path t P ra, bs ÞÑ Ξt : pS‚ Kg, Qgq ù ` S‚ KpXpEqr1sq, ¯Q ˘ made of Lie 8-morphisms that coincide with ¯Φ and ¯Ψ at t “ a and b, respectively, CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS129 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' and a piecewise rational path t P ra, bs ÞÑ Ht of Ξt-co-derivations of degree ´1 such that dΞt dt “ ¯Q ˝ Ht ` Ht ˝ Qg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Homotopy equivalence in the sense of the Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 is an equivalence rela- tion, and it is compatible with composition of Lie 8-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, we "glue" infinitely many equivalences, as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Convention 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the sequel, Qg and ¯Q will be in implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 is slightly more general than the equivalence relation [MZ12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In [MZ12], it is explained that Lie 8-oid morphisms are Maurer-Cartan elements in some Lie 8-algebroid g‘E of certain form, and they define equivalence as gauge-equivalence of the Maurer-Cartan elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This gauge equivalence corresponds to homotopies as above for which all functions are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, we do not require nilpotence unlike in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 of [MZ12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Last, we do not assume g to be of finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let g be a Lie algebra and pE, Qq a Lie 8-algebroid over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any Lie 8-morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq induces a weak symmetry action of g on the basic singular foliation F “ ρpΓpE´1qq of pE, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Homotopic Lie 8-morphisms Φ, Ψ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq induce equivalent weak symmetry actions ϱa, ϱb of g on the basic singular foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is a consequence of Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, take ϱ: g ÝÑ XpMq as in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=') We claim that ϱ is a weak symmetry action of g on F: Let x, y P g, and e P ΓpE´1q and f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' rΦ0pxq, Qs “ 0 entails, A Φ0pxqp0q ” Qp1qpfq ı , e E “ A Qp1q ´ Φ0pxqp0qrfs ¯ , e E ϱpxqrxQrfs, eys ´ xQrfs, ∇xpeqy “ ρpeqrϱpxqs, (by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=')) ϱpxqrρpeqsrfs ´ ρp∇xpeqqrfs “ ρpeqrϱpxqs By consequence, rϱpxq, ρpeqs “ ρp∇xpeqq P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, rϱpxq, Fs Ď F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7, there exists a skew-symmetric linear map η: ^2 g ÝÑ ΓpE´1q such that Φ1px, yqp´1q “ ιηpx,yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, the polynomial-degree zero of Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) evaluated at an arbitrary function f P O yields: Φ0prx, ysgqp0qpfq ´ rΦ0pxq, Φ0pyqsp0qpfq “ rQ, Φ1px, yqsp0qpfq ùñ Φ0prx, ysgqpfq ´ ” Φ0pxqp0q, Φ0pyqp0qı pfq “ ” Qp1q, Φ1px, yqp´1qı pfq ùñ ϱprx, ysgqrfs ´ rϱpxq, ϱpyqsrfs “ ” Qp1q, ιηpx,yq ı pfq “ ρpηpx, yqqrfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS130 Since f is arbitrary, this proves item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17, Φ „ Ψ implies for x P g that Ψpxq ´ Φpxq “ ¯Q ˝ Hpxq ` \x18\x18\x18\x18\x18 \x18 H ˝ Qgpxq “ rQ, Hpxqs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) with H : g ÝÑ X´1pEq a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let β : g ÝÑ ΓpE´1q be a linear map such that Hpxqp´1q “ ιβpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Taking the polynomial-degree zero of both sides in Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) and evaluating at f P O we obtain that pϱapxq ´ ϱbpxqq rfs “ ” Qp1q, Hpxqp´1qı “ ” Qp1q, ιβpxq ı rfs “ ρpβpxqqrfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since f is arbitrary, this proves item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 tells us that Lie 8-morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq induces weak symmetry action on the base manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The aim of the next section is to look at the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It responds to the following question: Do any weak symmetry action of a Lie algebra on a singular foliation comes from a Lie 8-morphism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If so, can we extend uniquely?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now we define what we call "lift" of a weak symmetry action of a Lie algebra g on a singular foliation F to a Lie 8-algebroid pE, Qq over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation over M and pE, Qq a Lie 8-algebroid over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider a weak symmetry action ϱ: g ÝÑ XpMq of g on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that a Lie 8-morphism of differential graded Lie algebras Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq lifts the weak symmetry action ϱ to pE, Qq if for all x P g, f P O, Φ0pxqpfq “ ϱpxqrfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When Φ exists, we say then Φ is a lift of ϱ on pE, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 Main statements We now state the main theorem of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 showed that a Lie 8-morphism between a Lie algebra g and a Lie 8-algebroid pE, Qq induces a weak action of g on the basic foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this section, we show that any weak symmetry action of a Lie algebra g on a singular foliation F arises this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F a be a singular foliation over a smooth manifold (or an affine variety) M and g a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ϱ: g ÝÑ XpMq be a weak symmetry action of g on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following assertions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for any universal Lie 8-algebroid pE, Qq of the singular foliation F, there exists a Lie 8- morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq that lifts ϱ to pE, Qq, CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS131 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' any two such Lie 8-morphisms are homotopy equivalent over the identity of M, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' any two such lifts of any two equivalent weak symmetry actions of g on F are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Again, Lie 8-morphisms in item 1 of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 are g-actions on pE, Qq in [MZ12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The item 1 in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 means that there exists a linear map Φ0 : g ÝÑ X0pEq such that Φ0pxqrfs “ ϱpxqrfs, and rQ, Φ0pxqs “ 0, @x P g, f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) This morphism is not a graded Lie algebra morphism, but there exist a linear map Φ1 : ^2g ÝÑ X´1pEq such that for all x, y, z P g, Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqs “ rQ, Φ1px, yqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) Also, Φ1 prx, ysg, zq ´ rΦ0pxq, Φ1py, zqs` ö px, y, zq “ rQ, Φ2px, y, zqs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) for some linear map ^3g ÝÑ X´2pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These sets of compatibility conditions continue to higher multilinear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any symmetry X P XpMq of the singular foliation F can be lifted to a degree zero vector field Z P X0pEq that commutes with Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' such that rZ, Qs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To construct Z, it suffices to apply Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 for g “ R and take Z to be the image of 1 through Φ0 : R ÝÑ X0pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 has the following consequences: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for any admissible t, the flow ΦZ t : E ÝÑ E of Z induces an isomorphism of vector bundles E´1 ÝÑ E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since rQ, Zs “ 0, the following diagram commutes, ΓpE´1q ρ � pΦZ t qp0q � ΓpE´1q ρ � XpMq pϕX t q˚ � XpMq where φX t is the flow of X at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consequently, for any open set U Ă M which is stable under ϕX t , there exists an invertible matrix Mt X with coefficients in OpUq that satisfies ` φX t ˘ ˚ ¨ ˚ ˚ ˝ X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Xn ˛ ‹‹‚“ Mt X ¨ ˚ ˚ ˝ X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Xn ˛ ‹‹‚, for some generators X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xn of F over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As announced earlier, we recover Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2, that is, ` φX t ˘ ˚ pFq “ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS132 Let pE, Qq and pE1, Q1q be two universal Lie 8-algebroids of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A direct consequence of Ricardo Campos’s Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 in [Cam19] is that the differential graded Lie algebras pX‚pEqr1s, r¨ , ¨s , adQq and ` X‚pEqr1s, r¨ , ¨s , adQ1˘ are homotopy equivalent over the identity of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This leads to the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ϱ: g ÝÑ XpMq be a weak symmetry action of a Lie algebra g on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, there exist Lie 8-morphisms, Φ: g ù pX‚pEqr1s, r¨ , ¨s , adQq and Ψ: g ù ` X‚pE1qr1s, r¨ , ¨s , adQ1˘ that lift ϱ, and Φ, Ψ make the following diagram commute up to homotopy g Φ � Ψ � pX‚pEqr1s, r¨ , ¨s , adQq � „ � ` X‚pE1qr1s, r¨ , ¨s , adQ1˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The composition of Φ with the horizontal map in the diagram (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) is a lift of the action ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is necessarily homotopy equivalent to Ψ by item p2q in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Proof of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 This section is devoted to the proof of the main results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation, and pE, Qq a universal Lie 8-algebroid of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We start with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every weak symmetry Lie algebra action of g on F there exists a linear map, Φ0 : g Ñ X0pEq, such that rQ, Φ0pxqs “ 0 and Φ0pxqrfs “ ϱpxqrfs for all x P g, f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x P g, let y ϱpxq P X0pEq be any arbitrary extension of ϱpxq P spFq to a degree zero vector field on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ϱpxq is a symmetry of F, the degree `1 vector field r y ϱpxq, Qs is also a longitudinal vector field on E, see Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In addition, r y ϱpxq, Qs is a adQ-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By item 1 of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10, there exists a vertical vector field Y pxq P apEq of degree zero such that rQ, Y pxq ` y ϱpxqqs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17) Let us set for x P g, Φ0pxq :“ Y pxq ` y ϱpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, we have, rQ, Φ0pxqs “ 0 and Φ0pxqrfs “ ϱpxqrfs for all x P g, f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We will need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume pE, Qq is a universal Lie 8-algebroid over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ¯Φ: pS‚ Kg, Qgq ÝÑ pS‚ KXpEqr1s, ¯Qq be a coalgebra morphism which is a Lie 8-morphism up to polynomial-degree n ě 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' `¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ ˘piq “ 0 for all integer i P t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, ¯Φ can be lengthened to a 8-morphism up to polynomial-degree n ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For convenience, we omit the variables x P S‚ Kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The identity, ¯Q ˝ `¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ ˘ ` `¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ ˘ ˝ Qg “ 0 CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS133 taken in polynomial-degree n ` 1 yields, 0 “ ` ¯Q ˝ p¯Φ ˝ Qg ´ ¯Q ˝ ¯Φq ˘pn`1q “ rQ, p¯Φ ˝ Qg ´ ¯Q ˝ ¯Φqpn`1qs, since Qp0q g “ 0 and `¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ ˘piq “ 0 for i P t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is clear that for all n ě 0 the map `¯Φ ˝ Qg ´ ¯Q ˝ ¯Φ ˘pn`1q : Sn`2 K g ÝÑ X´npEqr1s take value in vertical vector fields on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By virtue of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 there exists a map ζ : Sn`2 K g ÝÑ X´n´1pEqr1s such that rQ, ¯Φpn`1q ` ζs “ ¯Φpnq ˝ Qp1q g ´ ¯Qp1q ˝ ¯Φpnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18) By redefining the polynomial-degree n`1 of ¯Φ as ¯Φpn`1q :“ ¯Φpn`1q`ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One obtains a Lie 8-morphism up to polynomial-degree n ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof continues by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 gives the existence of a linear map Φ0 : g ÝÑ X0pEq such that, rQ, Φ0pxqs “ 0 for all x P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x, y P g, consider Λpx, yq “ Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19) Since ϱprx, ysgq ´ rϱpxq, ϱpyqs P F for all x, y P g, and since ρ: ΓpE´1q ÝÑ F surjective, we have ϱprx, ysgq ´ rϱpxq, ϱpyqs “ ρ pηpx, yqq for some element ηpx, yq P ΓpE´1q depending linearly on x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now we consider the vertical vector field of degree ´1, ιηpx,yq P X´1pUFq which is defined on ΓpE˚q as: ιηpx,yqpαq :“ xα, ηpx, yqy for all α P ΓpE˚q, and extended it by derivation on the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every f P O, ` Λpx, yq ´ rQ, ιηpx,yqs ˘ pfq “ pϱprx, ysgq ´ rϱpxq, ϱpyqs ´ ρpηpx, yqq rfs (by definition of Φ0) “ 0 (by definition of η) It is clear that Λpx, yq ` rQ, ιηpx,yqs is a adQ-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, ` Λpx, yq ` rQ, ιηpx,yqs ˘p´1q: for every α P ΓpE˚q, rQ, ιηpx,yqsp´1qpαq “ rQp0q, ιηpx,yqspαq “ (((((((( ( Qp0qrxα, ηpx, yqys ` (((((((( ( xQp0qrαs, ηpx, yqy “ 0, where the first term (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the second term) is cancelled by O-linearity of Qp0q (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for degree reason).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12, the degree zero vector field Λpx, yq ` rQ, ιηpx,yqs is of the form rQ, Υpx, yqs for some vertical vector field Υpx, yq P X´1pEq of degree ´1 with Υpx, yqp´1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all x, y P g, we define the Taylor coefficient Φ1 : ^2 g ÝÑ XpEq as Φ1px, yq :“ Υpx, yq ` ιηpx,yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, we have the following relation, Φ0prx, ysgq ´ rΦ0pxq, Φ0pyqs “ rQ, Φ1px, yqs, @x, y P g (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20) Consider for x, y, z P g, ϑpx, y, zq “ Φ1 prx, ysg, zq ´ rΦ0pxq, Φ1py, zqs` ö px, y, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21) Here, ö px, y, zq stands for circular permutation of x, y and z with Koszul sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For degree reason ϑpx, y, zq is O-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, ϑpx, y, zq is a adQ-cocycle: rQ, Φ1prrx, ysg, zsgqs ` ö px, y, zq “ ´ rΦ0 prx, ysgq , Φ0pzqs ` ö px, y, zq “ rrΦ0pzq, Qs, Φ1px, yqs ´ rrΦ1px, yq, Φ0pzqs, Qs` ö px, y, zq “ rQ, rΦ0pxq, Φ1py, zqss` ö px, y, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS134 Here, we have used the fact that rQ, Φ0pxqs “ 0 for all x P g, and the Jacobi identity for the Lie brackets r¨ , ¨sg and r¨ , ¨s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11, there exists a derivation of degree ´2 denoted by Φ2px, y, zq P X´2pEqr1s that satisfies, ϑpx, y, zq “ rQ, Φ2px, y, zqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22) So far, in the construction of the Lie 8-morphism, we have shown the existence of a Lie 8- morphism ¯Φ: S‚ Kg ÝÑ S‚ K pXpEqr1sq up to polynomial-degree 2 that is p¯Φ ˝ Qgqpiq “ p ¯Q ˝ ¯Φqpiq with i “ 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof continues by recursion or by applying directly Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves the part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Before proving item 3 of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 we will need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For convenience, we sometimes omit the variables in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any two Lie 8-morphisms Γ, Ω: pS‚ Kg, Qgq ù pS‚ KpXpEqr1sq, ¯Qq which coincide up to polynomial-degree n ě 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Γpiq “ Ωpiq, for 0 ď i ď n, their difference in polynomial-degree n ` 1, namely, Γpn`1q ´ Ωpn`1q : Sn`2 K g ÝÑ X´n´1pEqr1s is valued in adQ-coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, a direct computation yields ¯Q ˝ pΓ ´ Ωq “ pΓ ´ Ωq ˝ Qg ùñ ¯Qp0q ˝ pΓ ´ Ωqpn`1q ´ ppΓ ´ Ωq ˝ Qgqpn`1q looooooooooomooooooooooon “0 “ 0 ùñ rQ, Γpn`1q ´ Ωpn`1qs “ 0 ùñ Γpn`1q ´ Ωpn`1q “ rQ, Hpn`1qs (by item 1 of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) for some linear map Hpn`1q : Sn`2 K g ÝÑ X´n´2pEqr1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us show item 2 of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ, Ψ: g ÝÑ XpEqr1s be two different lifts of the action g ÝÑ XpMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by ¯Φ, ¯Ψ: S‚ Kg ÝÑ S‚ KpXpEqr1sq the unique comorphisms given respectively by the Taylor coefficients $ & % ¯Φprq : Sr`1 K g Φr ÝÑ X´rpEqr1s ¯Ψprq : Sr`1 K g Ψr ÝÝÑ X´rpEqr1s , for r ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23) For any x P g, the degree zero vector field Φ0pxq ´ Ψ0pxq P X0pEq is vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, we have, rQ, Φ0pxq ´ Ψ0pxqs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 there exists a vector field H0 P X´1pEq of degree ´1, such that Ψ0pxq ´ Φ0pxq “ rQ, H0pxqs g Ψ0´Φ0 � H0 � X´1pEqr1s adQ � X0pEqr1s (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24) Consider the following differential equation $ & % dΞt dt “ ¯Q ˝ Ht ` Ht ˝ Qg, t P r0, 1s Ξ0 “ ¯Φ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25) CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS135 where pΞtqtPr0,1s is as in Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6, and for t P r0, 1s, Ht is the unique Ξt-co-derivation where the only non-zero polynomial-degree is Hp0q “ H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25) gives a homotopy between ¯Φ and Ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When we consider the polynomial-degree zero component in Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25), one obtains dΞp0q t dt “ ¯Qp0q ˝ Hp0q t ` Hp0q t ˝ Qp0q g “ rQ, H0s “ Ψ0 ´ Φ0 “ ¯Ψp0q ´ ¯Φp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, Ξp0q t “ ¯Φp0q `tp¯Ψp0q ´ ¯Φp0qq, and ¯Φ „ Ξ1 with ¯Ψp0q “ Ξp0q 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Using Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9, the image of any element through the map ¯Ψp1q ´ Ξp1q 1 : S2 Kg ÝÑ X´1pEqr1s is a adQ-coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, ¯Ψp1q ´ Ξp1q 1 can be written as ¯Ψp1q ´ Ξp1q 1 “ rQ, Hp1qs, with Hp1q : S2 Kg ÝÑ X´2pEqr1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='26) Let us go one step further by considering the differential equation on r0, 1s given by $ & % dΘt dt “ ¯Q ˝ Ht ` Ht ˝ Qg Ξ0 “ ¯Ξ1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='27) Here Ht is the extension of Hp1q as the unique Θt-co-derivation where all its arities vanish except the polynomial-degree 1 which is given by Hp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In polynomial-degree zero, pΘp0q t qtPr0,1s is constant and has value Θp0q 1 “ ¯Ψp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In polynomial-degree one, we have, dΘp1q t dt “ ¯Qp0q ˝ Hp1q t “ rQ, Hp1qs “ ¯Ψp1q ´ Ξp1q 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, Θp1q t “ ¯Φp1q ` tp¯Ψp1q ´ Ξp1q 1 q with ¯Ψpiq “ Θpiq 1 for i “ 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We then continue this procedure by gluing all these homotopies as in the proof of item 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We obtain at last a Lie 8-morphism Ω such that ¯Φ „ Ω and Ωpiq “ ¯Ψpiq for i ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' That means Ω “ ¯Ψ, therefore ¯Φ „ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove item 3 of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given two equivalent weak symmetry actions ϱ, ϱ1 of g on a singular foliation F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ϱ, ϱ1 differ by a linear map g ÝÑ XpMq of the form x ÞÑ ρpβpxqq for some linear map β : g ÝÑ ΓpE´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φ, Φ1 : g ù pX‚pEqr1s, r¨ , ¨s , adQq be a lift into a Lie 8-morphism of the action ϱ and ϱ1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has for all x P g and f P O, ` Φ0pxq ´ Ψ0pxq ´ rQ, ιϕpxqs ˘ pfq “ ρpϕpxqqrfs ´ xQpfq, ϕpxqy “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since rQ, Φ0pxq ´ Ψ0pxq ´ rQ, ιϕpxqss “ 0, by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 there exists a vertical derivation pHpxq P X´1pEq of degree ´1 depending linearly on x P g such that Φ0pxq ´ Ψ0pxq “ rQ, pHpxq ` ιϕpxqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Hpxq :“ pHpxq ` ιϕpxq, for x P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof continues the same as for item 2 of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS136 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Particular examples We recall that for a regular foliation F on a manifold M, the Lie algebroid TF Ă TM, whose sections form F, is a universal Lie 8-algebroid of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Its corresponding Q-manifold is given by the leafwise De Rham differential on Γp^‚T ˚Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a regular foliation on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any weak symmetry action g ÝÑ XpMq, x ÞÝÑ ϱpxq, of F, can be lifted to Lie 8-morphism Φ: g ù pX‚pEqr1s, r¨ , ¨s , adQq given explicitly as follows: x P g ÞÝÑ Φ0pxq “ Lϱpxq P X0p^‚T ˚Fq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='28) x ^ y P ^2g ÞÝÑ Φ1px, yq “ ιχpx,yq P X´1p^‚T ˚Fq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='29) and ` Φi : ^i`1 g ÝÑ X´ip^‚T ˚Fq ˘ ” 0, for all i ě 2, where χpx, yq :“ ϱprx, ysgq ´ rϱpxq, ϱpyqs for x, y P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, LX stands for the Lie derivative on multi-forms w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t X P XpMq, and ιX is the internal product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on a manifold M together with a strict symmetry action ϱ: g ÝÑ XpMq such that g Ă F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, C8pMqg is a singular foliation which is the image of the transformation Lie algebroid g ˆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The universality theorem (see [LLS20, LGL22b]) provides the existence of a Lie 8-morphism ν : g ÝÑ UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us call its Taylor coefficients νn : ^n`1 g ÝÑ E´n´1, n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We may take for example the 0-th and 1-th Taylor coefficients of a Lie 8-morphism that lifts ϱ as: Φ0pxq :“ rQ, ιν0pxqs P X0pUFq, for x P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Φ1px, yq :“ rQ, ιν1px,yqsp´1q ´ ÿ kě0 rrQ, ιν0pxqs, ιν0pyqspkq P X´1pUFq, for x, y P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that in this case the action ϱ is equivalent to zero, therefore by item 3 of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 the Lie 8-morphism Φ is homotopic to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 Lifts of weak symmetry actions and Lie 8-algebroids In this section, g is a finite dimensional Lie algebra that we see as the trivial vector bundle over M with fiber g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following theorem says that any lift of strict symmetry action of g on a singular foliation F induces a Lie 8-algebroids with some special properties and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' See [MZ12], Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3, for a proof of the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, Qq be a Lie 8-algebroid over a singular foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any Lie 8-morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq with g of finite dimension induces a Lie 8-algebroid pE‘g, Q1q with Q1 :“ dCE ` Q ` ÿ kě1,i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ik“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',dimpgq 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='ξi1 d ¨ ¨ ¨ d ξikΦk´1pξi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξikq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='30) where dCE is the Chevalley-Eilenberg complex of g, and ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξdimpgq P g˚ is the dual basis of some basis ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξdimpgq P g and for all k ě 0, Φk : Sk`1g ÝÑ X´kpEqr1s is the k-th Taylor coefficients of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS137 In the dual point of view, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='30) corresponds to a Lie 8-algebroid over the complex ¨ ¨ ¨ ℓ1 ÝÑ E´3 ℓ1 ÝÑ E´2 ℓ1 ÝÑ g ‘ E´1 ρ1 ÝÑ TM (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='31) whose brackets satisfy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the anchor map ρ1 sends an element x ‘ e P g ‘ E´1 to ϱpxq ` ρpeq P ϱpgq ` TF, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the binary bracket satisfies ℓ2 pΓpE´1q, ΓpE´1qq Ă ΓpE´1q and ℓ2pΓpE´1q, xq Ă ΓpE´1q, @ x P g 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the g-component of the binary bracket on constant sections of g ˆ M is the Lie bracket of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conversely, if there exists a Lie 8-algebroid pE1, Q1q whose underlying complex of vector bundles is of the form (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='31) and that satisfies item 1, 2 and 3, then there is a Lie 8-morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq which is defined on a given basis ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξd of g by: Φk´1pξi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξikq “ pr ˝ r¨ ¨ ¨ rrQ1, ιξi1s, ιξi2s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ιξiks Ă XpEqr1s, k P N, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='32) where pr stands for the projection map XpE1qr1s ÝÑ XpEqr1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We explain the idea of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A direct computation gives the first implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conversely, let us denote by Q1 the homological vector fields of Lie 8-algebroid whose underlying complex of vector bundles is of the form (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map defined in Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='32) is indeed a lift into a Lie 8-morphism of the weak symmetry action ϱ: It is not difficult to check that, for any ξ P g, one has rQ, Φ0pξqs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The fact that Φ defines a Lie 8-morphism can be found using Voronov trick [Vor04, Vor05], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e, doing Jacobi’s identity inside the null derivation 0 “ pr ˝ r¨ ¨ ¨ rrrQ1, Q1s, ιξi1s, ιξi2s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ιξiks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='33) A direct computation of Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='33) falls exactly on the requirements of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us compute Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='33) for a small number of generators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g k “ 2, 3) in order to show how it works: from the identity ””“ Q1, Q1‰ , ιξi1 ı , ιξi2 ı “ 0, one obtains by using twice the Jacobi identity the following relation, “ Q1, ““ Q1, ξi1 ‰ , ξi2 ‰‰ ´ ““ Q1, ξi1 ‰ , “ Q1, ξi2 ‰‰ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='34) One should notice that rrQ1, ξi1s , ξi2s splits into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One part where the Chevalley-Eilenberg acts to give ““ dCE, ξi1 ‰ , ξi2 ‰ “ ιrξi1,ξi2sg, while the other part is ““ Q1 ´ dCE, ξi1 ‰ , ξi2 ‰ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, by putting them in Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='34), afterwards projecting on X pS‚pE˚qq, we get pr ˝ ” Q1, ιrξi1,ξi2sg ı ` pr ˝ rQ1, ”” Q1 ´ dCE, ξi1 ı , ξi2 ı s ´ pr ˝ ““ Q1, ξi1 ‰ , “ Q1, ξi2 ‰‰ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From here, we deduce that Φ0prξi1, ξi2sgq “ rQ, Φ1pξi1, ξi2qs ` rΦ0pξi1q, Φ0pξi2qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS138 Here is an application of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let g be a Lie algebra and G its Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pM, Fq a singular foliation together with a weak symmetry action ϱ: g ÝÑ XpMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following assertions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' On G ˆ M the C8pG ˆ Mq-module generated by $ & % pÐÝu , ϱpuqq u P g p0, Xq X P F (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='35) is a singular foliation that we denote by pXpGq ˆϱ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If pE, d, ϱq is a geometric resolution of pM, Fq, then ¨ ¨ ¨ ÝÑ p˚E´3 d ÝÑ p˚E´2 d ÝÑ g ‘ p˚E´1 ρ1 ÝÑ TpG ˆ Mq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='36) with ρ1pe, uq “ pÐÝu , ϱpuq ` ρpeqq, is a geometric resolution of pXpGqˆϱFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, p: GˆM ÝÑ M is the projection on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pE, Qq be a universal Lie 8-algebroid structure of pM, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let Φk : ^k`1 g ÝÑ X´kpEq be the Taylor coefficients of a Lie 8-morphism g ù XpEq that lifts ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then Q1 :“ dG dR ` Q ` ÿ kě1,i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',ik“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',dimpgq 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='ξi1 d ¨ ¨ ¨ d ξikΦk´1pξi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξikq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='37) with pξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξdimpgqq, pξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξdimpgqq be dual basis of g and g˚ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is a universal Lie 8-algebroid of pXpGq ˆϱ Fq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' whose coefficients are left invariant for the action of G on G ˆ M given by g ¨ ph, mq :“ pgh, mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conversely, a left invariant Lie 8-algebroid on (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='36) can be interpreted as Taylor coefficients of a lift of ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 A more general statement of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 We end the chapter with a generalization of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the previous section, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 is stated in the finite dimensional context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' it needs g to be finite dimensional and the existence of a geometric resolution for the singular foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this section we prove that given a weak symmetry action of a Lie algebra g (may be of infinite dimensional) on a Lie-Rinehart algebra F Ă XpMq (we do not require F being locally finitely generated), such Lie 8-algebroid described at the sections level of the complex (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='31) as Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We state the following theorem in the context of singular foliations, but the same statement and the same proof are valid word-by-word by replacing F by a Lie-Rinehart algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS139 Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let g be a (possibly infinite dimensional) Lie K-algebra and let ϱ: g ÝÑ XpMq be a weak symmetry action of g on a singular foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ppK´iqiě1, d, ρq be a free resolution of the singular foliation F over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The complex of trivial vector bundles over M ¨ ¨ ¨ d ÝÑ E´3 d ÝÑ E´2 d ÝÑ g ‘ E´1 ρ1 ÝÑ TM (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='38) where ΓpE´1q “ K´i, comes equipped with a Lie 8-algebroid structure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' whose unary bracket is d and whose anchor map ρ1, sends an element x ‘ e P g ‘ E´1 to ϱpxq ` ρpeq P ϱpgq ` TF, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the binary bracket satisfies ℓ2 pΓpE´1q, ΓpE´1qq Ă ΓpE´1q and ℓ2pΓpE´1q, Γpgqq Ă ΓpE´1q, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the g-component of the binary bracket on constant sections of g ˆ M is the Lie bracket of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When we have ϱpgq X TmF “ t0u for all m in M, Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='38) is a free resolution of the singular foliation C8pMqϱpgq ` F and we can apply directly the Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Otherwise, we need to show there is no obstruction in degree ´1 while doing the construction of the brackets if the result still needs to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) The complex of Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='38) being exact everywhere except in degree ´1 we cannot apply directly Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 in [LGL22b] but we can mimic the proof given for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 in [LGL22b] to construct the higher brackets when there is no obstruction in degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For convenience, let us denote R´1 :“ Γpgq ‘ ΓpE´1q and R´i :“ ΓpE´iq for i ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a natural number k ě 0, we consider the total complex ˆ z Page pkq ‚ pRq, D “ rd, ¨sRN ˙ of the following bicomplex .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ò Ò Ò ¨ ¨ ¨ Ñ HomO ´Äk`1 R |´k´3, R´3 ¯ dÑ HomO ´Äk`1 R |´k´3, R´2 ¯ dÑ HomO ´Äk`1 R |´k´3, dR´2 ¯ Ñ 0 δ Ò δ Ò δ Ò ¨ ¨ ¨ Ñ HomO ´Äk`1 R |´k´2, R´3 ¯ dÑ HomO ´Äk`1 R |´k´2, R´2 ¯ dÑ HomO ´Äk`1 R |´k´2, dR´2 ¯ Ñ 0 δ Ò δ Ò δ Ò ¨ ¨ ¨ Ñ HomO ´Äk`1 R |´k´1, R´3 ¯ dÑ HomO ´Äk`1 R |´k´1, R´2 ¯ dÑ HomO ´Äk`1 R |´k´1, dR´2 ¯ Ñ 0 Ò Ò Ò 0 0 0 "-3 column" "-2 column" "-1 column" (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='39) The map δ stands for the vertical differential which is defined for all Φ P HomO ´Äk`1 R, R ¯ by δpΦq pr1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , rk`1q :“ Φ ˝ d pr1 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' d rk`1q, @ r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , rk`1 P R, where here d acts as an O-derivation on r1 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' d rk`1 P Äk R and the horizontal differential given by Φ ÞÑ d ˝ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the line the bicomplex is exact, the total complex ˆ z Page pkq ‚ pRq, D “ rd, ¨sRN ˙ is also exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS140 Construction of the 2-ary bracket: its construction is almost the same as in [LGL22b] we adapt what has been done to our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We first construct a 2-ary bracket on R´1 to extend on every degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all k ě 1, let us denote by pep´kq i qiPIk a basis of ΓpE´kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The set tXi “ ρpep´1q i q P F | i P I1u is a set of generators of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, there exists elements ck ij P O and satisfying the skew-symmetry condition ck ij “ ´ck ji together with rXi, Xjs “ ÿ kPI ck ijXk @i, j P I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='40) By definition of weak symmetry one has rϱpξiq, ρpep´1q j qs P F and ϱprξi, ξjsqg ´ rϱpξiq, ϱpξjqs P F for all pi, jq P Ig ˆ I´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='41) Here, pξiqiPIg is a basis for g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ppK´iqiě1, d, ρq is a free resolution of F, there exists two O-bilinear maps χ: Γpgq ˆ ΓpE´1q Ñ ΓpE´1q, η: Γpgq ˆ Γpgq Ñ ΓpE´1q defined on generators ξi, ep´1q j by the relations rϱpξiq, ρpep´1q j qs “ ρpχpξi, ep´1q j qq and ϱprξi, ξjsgq ´ rϱpξiq, ϱpξjqs “ ρpηpξi, ξjqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We first define a naive 2-ary bracket on ΓpE´1q as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' an anchor map by ρ1pep´1q i q “ Xi, and ρ1pξiq “ ϱpξiq, for all i P I, Ig, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a degree `1 graded symmetric operation ˜ℓ2 on R‚ as follows: (a) ˜ℓ2 ´ ep´1q i , ep´1q j ¯ “ ř kPI ck ijep´1q k for all i, j P I´1, (b) ˜ℓ2 ´ ξi, ep´1q j ¯ “ χ ´ ξi, ep´1q j ¯ , (c) ˜ℓ2 pξi, ξjq “ rξi, ξjsg ` ηpξi, ξjq, (d) rℓ2 is zero on the other generators, (e) we extend ˜ℓ2 to R using O-bilinearity and Leibniz identity with respect to the anchor ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By paq, pbq, pcq, pdq, peq, ˜ℓ2 satisfies the Leibniz identity with respect to the anchor rρ and paq, pbq, pcq makes the latter a bracket morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map defined for all homogeneous r1, r2 P R‚ by rd, ˜ℓ2sRNpr1, r2q “ d ˝ ˜ℓ2 pr1, r2q ` ˜ℓ2 pdr1, r2q ` p´1q|r1|˜ℓ2 pr1, dr2q , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='42) is a graded symmetric degree `2 operation pRbRq‚ ÝÑ R‚`2, and rd, ˜ℓ2sRN|R´1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is O-bilinear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' for all f P O, r1, r2 P R rd, ˜ℓ2sRNpr1, fr2q ´ frd, ˜ℓ2spr1, r2q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We also have that ρprd, ˜ℓ2sRNpr1, fr2qq “ ρp˜ℓ2pdr1, r2qq “ 0, for all r1 P R´2, r2 P R´1, since ρ ˝ d “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, rd, ˜ℓ2sRN|R´2ˆR´1 P dR´2, because ppE´iqiě1, d, ρq is a geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, rd, ˜ℓ2sRN is a degree `2 element in the total complex z Page p1qpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-bilinear op- erator rd, ˜ℓ2sRN is D-closed in z Page p1qpRq, since rd, rd, ˜ℓ2sRNsRN|Rď´2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' So there exists τ2 P ‘jě2HomO ´Ä2 R´j´1, R´j ¯ such as Dpτ2q “ ´rd, ˜ℓ2sRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By replacing ˜ℓ2 by ˜ℓ2 ` τ2 we get a 2- ary bracket ℓ2 of degree `1 which is compatible with the differential map d and the anchor map rρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' SYMMETRIES OF SINGULAR FOLIATIONS THROUGH LIE 8-ALGEBROIDS141 Construction of higher brackets: notice that by construction of the 2-ary bracket ℓ2 one has, Jacpr1, r2, r3q P dR´2 for all r1, r2, r3 P R´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In other words, Jac P HomOpÄ3 R´1, dR´2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A direct computation shows dJacpr1, r2, r3q “ Jacpdr1, r2, r3q ` p´1q|r1|Jacpr1, dr2, r3q ` p´1q|r1|`|r2|Jacpr1, r2, dr3q for all r1, r2, r3 P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Which means, rJac, dsRNpr1, r2, r3q “ 0 for all r1, r2, r3 P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, DpJacq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows that, Jac is a D-coboundary, there exists an element ℓ3 “ ř jě2 ℓj 3 P z Page p2q 1 pRq with ℓj 3 P HompÄ3 R |´j´1, R´jq such that Dpℓ3q “ ´Jac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='43) We choose the 3-ary bracket to be ℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For degree reason, the remaining terms of the k-ary brackets for k ě 3 have trivial components on the column ´1 of the bicomplex (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' From this point, the proof continues exactly as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We return to Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, the Lie algebra g “ XpLq that acts on the singular foliation T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, we have ϱpgq X TmT “ t0u for all m in rL, Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can apply directly Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, to obtain a Lie 8-algebroid structure on the complex ¨ ¨ ¨ d ÝÑ ΓpE´3q d ÝÑ ΓpE´2q d ÝÑ g ‘ ΓpE´1q ρ1 ÝÑ XprL, Msq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='44) Notice that here the Lie algebra g is infinite dimensional, therefore we are not allowed to use the duality between Lie 8-algebroid and Q-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, we cannot use the explicit Formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='32) to define at lift Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, we have to rely on the existence theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 to assure the existence of a lift Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conclusion: We show that actions of a Lie algebra g on the leaf space M{F i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' weak symmetry actions, lift to Lie 8-algebra morphisms g ù XpEq on the DGLA of vector fields on an universal Lie 8-algebroid pE, Qq, provided it exists, in a unique up to homotopy manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We explain how to use chapter 4 to get rid of all finite ranks/dimensions assumptions, using the universal Lie 8-algebroids of Lie-Rinehart algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 9 On weak and strict symmetries: an obstruction theory In this chapter, we apply the main theorems of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 of Chapter 8 to define a class obstructing the existence of strict symmetry action equivalent to a given weak symmetry action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Introduction Recall that Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 assures that any weak symmetry action ϱ: g Ñ XpMq of a Lie algebra g on a singular foliation F admits a lift to a Lie 8-morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) To understand the compatibility conditions between the low terms of Φ see Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By playing with the definition of Φ we make the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' What does Φ induces on the linear part of pE, Qq?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have seen in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 that the 0-th Taylor coefficient of the lift Φ induces a linear map x P g ÞÝÑ p∇x : E´i ÝÑ E´iq for every i ě 1 that satisfies ∇xpfeq “ f∇xpeq ` ϱpxqrfse, for f P O, e P ΓpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, for every x P g and e P ΓpE´1q, ρp∇xpeqq “ rϱpxq, ρpeqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 142 CHAPTER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY 143 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A direct computation gives 0 “ rrQ, Φ0pxqs , ιesp´1q “ rQ, rΦ0pxq, ιessp´1q ´ rΦ0pxq, rQ, ιessp´1q “ ” Qp0q, rΦ0pxq, ιesp´1qı ´ ” Φ0pxqp0q, rQ, ιesp´1qı “ ” Qp0q, ι∇xpeq ı ´ ” Φ0pxqp0q, ιℓ1peq ı “ ιℓ1˝∇xpeq ´ ι∇x˝ℓ1peq We have used graded Jacobi identity and the dual correspondence between Lie 8-algebroids and NQ-manifolds (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We recapitulate in the following commutative diagram: ¨ ¨ ¨ d � ΓpE´2q d � ∇x � ΓpE´1q ρ � ∇x � F adϱpxq � ¨ ¨ ¨ d � ΓpE´2q d � ΓpE´1q ρ � F (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) which means, ℓ1 ˝ ∇x “ ∇x ˝ ℓ1 and ρ ˝ ∇x “ adϱpxq ˝ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, ℓ1 stands for the corresponding unary bracket of pE, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, for X P XpMq, adX :“ rX, ¨ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x, y P g, and e P ΓpEq, the relation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) yields rΦ0prx, ysgq ´ rΦ0pxq, Φ0pyqs , ιesp´1q “ rrQ, Φ1px, yqs , ιesp´1q ι∇rx,ysgpeq ´ rΦ0pxq, rΦ0pyq, ιessp´1q ` rΦ0pyq, rΦ0pxq, ιessp´1q “ ” ι∇rx,ysgpeq ´ ” Φ0pxqp0q, ι∇ypeq ı ` ” Φ0pyqp0q, ι∇xpeq ı “ rrQ, Φ1px, yqs , ιesp´1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus rrQ, Φ1px, yqs , ιesp´1q “ ι∇rx,ysgpeq ´ ι∇x˝∇ypeq ` ι∇y˝∇xpeq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) On the other hand, Φ1px, yq admits a polynomial decomposition Φ1px, yqp´1q ` ÿ iě0 Φ1px, yqpiq “ ιηpx,yq ` ÿ iě0 Φ1px, yqpiq and that «ÿ iě0 Φ1px, yqpiq, ιe ffp´1q “ ” Φ1px, yqp0q, ιe ı “ ιγpx,yqpeq, CHAPTER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY 144 for some linear map γpx, yq: ΓpE´‚q ÝÑ ΓpE|e|´1q depending linearly on x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, rrQ, Φ1px, yqs , ιesp´1q “ ““ Q, ιηpx,yq ‰ , ιe ‰p´1q ` «« Q, ÿ iě0 Φ1px, yqpiq ff , ιe ffp´1q “ ιℓ2pηpx,yq,eq ` « Q, «ÿ iě0 Φ1px, yqpiq, ιe ffffp´1q ´ «ÿ iě0 Φ1px, yqpiq, rQ, ιes ffp´1q “ ιℓ2pηpx,yq,eq ` » –Qp0q, «ÿ iě0 Φ1px, yqpiq, ιe ffp´1qfi fl ´ ” Φ1px, yqp0q, rQ, ιesp´1qı “ ιℓ2pηpx,yq,eq ` ” Qp0q, ιγpx,yqpeq ı ´ ” Φ1px, yqp0q, ιℓ1peq ı “ ιℓ2pηpx,yq,eq ` ιℓ1pγpx,yqpeqq ´ ιγpx,yqpℓ1peqq By equating the latter with (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) we can recapitulate as follows: Conclusion: In general, the map g ÝÑ DerpEq, x ÞÑ ∇x is not a Lie algebra morphism even when the action ϱ is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In fact, there exists a bilinear map γ : ^2 g ÝÑ EndpEqr1s of degree 0 that satisfies ∇rx,ysg ´ r∇x, ∇ys “ γpx, yq ˝ ℓ1 ´ ℓ1 ˝ γpx, yq ` ℓ2pηpx, yq, ¨ q, here ℓ2 is the corresponding 2-ary bracket of pE, Qq, and η: ^2 g ÝÑ ΓpE´1q is such that ϱprx, ysgq ´ rϱpxq, ϱpyqs “ ρpηpx, yqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, Equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) taken in polynomial-degree ´1 implies, that, for all α P ΓpE˚ ´1q Φ1prx, ysg, zqp´1q ´ rΦ0pxqp0q, Φ1py, zqp´1qs` ö px, y, zq “ rQp0q, Φ2px, y, zqp´1qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ùñ ιηprx,ysg,zq ´ rΦ0pxqp0q, ιηpy,zqs` ö px, y, zq “ rQp0q, ιζpx,y,zqs, with ζ : ^3 g Ñ ΓpE´2q ùñ xα, ηprx, ysg, zqy ´ ´ Φ0pxqp0qrxα, ηpy, zqys ´ xΦ0pxqp0qpαq, ηpy, zqy ¯ ` ö“ xQp0qrαs, ζpx, y, zqy, ùñ xα, ηprx, ysg, zqy ´ ` (((((((( ( ϱpxqrxα, ηpy, zqys ´ (((((((( ( ϱpxqrxα, ηpy, zqys ` xα, ∇xηpy, zqy ˘ ` ö px, y, zq “ xα, ℓ1pζpx, y, zqqy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have used Equations (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence, xα, ηprx, ysg, zq ´ ∇xηpy, zqy` ö px, y, zq “ xα, ℓ1pζpx, y, zqqy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since α is arbitrary, one obtains ∇xηpy, zq ´ ηprx, ysg, zq` ö px, y, zq “ ℓ1pζpx, y, zqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) CHAPTER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY 145 In particular, if m P M is such that ℓ1|m “ 0 and ℓ2|m “ 0, then the map x ÞÑ ∇x defines an action on the isotropy Lie algebra gm of F, since ∇x preserves the kernel of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If in addition ηpx, yq P ker ρm, then η|m : ^2 g ÝÑ gm is a cocycle of Chevalley-Eilenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that by using the duality which is given in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, the linear map x ÞÑ ∇x is simply x ÞÑ ℓ1 2px, ¨q, where ℓ1 2 is the 2-ary bracket between sections of g and E of the Lie 8-algebroid pQ1, E ‘ gq over the complex ¨ ¨ ¨ ℓ1 ÝÑ E´3 ℓ1 ÝÑ E´2 ℓ1 ÝÑ g ‘ E´1 ρ1 ÝÑ TM (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) like in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, for α P ΓpE˚ ´1q and e P ΓpE´1q, Φ0pxqp0q “ pr ˝ rQ1, ιxsp0q “ pr ˝ rQ1p1q, ιxs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that: xΦ0pxqp0qα, ey “ xQ1p1qα, x d ey “ ρ1pxqrxα, eys ´ \x18\x18\x18\x18\x18\x18 ρ1peqrxα, xys ´ xα, ℓ1 2px, eqy “ ϱpxqrxα, eys ´ xα, ℓ1 2px, eqy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 An obstruction theory Let us start with some generalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume we are given a Lie algebra g, a weak symmetry action ϱ: g ÝÑ XpMq of g on a singular foliation F, together with η: ^2 g ÝÑ ΓpE´1q such that x, y P g ϱprx, ysgq ´ rϱpxq, ϱpyqs “ ρpηpx, yqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) an universal Lie 8-algebroid pE, QEq of F, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 assures ϱ: g Ñ XpMq admits a lift to a Lie 8-morphism Φ: pg, r¨ , ¨sgq ù pX‚pEqr1s, r¨ , ¨s , adQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) Equivalently, if g is of finite dimension, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) corresponds (by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) to a Lie 8-algebroid pE1, Q1q over M such that pE, QEq is included as a sub-Lie 8-algebroid in a Lie algebroid pE1, Qq over M, its underlying complex is, E1 ´1 :“ g ‘ E´1, and for any i ě 2, E1 ´i “ E´i, namely ¨ ¨ ¨ d ÝÑ E´3 d ÝÑ E´2 d ÝÑ g ‘ E´1 ρ1 ÝÑ TM, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) CHAPTER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY 146 we have, ℓ1 2px ‘ 0, y ‘ 0q “ rx, ysg ‘ ηpx, yq and ℓ1 2px, ΓpE´1qq Ă ΓpE´1q for all x P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is important to note that the Lie 8-algebroid pE1, Q1q can be constructed directly out of the weak symmetry action ϱ: g ÝÑ XpMq, even if g is of infinite dimension (see Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In what follows, we will use the 8-algebroid which is described on the complex (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let m P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume that the underlying complex pE, ℓ1q is minimal at a point m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ℓ1|m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map ν : g ÝÑ End ´ E´1|m ¯ , x ÞÝÑ ℓ1 2px , ¨q|m satisfies (a) νprx, ysgq ´ rνpxq, νpyqs ` ℓ2p ¨, ηpx, yqq|m “ 0, (b) νpzq ` ηpx, yq|m ˘ ´ ηprx, ysg, zq|m` ö px, y, zq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ℓ1|m “ 0, E1 ´1|m is a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Jacobi identity on elements x, y P g, e P ΓpE´1q, evaluated at the point m, implies that νprx, ysgqpe|mq ´ rνpxq, νpyqspe|mq ` ℓ2pηpx, yq, eq|m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Likewise, Jacobi identity on elements x, y, z P g and since ℓ1|m “ 0 give: ℓ1 2pℓ1 2px, yq, zq|m` ö px, y, zq “ 0 ùñ ℓ1 2prx, ysg, zq|m ` ℓ1 2pηpx, yq, zq|m` ö px, y, zq “ 0, ùñ νpzq ` ηpx, yq|m ˘ ´ ηprx, ysg, zq|m` ö px, y, zq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here we have used the definition of ℓ1 2 on degree ´1 elements and Jacobi identity for the bracket r¨ , ¨sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2, E´1 is equipped with a g-module structure when ηpx, yq|m is for all x, y P g valued in the center the Lie algebra E´1|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following proposition generalizes this remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let m P M and assume that the underlying complex pE, ℓ1q of pE, Qq is minimal at m, for all x, y P g, ηpx, yq|m is valued in the center1 ZpE´1|mq of E´1|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the restriction of the 2-ary bracket ℓ1 2 : g b ZpE´1|mq ÝÑ ZpE´1|mq endows ZpE´1|mq with a g-module structure which does not depend neither on the choices of weak symmetry action ϱ, a universal Lie 8-algebroid of F, nor of the Lie 8-morphism Φ: g ÝÑ XpEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1In particular, when the 2-ary bracket ℓ2 is zero at m, on elements of degree ´1 we have, ZpE´1|mq “ E´1|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY 147 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the restriction of the map η: ^2 g ÝÑ ΓpE´1q at m η|m : ^2 g ÝÑ ZpE´1|mq is a 2-cocycle for the Chevalley-Eilenberg complex of g valued in ZpE´1|mq, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the cohomology class of this cocycle does not depend on the representatives of the equivalence class of ϱ, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if ϱ is equivalent to a strict symmetry action, then η|m is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We may assume that, ℓ2|m “ 0 on E´1|m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', ZpE´1|mq “ E´1|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The first clause of item p1q follows from item paq of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 when ℓ2|m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easy to see that if we change the action ϱ to ϱ ` ρ ˝ β for some vector bundle morphism β : g ÝÑ E´1, the new 2-ary bracket between sections of g and E´1 made in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 is modified by px, eq ÞÑ ℓ1 2px, eq ` ℓ2pβpxq, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, under the assumption, ℓ2|m “ 0, we obtain the last clause of item p1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item p2q follows from Item (b) of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 that tells that η|m : ^2 g ÝÑ E´1|m is a 2-cocycle for the Chevalley-Eilenberg complex of g valued in E´1|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ϱ1 be a weak action of g on F which is equivalent to ϱ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' there exists a vector bundle morphim β : g ÝÑ E´1 such that ϱ1pxq “ ϱpxq ` ρpβpxqq for all x P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let η1 : ^2 g ÝÑ E´1 be such that ϱ1prx, ysgq ´ rϱ1pxq, ϱ1pyqs “ ρpη1px, yqq for all x, y P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Following the constructions in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4, this implies that η1px, yq “ ηpx, yq ` βprx, ysgq ´ ℓ1 2px, βpyqq ` ℓ1 2py, βpxqq ´ ℓ2pβpxq, βpyqqq, for all x, y P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) Hence, if η1 |m P H2pg, E´1|mq is exact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' there exists a linear map λ: g ÝÑ E´1|m such that dCEpλq “ η1 |m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Using Equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) and ℓ2|m “ 0, one gets dCEpβ|m ` λq “ η|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves items p3q and p4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When ℓ2|m ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The weak symmetry action ϱ is equivalent to strict one if the Maurer-Cartan-like equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) has no solution with η1 |m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us choose a universal Lie 8-algebroid pE, Qq such that pE, ℓ1q is minimal at a point m P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Such a structure always exists (see Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14 in [LLS20] the isotropy Lie algebra gm of the singular foliation F at the point m P M is isomorphic to kerpρmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following is a direct consequence of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let m P M be a point of M Assume that the isotropy Lie algebra gm of F at m is Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, for any weak symmetry action ϱ of a Lie algebra action g on F such that ϱprx, ysgq ´ rϱpxq, ϱpyqs P Fpmq for all x, y P g 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' gm is a g-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The bilinear map, η|m : ^2 g Ñ gm, is a Chevalley-Eilenberg 2-cocycle of g valued in gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Its class clpηq P H2pg, gmq does not depend on the choices made in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Furthermore, clpηq is an obstruction of having a strict symmetry action equivalent to ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We return to Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 with m P M a leaf of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the isotropy Lie algebra gk m is Abelian for every k ě 2 the following assertions hold by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5: CHAPTER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY 148 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For each k ě 1, the vector space gk`1 m is a gk m-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The obstruction of having a strict symmetry action equivalent to ϱk is a Chevalley-Eilenberg cocycle valued in gk`1 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is a particular case of this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F :“ I3 0XpRnq be the singular foliation generated by vector fields vanishing to order 3 at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The quotient g :“ I2 0XpRnq I3 0XpRnq is a trivial Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is a weak symmetry action of g on F which assigns to an element in g a representative in I2 0XpRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this case, the isotropy Lie algebra of F at zero is Abelian and ℓ1 2pg, g0q|0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, the action of g on g0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can choose η: ^2 g ÝÑ g0 such that η ´ x2 i B Bxi , x2 i B Bxj ¯ “ 2eij, with eij a constant section in a set of generators of degree ´1 whose image by the anchor is x3 i B Bxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, η|0 ´ x2 i B Bxi , x2 i B Bxj ¯ ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that the class of η is not zero at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, by item 2 of Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 the weak symmetry action of g on F is not equivalent to a strict one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider again the Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 and let pE, ℓ‚, ρq be a universal Lie 8-algebroid of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Recall that a flat Ehresmann connection is a horizontal distribution whose sections are closed under the Lie bracket of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let m P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume in Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 that the section ϱH : XpLq Ñ Fproj satisfies ϱHpra, bsq ´ rϱHpaq, ϱHpbqs “ ρpηpa, bqq P T pmq for some bilinear map η: ^2 XpLq Ñ E´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5, the isotropy Lie algebra gT m of T at m is a XpLq-module, and η is a cocycle of Chevalley-Eilenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This provides an obstruction for the F-connection to be flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, we have the following consequence of Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 for Lie algebra actions on affine vari- eties, as in Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Before going to Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12 let us write definitions and some facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Settings: Let W be an affine variety realized as a subvariety of Cd, and defined by some ideal IW Ă Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by XpWq :“ DerpOW q the Lie algebra of vector fields on W, where OW is coordinates ring of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A point p P W is said to be strongly singular if for all f P IW , dpf ” 0 or equivalently if for all f P IW and X P XpCdq, one has Xrfsppq P Ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any singular point of a hypersurface W defined by a polynomial ϕ P Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds is strongly singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The lemma below is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In a strongly singular point, the isotropy Lie algebra of the singular foliation F “ IW XpCdq is Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ϱ: g ÝÑ XpWq be a Lie algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any extension rϱ as in Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 is a weak symmetry action for the singular foliation F “ IW XpCdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY 149 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any strongly singular point p in W if the class clpηq does not vanish the strict action ϱ: g ÝÑ DerpOW q can not be extended to the ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us give an example of a Lie algebra action on an affine variety that do not extend to the ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We hope to construct an example as follows Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W Ă Cd be an affine variety generated by a regular homogeneous polynomial ϕ P O “ Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds of degree ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume there exists two vector fields X, Y P XpCdq that satisfy Xrϕs “ fϕ, Y rϕs “ gϕ, with f, g P I0, and such that rX, Y s “ ϕZ, for some Z P XpCdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the action of the trivial g “ R2 on W that sends its canonical basis to X, and Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is a weak symmetry action on the singular foliation Fϕ :“ xϕyXpCdq and induces a Lie algebra map, g ÝÑ XpWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that the universal Lie algebroid of Fϕ is a Lie algebroid (see Example of [LGL22b]) because, 0 � Oµ bO XpCdq ϕ B Bµ bOid � Fϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' is a O-module isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here µ is a degree ´1 variable, so that µ2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The universal algebroid structure over that resolution is given on the set of generators by: ℓ2 ˆ µ bO B Bxa , µ bO B Bxb ˙ :“ Bϕ Bxa µ bO Bxb ´ Bϕ Bxb µ bO Bxa and ℓk :“ 0 for every k ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Write Z “ dÿ i“1 fi B Bxi , with pfiqi“1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=',d Ă O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have, ηpe1, e2q :“ dÿ i“1 fiµ bO B Bxi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' where e1, e2 is the canonical basis of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This Lie 8-algebroid structure satisfies all the assuptions of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume that the vector field Z does not vanish at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the action is trivial at zero and η|0 ‰ 0, therefore its class is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By consequence, such action cannot be extended to ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us make it explicit Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let W Ă C2 be the affine variety generated by the polynomial ϕ “ FG with F, G P Crx, ys “: O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We consider the vector fields U “ FXG, V “ GXF P XpC2q, where XF and XG are Hamiltonian vector fields w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t the Poisson structure tx, yu :“ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that U, V are tangent to W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Urϕs, V rϕs P xϕy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easily checked that rU, V s “ ϕXtF,Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The action of the trivial Lie algebra g “ R2 on W that sends its canonical basis pe1, e2q to U, and V respectively, is a weak symmetry action on the singular foliation Fϕ :“ xϕyXpC2q, and induces a Lie algebra map, ϱ: g ÝÑ XpWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) A universal Lie algebroid of Fϕ is a Lie algebroid (see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19 of [LGL22b]) because, 0 � Oµ bO XpC2q ϕ B Bµ bOid � Fϕ CHAPTER 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ON WEAK AND STRICT SYMMETRIES: AN OBSTRUCTION THEORY 150 is a O-module isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here µ is a degree ´1 variable, so that µ2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The universal algebroid structure over that resolution is given on the set of generators by: ℓ2 ˆ µ bO B Bx, µ bO B By ˙ :“ Bϕ Bx µ bO B By ´ Bϕ By µ bO B Bx (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) and ℓk :“ 0 for every k ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Write XtF,Gu “ BtF, Gu By B Bx ´ BtF, Gu Bx B By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, we can put ηpe1, e2q :“ BtF, Gu By µ bO B Bx ´ BtF, Gu Bx µ bO B By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) Take for example, Fpx, yq “ y ´ x2 and Gpx, yq “ y ` x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The isotropy Lie algebra gp0,0q of Fϕ is Abelian, since zero is a strong singular point of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 (1), gp0,0q is a R2-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A direct computation shows that the action on gp0,0q is not trivial, but takes value in O µ bO B Bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Besides, Equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) applied to tF, Gu “ 4x gives ηpe1, e2q “ ´4 µ bO B By.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) If η|p0,0q were a coboundary of Chevalley Eilenberg, we would have (in the notations of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) that ηpx, yq|p0,0q “ βprx, ysR2q ´ ℓ1 2px, βpyqq ` ℓ1 2py, βpxqq P O µ bO B Bx, for all x, y P g (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) for some linear map β : g ÝÑ gp0,0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, Equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) is impossible by Equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) and since η|p0,0q ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In orther words, its class clpηq does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12 (2), the action ϱ given in Equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) cannot be extended to ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Conclusion: We applied the existence of a Lie 8-morphism to the question of lifting to M an g-action on M{F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of making strict a weak g-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We apply this question to several geometric issues, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' neighbourhood of leaves, Lie algebra actions on an affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 10 Bi-submersion towers 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Symmetries of bi-submersions In this chapter, we introduce the notion “bi-submersion towers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The work contained in this chapter is entirely original, except for the notion below that arose in a discussion between C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Laurent-Gengoux, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ryvkin, and I, and will be the object of a separate study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us firstly recall the definition of bi-submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The concept of bi-submersion over singular folia- tions has been introduced in [AS09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let M be a manifold endowed with a singular foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A bi-submersion B s � t � M over F is a triple pB, s, tq where: B is a manifold, s, t: B Ñ M are submersions, respectively called source and target, such that the pull-back singular foliations s´1F and t´1F are both equal to the space of vector fields of the form ξ ` ζ with ξ P Γpkerpdsqq and ζ P Γpkerpdtqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Namely, s´1F “ t´1F “ Γpkerpdsqq ` Γpkerpdtqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) In that case, we also say that pB, s, tq is a bi-submersion over pM, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation over a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x P M and X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xn P F inducing generators for Fx :“ F{IxF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We know from [AS09] that there is an open neighborhood W of px, 0q P M ˆ Rn such that pW, t, sq is a bi-submersion over F, where spx, yq “ x and tpx, yq “ expx ˜ nÿ i“1 yiXi ¸ “ ϕ řn i“1 yiXi 1 pxq, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) where for X P XpMq, ϕX 1 denotes the time-1 flow of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Such bi-submersions are called path holonomy bi-submersions [AZ13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 151 CHAPTER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' BI-SUBMERSION TOWERS 152 Now we can introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A bi-submersion tower over a singular foliation F on M, is a (finite or infinite) sequence of manifolds and maps as follows TB : ¨ ¨ ¨ si`1 � ti`1 � Bi`1 si � ti � Bi si´1 � ti´1 � ¨ ¨ ¨ s1 � t1 � B1 s0 � t0 � B0, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) together with a sequence Fi of singular foliations on Bi, with the convention that B0 “ M and F0 “ F, such that for all i ě 1, Fi Ă Γpker dsi´1q X Γpker dti´1q, for each i ě 1, Bi`1 si � ti � Bi is a bi-submersion over Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A bi-submersion tower over pM, Fq shall be denoted as pBi`1, si, ti, Fiqiě0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The bi-submersion tower over F in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) is said to be of length n P N if Bj “ Bn, sj “ tj “ id and Fj “ t0u for all j ě n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us spell out some consequences of the axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For i ě 1, two points b, b1 P Bi of the same leaf of Fi satisfy si´1pbq “ si´1pb1q and ti´1pbq “ ti´1pb1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, for all b P Bi, TbFi Ă pker dsi´1q|b X pker dti´1q|b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us explain how such towers can be constructed out of a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 in [AS09], there always exists a bi-submersion B1 s0 � t0 � M over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The C8pB1q-module Γpker ds0q X Γpker dt0q is closed under Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When it is locally finitely generated, it is a singular foliation on B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, it admits a bi-submersion B2 s1 � t1 � B1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, we have obtained the two first terms of a bi-submersion tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can then continue this construction provided that Γpker ds1q X Γpker dt1q is locally finitely generated as a C8pB2q-module, and that it is so at each step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A bi-submersion tower TB “ pBi`1, si, ti, Fiq over pM, Fq is called exact bi- submersion tower over pM, Fq when Fi`1 “ Γpkerpdsiqq X Γpkerpdtiqq for all i ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is called a path holonomy bi-submersion tower (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' path holonomy atlas bi-submersion tower) if Bi`1 si � ti � Bi is a path holonomy bi-submersion (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a path holonomy atlas) for Fi for each i ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When a path holonomy bi-submersion tower is exact, we speak of exact path holonomy bi-submersion tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following theorem gives a condition which is equivalent to the existence of a bi-submersion tower over a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof uses Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17 which is stated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following items are equivalent: 1In real analytic case, the module Γpker ds1q X Γpker dt1q is locally finitely generated because of the noetherianity of the ring of germs of real analytic functions [Fri67, Siu69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' BI-SUBMERSION TOWERS 153 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' F admits a geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There exists an exact path holonomy bi-submersion tower over pM, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For smooth maps φ, ψ: M ÝÑ N, we denote by ψΓpker dφq the space of ψ-projectable vector fields in Γpker dφq Ă XpMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1 ñ 2 : Assume that F admits a geometric resolution pE, d, ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, ρpΓpE´1qq “ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pB1, s0, t0q be a path holonomy bi-submersion over pM, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let b P B1 and Ub an open neighborhood of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , erq be a local trivialization of E´1 on the open subset U “ t0pUbq Ă M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define a map on generators by ÐÝ‚ : ΓUpE´1q ÝÑ s0ΓUbpker dt0q Ă XpB1q (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) ei ÞÝÑ ÐÝ ei :“ ÐÝÝ ρpeiq and extend by s0-linear map on U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ÐÝ‚ is additive and for every i P t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ru and f P C8pUq one has ÐÝ fei “ ps0q˚pfqÐÝ ei .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17, the map (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since for every i ě 1, ÐÝ ei is s0-related to ρpeiq, in particular the map (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) restricts to a surjective map ker ρ|U ÝÑ ker ´ ds0|ΓUbpker dt0q ¯ “ Γpker ds0q X Γpker dt0q|Ub Ă XpUbq (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) e ÞÝÑ ÐÝe By exactness in degree ´1, ker ρ|U “ dpΓUpE´2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, ker ρ is locally finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By subjectivity of the map (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5), Γpker ds0q X Γpker dt0q “: F1 is also locally finitely generated, in par- ticular F1 is a singular foliation on B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, one can take a path holonomy bi-submersion pB2, s1, t1q over pB1, F1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof continues the same as the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us make a step further for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let b P B2 and Ub an open neighbourhood of b Let pe1, e2, e3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='q be a local trivialization of E´2 on the open subset U “ t1pUbq Ă B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Just like in the first step, define the surjective s1-linear map, ΓUpE´2q ÝÑ s1ΓUbpker dt1q Ă XpB2q (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) e ÞÝÑ ÐÝ ei :“ ÐÝÝ dpeiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' which restricts to a surjective map ker pd: ΓUpE´2q Ñ ΓUpE´1qq ÝÑ ker ` ds1|Γpker dt1q ˘ “ Γpker ds1q X Γpker dt1q Ă XpB2q, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) e ÞÝÑ ÐÝe since 0 “ dpeq “ ds1pÐÝe q for any e P ker ρ|U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By exactness in degree ´2, the C8pUq-module ker pd: ΓUpE´2q Ñ ΓUpE´1qq “ dpΓUpE´3qq is (locally) finitely generated, hence F2 :“ Γpker ds1q X Γpker dt1q is a singular foliation on B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof continues by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2 ñ 1 is proven by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 and Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' BI-SUBMERSION TOWERS 154 Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume that there exists a bi-submersion tower TB “ pBi, ti, si, Fiqiě0 over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, ¨ ¨ ¨ � ker ds2 � dt2 � ker ds1 � dt1 � ker ds0 � dt0 � TM � ¨ ¨ ¨ � B3 t2 � B2 t1 � B1 t0 � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) is a complex of vector bundles, which is exact on the sections level2 if TB is an exact bi-submersion tower, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if Fi “ Γpker dsi´1q X Γpker dti´1q for all i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any element b P Bi`1 and any vector v P ker dsi Ă TbBi`1 one has dtipvq P TtipbqFi, (since Γpker dsiq Ă t´1 i pFiq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ùñ dtipvq P pker dsi´1 X ker dti´1q |tipbq by Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ùñ dtipvq P ker dsi´1 and dti´1 ˝ dtipvq “ 0, for all i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This shows the sequence (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) is a well-defined complex of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us prove that it is exact when Fi “ Γpker dsi´1qXΓpker dti´1q for all i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ξ P Γ pker dsi´1q be a ti´1-projectable vector field that projects to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' dti´1pξq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that ξ P Γpker dsi´1q X Γpker dti´1q “ Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since ti is a submersion, there exists a ti-projectable vector field ζ P t´1 i pFiq that satisfies dtipζq “ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The vector field ζ can be written as ζ “ ζ1`ζ2 with ζ1 P Γ pker dtiq and ζ2 P Γ pker dsiq, because t´1 i pFiq “ Γpker dsiq`Γpker dtiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has, dtipζ2q “ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A similar argument shows that the map, Γpker ds0q dt0 ÝÑ t˚ 0F, is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves exactness in all degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One of the consequence of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 is that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If there exists a sequence of maps M � ε0 � B1 � ε1 � B2 � ε2 � ¨ ¨ ¨ (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) where for all i ě 0, εi is a section for both si and ti then the pull-back of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) on M through the sections pεiqiě0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ¨ ¨ ¨ dt3� ε˚ 2,0 ker ds2 dt2 � ε˚ 1,0 ker ds1 dt1 � ε˚ 0 ker ds0 dt0 � TM (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) is a complex of vector bundles, with the convention εn,0 “ εn ˝ ¨ ¨ ¨ ˝ ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If TB is an exact bi-submersion tower then, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) is a geometric resolution of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In case that TB is an exact path holonomy bi-submersion tower, such a sequence (1) always exists, since the bi-submersions pBi`1, si, tiq are as in Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For such bi-sumersions, the zero section x ÞÑ px, 0q is a section for both si and ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2Let us explain the notion of exactness at the level of sections when the base manifolds are not the same: what we mean is that for all n ě 0, Γpker dtnq X Γpker dsnq “ ptn`1q˚pΓpker dsn`1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Equivalently, it means that the pull-back of the vector bundles in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) to any one of the manifold Bm with m ě n is exact at the level of sections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e Γpt˚ n`1,m ker dsn`1q dtn`1 � Γpt˚ n,m ker dsnq dtn � Γpt˚ n´1,m ker dsn´1 q (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) is a short exact sequence of C8pBmq-modules, with tn,m “ tn ˝ ¨ ¨ ¨ ˝ tm for all m ě n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' BI-SUBMERSION TOWERS 155 Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Under the assumptions of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7, assume the tower of bi-submersion TB is of length n ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, the pull-back of the sequence of vector bundles ker dsn � dtn � t˚ n ker dsn´1 � � ¨ ¨ ¨ � dt2 � t˚ 2,n ker ds1 � dt1 � TBn`1 ˆTM ker ds0 � pr1 � TBn`1 � Bn`15 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) is a geometric resolution of the pull-back foliation t´1 0,npFq Ă XpBn`1q, where pr1 is the projection on TBn`1 and for i ě 1, ti,j is the composition ti ˝ ¨ ¨ ¨ ˝ tj : Bj`1 Ñ Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7, the complex in Equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, the projection of the fiber product TBn`1 ˆTM ker ds0 to TBn`1 induces the singular foliation t´1 0,npFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Lift of a symmetry to the bi-submersion tower Let us investigate what an action ϱ: g Ñ XpMq of a Lie algebra g on pM, Fq would induce on a bi-submersion tower TB over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We start with some vocabulary and preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pB, s, tq be a bi-submersion of a singular foliation F on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We call lift of a vector field X P XpMq to the bi-submersion pB, s, tq a vector field r X P XpBq which is both s-projectable on X and t-projectable on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The coming proposition means that the notion of lift to a bi-submersion only makes sense for symmetries of the singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If a vector field on M admits a lift to a bi-submersion pB, s, tq, then it is a symmetry of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let r X P XpBq be a lift of X P XpMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since r X is s-projectable, r r X, Γpker dsqs Ă Γpker dsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since r X is t-projectable, r r X, Γpker dtqs Ă Γpker dtq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Hence: r r X, s´1pFqs “ r r X, Γpker dsq ` Γpker dtqs “ r r X, Γpkerpdsqs ` r r X, Γpker dtqs Ă Γpker dsq ` Γpker dtq “ s´1pFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In words, r X is a symmetry of s´1F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since r X projects through s to X, X is a symmetry of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We investigate the existence of lifts of symmetries of F to bi-submersions over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For a given X P spFq, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the lift r X to a given bi-submersion is not unique, even when it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, two different lifts of a X P spFq to a bi-submersion pB, s, tq differ by an element of the intersection Γpkerpdsqq X Γpkerpdtqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' r X is a symmetry of Γpkerpdsqq X Γpkerpdtqq: r r X, Γpkerpdsqq X Γpkerpdtqqs Ă Γpkerpdsqq X Γpkerpdtqq, since r X is s-projectable and t-projectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' BI-SUBMERSION TOWERS 156 As the following example shows, the lift of a symmetry to a bi-submersion may not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the trivial foliation F :“ t0u on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any diffeomorphism φ: M ÝÑ M, pM, id, φq is a bi-submersion over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Every vector field X P XpMq is a symmetry of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If it exists, its lift has to be given by, r X “ X since the source map is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But r X “ X is t-projectable if and only if X is φ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A non φ-invariant vector field X therefore admits no lift to pM, id, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, internal symmetries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' elements in F admit lifts to any bi-submersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pB, s, tq be any bi-submersion of a singular foliation F on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Every internal symmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' every vector field in F, admits a lift to pB, s, tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let X P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since s: B ÝÑ M is a submersion, there exists Xs P XpBq s-projectable on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since t is a submersion, there exists Xt P XpBq t-projectable on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction Xs P s´1pFq and Xt P t´1pFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Using the property (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) of the bi-submersion pB, s, tq, the vector fields Xs and Xt decompose as $ & % Xs “ Xs s ` Xs t with Xs s P Γpkerpdsqq, Xs t P Γpkerpdtqq, Xt “ Xt s ` Xt t with Xt s P Γpkerpdsqq, Xt t P Γpkerpdtqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, Xs t is s-projectable to X and t-projectable to 0 while Xt s is s-projectable to 0 and t-projectable to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows that, r X :“ Xt s ` Xs t , is a lift of X to the bi-submersion pB, s, tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us make the Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14 more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pB, s, tq be any bi-submersion of a singular foliation F on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Every vector field X P F admits a lift r X P XpBq on pB, s, tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, r X can be decomposed as r X “ ÝÑ X ` ÐÝ X where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the vector field ÝÑ X P XpBq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ÐÝ X P XpBq) is t-related (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' s-related) with X P XpMq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the vector field ÝÑ X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ÐÝ X) is tangent to the fibers of s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Take ÝÑ X :“ Xt s P Γpker dsq and ÐÝ X :“ Xs t P Γpker dtq in the Proof 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Upon choosing generators for F, F ÝÑ Γpker dtq X ÞÝÑ ÐÝ X can not be a O-linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following lemma is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pB, s, tq be any bi-submersion of a singular foliation F on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any t-projectable vector field of Γpker dsq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' s-projectable of Γpker dtq) is of the form ÝÑ X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ÐÝ X) for some X P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' BI-SUBMERSION TOWERS 157 We can now state one of the important results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It uses several concepts introduced in [AS09], which are recalled in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any symmetry X P spFq admits a lift 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to any path holonomy bi-submersion pB, s, tq, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to Androulidakis-Skandalis’ path holonomy atlas, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to a neighborhood of any point in a bi-submersion through which there exists a local bisection that induces the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In cases (1) or (2) in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18, a linear lift X Ñ r X can be defined on the whole space spFq of symmetries of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' As an immediate consequence of Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12, we obtain that for all X, Y P spFq, Č rX, Y s ´ r r X, rY s P Γpker dsq X Γpker dtq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) Proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let X P spFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume that pB, s, tq “ pW, s0, t0q is a path holonomy bi-submersion associated to some generators X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Xn P F as in Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Fix pu, y “ py1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ynqq P W Ă M ˆ Rn, set Y :“ řd i“1 yiXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since dϕY 1 pXq “ pϕY 1 q˚pXq P X ` F, there exists Zy P F depending in smoothly on y such that dt0pX, 0q “ X ` Zy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Take rZy P t´1 0 pFq such that dt0p rZyq “ Zy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has, dt0 ´ pX, 0q ´ rZy ¯ “ X “ ds0pX, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can write rZy “ rZ1 y ` rZ2 y, with rZ1 y P Γpker ds0q, rZ2 y P Γpker dt0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By construction, r X :“ pX, ´ rZ1 yq is a lift of X to the bi-submersion pW, s0, t0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If XB P XpBq and XB1 P XpB1q are two lifts of the symmetry X on the path holonomy bi- submersions pB, s, tq and pB1, s1, t1q respectively, then pXB, XB1q is a lift of X on the composition bi-submersion B ˝ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item 2, since the path holonomy atlas is made of fibered prod- ucts and inverse of holonomy path holonomy bi-submersions [AS09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also Xa is a symmetry of pB, t, sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Item 2 in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10 of [AS09] states that if the identity of M is carried by pB, s, tq at some point v P B, then there exists an open neighborhood V Ă B of v that satisfies s|V “ s0 ˝ g and t|V “ t0 ˝ g, for some submersion g: V ÝÑ W, for W of the form as in item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, for all X P spFq there exists a vector field r X P XpV q fulfilling ds|V p r Xq “ dt|V p r Xq “ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A symmetry of the tower of bi-submersion TB “ pBi`1, si, ti, Fiqiě0 is a family X “ pXiqiě0, with the i-th component Xi in spFiq, such that dsi´1pXiq “ dti´1pXiq “ Xi´1 for all i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote by spTBq the Lie algebra of symmetries of TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The next theorem gives a class of bi-submersion tower to which any symmetry of the base singular foliation F lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' BI-SUBMERSION TOWERS 158 Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let TB be a path holonomy bi-submersion tower (or an exact path holonomy atlas bi-submersion tower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A vector field X P XpMq is a symmetry of F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' rX, Fs Ă F, if and only if it is the component on M of a symmetry of TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is a direct consequence of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 and of item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is due to the fact that the tower TB is generated by path holonomy bi-submersions, and then we can lift symmetries at every stage of the tower TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pXiqiě0 be a lift of X0 :“ X P spFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For i ě 1, ∇i X :“ adXi preserves Γpker dsi´1q, since Xi is si´1-projectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Altogether, they define a chain map p∇i Xqiě0 at the section level of the complex (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8), on projectable vector fields in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9), since for every i ě 0 and any ti-projectable vector field ξ P ker dsi, dtiprXi`1, ξsq “ rdtipXi`1q, dtipξqs “ rXi, dtipξqs, that is dti ˝ ∇i`1 X “ ∇i X ˝ dti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In [GZ21], under some assumptions, it is shown that if a Lie group G acts on a foliated manifold pM, Fq, then it acts on its holonomy groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is likely that this result follows from Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21, this will be addressed in another study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Lifts of actions of a Lie algebra on a bi-submersion tower We end the section with the following constructions and some natural questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let TB “ pBi`1, si, ti, Fiqiě0 be an exact path holonomy bi-submersion tower over a singular foliation pM, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21, any vector field X P spFq lifts to a symmetry pXiqiě0 of TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Once a lift is chosen, we can define a linear map, X P spFq ÞÑ pXiqiě1 P spTBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ϱ: g Ñ XpMq be a strict symmetry action of a Lie algebra g on pM, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x P g, there exists pϱpxqiqiě0, with ϱpxqi P spFiq Ă XpBiq a symmetry of TB such that ϱpxq0 “ ϱpxq P spFq, by Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Consider the composition, x P g ÞÝÑ ϱpxq P spFq ÞÝÑ pϱpxqiqiě0 P spTBq ÞÑ ϱpxq1 P XpB1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all x, y P g, rϱpxq, ϱpyqs1 ´ rϱpxq1, ϱpyq1s “ dt1pC1px, yqq with C1px, yq P Γpker ds1 Ñ B2q a t1-projectable vector field, for some bilinear map C1 : ^2 g ÝÑ Γpker ds1 Ñ B2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This follows from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7, because rϱpxq, ϱpyqs1 ´rϱpxq1, ϱpyq1s P Γpker ds0qXΓpker dt0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' CHAPTER 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' BI-SUBMERSION TOWERS 159 Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The map C1 : ^2 g ÝÑ Γpker ds1 Ñ B2q of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='24 satisfies for all x, y, z P g, C1prx, ysg, zq ` ∇2 ϱpxqpC1py, zqq` ö px, y, zq “ dt2pC2px, y, zqq (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) for some tri-linear map C2 : ^3 g ÝÑ Γpker ds2 Ñ B3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, ∇2 is as in Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For x, y, z P g, dt1 pC1prx, ysg, zqq ` ö px, y, zq “ rϱprx, ysgq, ϱpzqs1 ´ rϱprx, ysgq1, ϱpzq1s` ö px, y, zq “ ((((((((( ( rrϱpxq, ϱpyqs, ϱpzqs1 ´ rrϱpxq, ϱpyqs1, ϱpzq1s` ö px, y, zq “ ´(((((((((( ( rrϱpxq1, ϱpyq1s, ϱpzq1s ` rdt1pC1px, yqq, ϱpzq1s` ö px, y, zq “ dt1prC1px, yq, ϱpzq2sq` ö px, y, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have used Jacobi identity and dt1pϱpzq2q “ ϱpzq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that dt1 ` C1prx, ysg, zq ´ rC1px, yq, ϱpzq2s` ö px, y, zq ˘ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) Again Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7 implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is a natural question: Question: Can we continue the construction in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='25 to a Lie 8-morphism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There is another natural type of questions: Question: Can a strict symmetry action ϱ: g Ñ XpMq of a Lie algebra g on a singular foliation pM, Fq lift to a given geometric resolution pE‚, d, ρq of F?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Discussion Can we answer this question with what we have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We know by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 that ϱ lifts on any universal Lie 8-algebroid pE, Qq of F to a Lie 8-morphism Φ: g ÝÑ X‚pEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If we can choose at least the polynomial-degree zero of the Taylor coefficient Φ1 : ^2 g Ñ X´1pEq to be zero, then the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If yes, can we assume that the previous action to preserve the dg-almost Lie algebroid?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Again, if yes, then the polynomial-degree `1 can be chosen to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It seems that being able to suppress the higher Taylor coefficients has a strong geometric meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We intend to study that problem in a subsequent paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Question: Can a bi-submersion tower be equipped with a (local) Lie 8-groupoid structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Appendices 160 APPENDIX A Tensor algebra We have used almost everywhere in the thesis, modules or vector spaces that arise as the quotient of the tensor algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is worth it to dedicate a section to recall the construction and some basic facts on tensor algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this chapter we assume that the reader is familiar with the notion of O-modules, graded O-modules, and vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is known that many algebras such as the exterior algebra, symmetric algebra, Clifford algebras [Tod11], universal enveloping algebras [Bek] and many other algebras are the quotient of tensor al- gebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These make the tensor algebra a fundamental and a very useful notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this thesis we have dealt a lot with O-multilinear maps, it simply means that these maps are O-linear w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t each argument while we fix the other arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The tensor algebra is used to characterize multilinear relations between algebraic objects related to modules or vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There are several ways to construct the tensor algebra, we refer the reader e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to chapter 16 of [Lan05] or [And, Kei05] for more details on the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When O “ K the reader may replace "O-module" by "K-vector space" and the construction of the tensor algebra that we give below works the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Tensor product Let V and W and Z be O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The tensor product V bO W of V and W over O is the O-module which is defined by the following universal property Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' There exists a O-bilinear map, p: V ˆ W Ñ V bO W, that satisfies the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' any O-bilinear map B : V ˆ W Ñ Z admits a unique O-linear map γ : V bO W Ñ Z, such that 161 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' TENSOR ALGEBRA 162 γ ˝ p “ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In other words, such that the following diagram commutes: V ˆ W p � B � V bO W γ � Z (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Furthermore, if there is another O-module Q and a O-bilinear map, p1 : V ˆ W Ñ Q with the property (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1), then there exists a unique isomorphism i: V bO W Ñ Q such that p1 “ i ˝ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We first show "uniqueness" when such O-module exits, then show existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Uniqueness: Suppose that there exist two pairs pVbW, p: VˆW Ñ VbOWq and pV ˜bW, ˜p: VˆW Ñ V ˜bOWq satisfying the property A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By using two times A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, we deduce the existence of a unique O-linear map i: V bO W Ñ V ˜bW, and another one j : V ˜bOW Ñ VbW such that i ˝ p “ ˜p and j ˝ ˜p “ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that j ˝ i ˝ p “ p and i ˝ j ˝ ˜p “ ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By unicity of the factorization in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1, we must have i ˝ j “ id and j ˝ i “ id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This proves item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Existence: Let us consider the free O-module P generated by elements of the Cartesian product V ˆ W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' elements of P are formal finite sums of elements of the form fpv, wq, with f P O and v P V, w P W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Next, we divide by the submodule R Ă P of P generated by elements of the following types: pv1 ` v2, wq ´ pv1, wq ´ pv2, wq, pv, w1 ` w2q ´ pv, w1q ´ pv, w2q, pfv, wq ´ fpv, wq, and pv, fwq ´ fpv, wq which are the relations that elements of the tensor product must satisfy with v, v1, v2 P V and w, w1, w2 P W and f P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In those notations, the tensor product of V and W over O is defined as the quotient V bO W :“ P{R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For v P V, w P W, we denote the class of pv, wq P V ˆ W by v b w P V bO W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This quotient comes with the natural map, p: V ˆ W Ñ V bO W, pv, wq ÞÑ v b w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any O-linear map B : V ˆ W Ñ Z, the O-linear map q: P Ñ Z which is given on the basis of P by px, yq ÞÑ qppx, yqq :“ Bpx, yq clearly goes to quotient to define a O-linear map sq: V bO W Ñ Z, since e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' q ppv1 ` v2, wq ´ pv1, wq ´ pv2, wqq “ qppv1 ` v2, wqq ´ qppv1, wqq ´ qppv2, wqq “ Bpv1 ` v2, wq ´ Bpv1, wq ´ Bpv2, wq “ 0, by bilinearity of B also, qppfv, wq ´ fpv, wqq “ qppfv, wqq ´ fqppv, wqq, by O-lineraity of q “ Bpfv, wq ´ fBpv, wq “ 0, by O-bilineraity of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We did everything to get, sq ˝ p “ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, we can take γ “ sq to satisfy (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' TENSOR ALGEBRA 163 Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let V, W, Z be O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have the following isomorphisms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' V bO W » W bO V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' O bO V » V, and for any ideal I Ă O, Also, O I bO V » V IV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' pV bO Wq bO Z » V bO pW bO Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For item 1, the map pv, wq P V ˆ W ÞÑ w b v P V bO W goes to quotient to the twist map v b w ÞÑ w b v and give the isomorphism whose inverse is w b v P W bO V ÞÑ v b w P V bO W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For item 2, the map pf, vq P O ˆ V Ñ fv also induces an isomorphism whose inverse is v P V ÞÑ 1 b v P O bO V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A similar map gives the second clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For item 3, the isomorphism is, obviously, pv1 bv2qbv3 ÞÑ v1 bpv2 bv3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is trivially extended to more O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A similar construction as in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 can be done for a finite family of O- modules V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , Vr, as in the case of bilinear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then V1 bO ¨ ¨ ¨ bO Vr is a universal object that factorizes r-multilinear maps, defined on V1 ˆ ¨ ¨ ¨ ˆ Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We naturally have, pV1 bO V2q bO V3 » V1 bO pV2 bO V3q » V1 bO V2 bO V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' So, in this thesis, we make no difference in how we denote the elements pv1 b v2q b v3, v1 b pv2 b v3q, or v1 b v2 b v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is important to notice that if C and C1 are graded O-algebras, then C bO C1 is a graded O-algebra with product pc1 b c1 1qpc2 b c1 2q :“ p´1q|c1 1||c2|c1c2 b c1 1c1 2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) for homogeneous elements c1, c2 P C and c1 1, c1 2 P C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, if C and C1 are unitary with units 1C and 1C1, respectively, then the algebra C bO C is also unitary with unit 1C b 1C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, to avoid explosion of notations, we denote the product of two elements a, b by ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, we will not make notational distinctions between the unit elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Convention A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For Φ: C Ñ C1 and Ψ: C2 Ñ C3 two homogeneous morphisms of Z-graded O- modules, then Φ b Ψ : C bO C2 Ñ C1 bO C3 stands for the following morphism: pΦ b Ψqpx b yq “ p´1q|Ψ||x|Φpxq b Ψpyq, for all homogeneous x P C, y P C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In virtue of Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 and Convention A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5, if Φ: C Ñ C1 and Ψ: C2 Ñ C3 are graded algebra morphisms, then ΦbΨ is also a graded algebra morphism w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t to the product defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2): Φ b Ψppc1 b c1 1qpc2 b c1 2qq “ p´1q|c1 1||c2|Φ b Ψpc1c2 b c1 1c1 2q “ p´1q|c1 1||c2|Φpc1c2q b Ψpc1 1c1 2q “ p´1q|c1 1||c2|Φpc1qΦpc2q b Ψpc1 1qΨpc1 2q “ ` Φpc1q b Φpc1 1q ˘ ` Ψpc2q b Ψpc1 2q ˘ “ ` Φ b Ψpc1 b c1 1q ˘ ` Φ b Ψpc2 b c1 2q ˘ , since Φ, Ψ are of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' TENSOR ALGEBRA 164 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 The tensor algebra of a linear space Let V be an O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For k P N0, the k-th tensor power T k OV over O of V (elements of polynomial- degree k) is the tensor product of V with itself k times, namely T k OV :“ V bO ¨ ¨ ¨ bO V loooooooomoooooooon k times we also adopt the convention T 0 OV » O, and V » T 1 OV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This leads to consider the O-module con- structed as the direct sum of the tensor powers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' T ‚ OV :“ 8 à k“1 T k OV “ O ‘ V ‘ pV bO Vq ‘ pV bO V bO Vq ‘ ¨ ¨ ¨ Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-module T ‚ OV comes equipped with a graded unital O-algebra structure, which is induced by the canonical map T k OV ˆ T ℓ OV ÝÑ T k OV bO T ℓ OV » T k`ℓ O V, that is extended by bilinearity to all T ‚ OV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We denote this product by b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The unit element is 1 P O » T 0 OV, in particular, we have 1 b v “ 1 ¨ v “ v for every v P V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 T ‚ OV as a co-algebra In this section, we consider an O-module V which can be possibly graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, the construction is independent whether the module is graded or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The notion of co-algebra structure is important in the context of the thesis, since it allows dealing with infinite dimension objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It appears all along the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For this concept, see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A natural way to construct a co-product (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g [JLL, Kas12]) structure 1 on T ‚ OV ∆: T ‚ OV ÝÑ T ‚ OV â T ‚ OV is to define it on elements v P V » T 1 OV of polynomial-degree 1, and on the unit element 1 P O » T 0 OV and extend it to a (degree 0) O-algebra morphism to the whole T ‚ OV, namely for v1 b ¨ ¨ ¨ b vk P T k OV, ∆pv1 b ¨ ¨ ¨ b vkq “ ∆pv1q ¨ ¨ ¨ ∆pvkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The one which is defined by v ÞÑ v b 1 ` 1 b v 1 ÞÑ 1 b 1 endows T ‚ OV with a co-associative (co-commutative) co-algebra structure, that is, it satisfies the axioms of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, the maps ∆ b id, id b ∆: T ‚ OV b T ‚ OV Ñ T ‚ OV b T ‚ OV b T ‚ OV 1here b is an "outer" tensor product, it should not be confused with the internal tensor product in T ‚ OV that denotes also its graded algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' TENSOR ALGEBRA 165 are algebra morphisms, so are p∆ b idq ˝ ∆ and pid b ∆q ˝ ∆, therefore it suffices to check that they coincide on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us check that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For v P V, p∆ b idq ˝ ∆pvq “ ∆ b idpv b 1 ` 1 b vq “ ∆pvqb1 ` ∆p1qbv “ pv b 1 ` 1 b vq b 1 ` p1 b 1q b v “ pv b 1q b 1 ` p1 b vq b 1 ` p1 b 1q b v “ pid b ∆q ˝ ∆pvq, by associativity of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By extending ∆ to an algebra morphism, one gets explicit expressions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For v b w P T 2 OV, one has ∆pv b wq “ ∆pvq∆pwq “ pv b 1 ` 1 b vqpw b 1 ` 1 b wq “ pv b wq b 1 ` v b w ` p´1q|v||w|w b v ` 1 b pv b wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have used Formula (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) and 1 b v “ 1 ¨ v “ v, each time b is the tensor symbol in T ‚ OV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If V is not graded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', concentrated in degree zero, there is no sign p´1q|v||w|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More generally, for every v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , vn P V, ∆pv1 b ¨ ¨ ¨ b vnq “ n´1 ÿ i“1 ϵpσq ÿ σPSpi,n´iq vσp1q b ¨ ¨ ¨ b vσpiq â vσpi`1q b ¨ ¨ ¨ b vσpnq, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) where σ P Spi, n ´ iq is a pi, n ´ iq-shuffle and ϵpσq is the Koszul sign associated to the n-uplet v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' See Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B Homological algebra The goal of this chapter is to introduce some important results on homological algebra, which are used in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Although, these are classical notions in commutative algebra, I think it is important to make a brush-up on them for the readability of the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Most of the notions of this chapter can be found in [Cha14, Eis95, Hid89, Mic07], I also have learned a lot from the Lecture notes [Las18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Complexes of modules We recall that a module over O is like a vector space in the sense that all the axioms still hold, except that the underlying field is replaced by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this section, when O “ K, the reader may replace "module" by "vector space".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For us, a Z-graded module over O is a module V endowed with a direct sum decomposition V “ ‘iPZVi of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We simply say "O-modules" for graded modules which are concentrated in degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every i P Z, elements of Vi are said to be of degree i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let V and W be graded Z-modules over O, a O-linear map L: V Ñ W is said to be homogeneous or a morphism of Z-graded O-modules of degree |L| :“ ℓ P Z, if LpVkq Ď Vk`ℓ for all k P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The set of all O-linear maps of degree ℓ from V to W form an O-module that we denote by Homℓ OpV, Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that HomOpV, Wq :“ À ℓPZ Homℓ OpV, Wq is a graded module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, this graded module comes equipped with natural graded Lie bracket given by the graded commutator, namely rF, Gs :“ F ˝ G ´ p´1q|F||G|G ˝ F (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1) for any homogeneous elements F, G P HomOpV, Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easily checked that the bracket satisfies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' rF, Gs “ ´p´1q|F||G|rG, Fs (graded skew-symmetry) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' p´1q|F||H|rF, rG, Hss`p´1q|H||G|rH, rF, Gss`p´1q|G||F|rG, rH, Fss “ 0, (graded Jacobi identity) for homogeneous O-linear maps F, G, H P HomOpV, Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 166 APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 167 Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A complex of O-modules pV‚, dq is a graded module V “ À iPZ Vi together with a squared to zero O-linear map d: V Ñ V of degree `1 called the differential map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In other words, it is a sequence ¨ ¨ ¨ ÝÑVi´1 d ÝÑ Vi d ÝÑ Vi`1ÝÑ ¨ ¨ ¨ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) O-linear maps such that d2 “ d ˝ d “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every i P Z, elements of Vi are called cochains of degree i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We say that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2) is bounded below/above if Vi “ 0 for i ď n{i ě n, for some n P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A subcomplex of a complex pV, dq a collection of O-modules pV1 i Ď ViqiPZ such that dpV1 iq Ă Vi`1 for each i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular pV1, d1 “ d|V1q is a complex of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, it induces a complex ` V{V1, d ˘ called the quotient complex where for i P Z, pV{V1qi :“ Vi{V1 i and d: Vi{V1 i ÝÑ Vi`1{V1 i`1 is determined uniquely by the universal property of the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pV, dq be a complex of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Denote by di : Vi Ñ Vi`1 the restriction of d to Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then d2 “ 0 means that di ˝ di´1 for each i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, Imdi´1 Ď ker di for every i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This leads us to the next definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pV, dq be a complex of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For each i P Z, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a i-cocycle of pV, dq is an element of kerpVi d ÝÑ Vi`1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' a i-coboundary of pV, dq is an element of ImpVi´1 d ÝÑ Viq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the i-th cohomology group of pV, dq is the quotient HipVq :“ kerpVi d ÝÑ Vi`1q ImpVi´1 d ÝÑ Viq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The complex pV, dq is exact at i if HipVq “ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is said to be exact or acyclic if it is exact at every degree i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pV, dVq and pW, dWq be complexes of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A chain map or complex of O-modules morphism between the complexes pV, dq and pW, dq is a O-linear map L: V Ñ W of degree 0, which commutes with the differentials, that is a collection of O-linear map L‚ : V‚ ÝÑ W‚, such that the following diagram commutes ¨ ¨ ¨ � Vi Li � dV � Vi`1 Li`1 � � ¨ ¨ ¨ ¨ ¨ ¨ � Wi dW � Wi`1 � ¨ ¨ ¨ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' dW ˝ Li “ Li`1 ˝ dV for every i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 168 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A homotopy between two chain maps K‚, L‚ : V‚ ÝÑ W‚ is the datum thi : Vi ÝÑ Wi´1uiě1 of O-linear maps, that satisfies for each i P Z, Ki ´ Li “ dW ˝ hi ` hi´1 ˝ dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These maps are displayed in the following diagram as ¨ ¨ ¨ � Vi´1 Ki´1´Li´1 � dV � Vi hi � Ki´Li � dV � Vi`1 Ki`1´Li`1 � hi`1 � � ¨ ¨ ¨ ¨ ¨ ¨ � Wi´1 dW � Wi dW � Wi`1 � ¨ ¨ ¨ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4) (a) When there is a homotopy between two chain maps, L‚, K‚ : V‚ ÝÑ W‚, we often write L „ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can check that „ is indeed an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) Two complexes of O-modules pV, dVq and pW, dWq are said to be homotopy equivalent, if there exist chain maps L‚ : V‚ ÝÑ W‚ and K‚ : W‚ ÝÑ V‚ such that L ˝ K „ idW‚ and K ˝ L „ idV‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Likewise, one can check that homotopy equivalence between complexes of O-modules is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that in particular, if L‚ : V‚ ÝÑ W‚ is a chain map that is an O-linear iso- morphism, then its inverse is also a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, they define a homotopy equivalence between pV, dVq and pW, dWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let L‚ : V‚ ÝÑ W‚ be a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For every i P Z, we have L ˆ kerpVi dV ÝÑ Vi`1q ˙ Ď kerpWi dW ÝÑ Wi`1q, and L ˆ ImpVi´1 dV ÝÑ Viq ˙ Ď ImpWi´1 dW ÝÑ Wiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' L induces naturally a well-defined O-linear map HpLq: HpVq Ñ HpWq, rvs ÞÑ rLpvqs: for every v P kerpVi dV ÝÑ Vi`1q and v0 P Vi´1 HpLqprv ` dVpv0qsq “ rLpv ` dVpv0qqs “ rLpvq ` dW ˝ Lpv0qqs “ rLpvqs “ HpLqprvsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Notice that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' homotopic chain maps induce the same map on cohomology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' if the complexes of O-modules pV, dVq and pW, dWq are homotopy equivalent through L, then HpLq: HpVq Ñ HpWq is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If pV1, d1q is an acyclic subcomplex of a complex pV, dq, then H‚pV{V1q » H‚pVq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The projection p‚ : V ÝÑ V‚{V1 ‚ is a chain map from pV, dq and pV{V1, dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We claim p‚ induces an isomorphism on the cohomology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' To show injectivity of Hppq: H‚pVq ÝÑ H‚pV{V1q, let e P ker d such that ppeq “ e P Im d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' there is u P V|e|´1 such that e “ dpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows that e ´ du P V1 |e|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies, d1pe ´ duq “ dpe ´ duq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5) By Exactness of pV1, d1q, ùñ e ´ du “ d1pvq, (for some v P V1 |e|´1 Ă V|e|´1) ùñ e “ dpu ` vq P Im d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 169 This proves injectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Surjectivity goes as follows: let e P Vi such that dpeq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' dpeq “ 0 ùñ dpeq P V1 i`1 We have, d1pdpeqq “ d ˝ dpeq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By exactness of pV1, d1q, we can write dpeq “ d1pvq for some v P V1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This implies that, e ´ v “ u P ker d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, Hppqprusq “ rppuqs “ rppeq ´ ppvqqs “ rppeqs “ res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Chevalley-Eilenberg complex The following example of complex is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We have used it several times in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Especially in Chapter 9 to define obstruction classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let us recall the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We refer the reader e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' to [Wag10] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pg, r¨ , ¨sgq be a Lie algebra and V a K-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A representation or action of g on V is a Lie algebra morphism ν : pg, r¨ , ¨sgq ÝÑ pEndpV q, r¨ , ¨sq (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) where pEndpV q, r¨ , ¨sq denotes the vector space EndpV q of endomorphisms of V together with the Lie bracket r¨ , ¨s which is the commutator: rα, βs “ α ˝ β ´ β ˝ α, @ α, β P EndpV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this case, V is then called a g-module (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t to ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the literature, the action ν is often denoted by ¨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6) means that for all x, y P g, νprx, ysgq “ rνpxq, νpyqs “ νpxq ˝ νpyq ´ νpyq ˝ νpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here are two important examples of g-modules that we often use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The adjoint action: g acts on itself by the Lie bracket, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' νpxqpyq :“ rx, ysg for all x, y P g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, ν : g ÝÑ Endpgq, x ÞÝÑ rx, ¨ sg “: adx is a Lie algebra morphism: adrx,ysg “ rrx, ysg, ¨ sg “ ´rry, ¨ sg, xsg ´ rr¨ , xsg, ysg, pby identity of Jacobiq “ rx, ry, ¨ sgsg ´ ry, rx, ¨ sgsg “ adx ˝ ady ´ ady ˝ adx “ radx, adys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The trivial action: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' K is a g-module through the action νpxqpλq :“ 0 for all x P g and all λ P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ‘ Now let us recall the definition of the Chevalley-Eilenberg complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 170 Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let ν : g ÝÑ EndpV q be an action of g on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Chevalley-Eilenberg complex of g valued in V is the complex ¨ ¨ ¨ ÝÑHomKp^i´1g, V q dCE ÝÑ HomKp^ig, V q dCE ÝÑ HomKp^i`1g, V qÝÑ ¨ ¨ ¨ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) whose i-th cochains space is defined to be HomKp^ig, V q, the vector space of i-linear skew-symmetric linear maps from g ˆ ¨ ¨ ¨ ˆ g looooomooooon i-times to V , under the convention HomKp^0g, V q » V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The differential map is defined for µ P HomKp^ig, V q by ´ dCEµ ¯ px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xi`1q “ i`1 ÿ k“1 p´1qk´1νpxkqpµpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , pxk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xi`1qq ` ÿ 1ďkălďi`1 p´1qk`lµprxk, xlsg, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , pxkl, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xi`1q where pxk means xk is missing in the k-th place also pxkl means that xk, xl are missing in the k-th and l-th place respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It follows from the identity of Jacobi that dCE ˝ dCE “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all x, y, z P g we have, dCEpxq “ νpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For µ P HomKpg, V q, ` dCEµ ˘ px, yq “ νpxqpµpyqq ´ νpyqpµpxqq ´ µprx, ysgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For η P HomKp^2g, V q, ´ dCEη ¯ px, y, zq “ νpxqpηpy, zqq ´ νpyqpηpx, zqq ` νpzqpηpx, yqq ´ ηprx, ysg, yq ` ηprx, zsg, yq ´ ηpry, zsg, xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When g is of finite dimension, the Chevalley-Eilenberg complex (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7) is canonically isomorphic to the complex p^‚g˚ b V, dq and for ξ “ ξ1 ^ ¨ ¨ ¨ ^ ξk P ^kg˚, dpξ b vq “ nÿ i“1 pξ ^ ξiq b pξi ¨ vq ´ dgpξq b v (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8) where ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , ξn is a basis of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In the formula (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8), dg is the Chevalley-Eileberg differential of g w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='t the trivial action on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 Operations on complexes We have used and adapted the following lemma many times in this thesis, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 171 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pV‚, dVq and pW‚, dWq be complexes of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, the pair pHom‚ OpV, Wq , Bq is a complex of O-modules, where the differential map is given by BpFq :“ dW ˝ F ´ p´1q|F|F ˝ dV, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9) for all homogeneous element F P HomOpV, Wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-linear map, B: HomOpV, Wq Ñ HomOpV, Wq, is clearly of degree `1, since the maps dV and dW are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Moreover, for every F P HomOpV, Wq B2pFq “ BpdW ˝ F ´ p´1q|F|F ˝ dVq “ dW ˝ dW ˝ F loooooomoooooon “0 ´p´1q|F|`1(((((( dW ˝ F ˝ dV ´ p´1q|F| ˜ (((((( dW ˝ F ˝ dV ´ p´1q|F|`1 F ˝ dV ˝ dV looooomooooon “0 ¸ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It’s worth it to notice that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' the cocycles F of pHom‚ OpV, Wq, Bq are those that satisfy dV ˝ F “ p´1q|F|F ˝ dW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, the chain maps between pV, dVq and pW, dWq are the 0-cocyles of pHom‚ OpV, Wq , Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' two chain maps F‚, G‚ : V‚ ÝÑ W‚ are homotopic if and only if G ´ F is 0-coboundary of pHom‚ OpV, Wq , Bq, that is, there exists a O-linear map H P Hom´1 O pV, Wq of degree ´1 such that F ´ G “ dW ˝ H ` H ˝ dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' when pV‚, dVq “ pW‚, dWq, we have ` Hom‚ OpV, Vq, B “ rdV, ¨ s ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Direct sum and tensor product of complexes Let pV‚, dVq and pW‚, dWq be complexes of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, the tensor product V‚ bO W‚ together with the grading pV bO Wqk “ à redi`j“k Vi bO Wj for k P Z, comes equipped with a differential map classically defined by B “ dV b id ` id b dW, namely, Bpv b wq “ dVpvq b w ` p´1q|v|v b dWpwq, for all homogeneous elements v, w P V bO W, is complex of O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The operator B is indeed of degree `1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is easily checked that B2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 172 Bi-complex Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A bi-complex or double complex is a collection of O-modules V “ pVi,jqi,jPZ together with two families of O-linear maps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' dh i,j : Vi,,j Ñ Vi`1,j, such that dh i,j ˝ dh i´1,j “ 0, for i, j P Z called horizontal differential map 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' dv i,j : Vi,j Ñ Vi,j`1, such that dv i,j ˝ dv i,j´1 “ 0, for all i, j P Z called vertical differential map that obey for all i, j P Z the identity dh i,j`1 ˝ dv i,j “ dv i,j ˝ dh i`1,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In this thesis we only consider first quadrant bi-complexes that is, Vi,j “ 0 for all i P Zď0 and j P N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' These can be represented as the commutative diagram .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ò Ò Ò ¨ ¨ ¨ Ñ Vi,j dh Ñ Vi,j dh Ñ Vi,j Ñ 0 dv Ò dv Ò dv Ò ¨ ¨ ¨ Ñ Vi,j dh Ñ Vi,j dh Ñ Vi,j Ñ 0 dv Ò dv Ò dv Ò ¨ ¨ ¨ Ñ Vi,j dh Ñ Vi,j dh Ñ Vi,j Ñ 0 Ò Ò Ò 0 0 0 "-2 column" "-1 column" "last column" (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) One associate to the bi-complex (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10) the so-called total complex which is defined by the anti- diagonals of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10), namely ´ Tr :“ À i`j“r Vi,j ¯ r with total differential D: Tr Ñ Tr`1 defined by Dpτijq :“ dhpτijq ´ p´1qrdvpτijq, for τij P Vi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Indeed, D2 “ pdh ´ p´1qr`1dvq ˝ pdh ´ p´1qrdvq “ pdhq2 loomoon “0 ´\x18\x18\x18\x18\x18\x18 \x18 p´1qrdh ˝ dv ´ ((((((( ( p´1qr`1dv ˝ dh ´ pdvq2 loomoon “0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16 (Acyclic Assembly Lemma [Cha14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pVi,jqi,jPZ be a first quadrant bi-complex like in the notation above such that the rows are exact, then the total complex pT‚, Dq is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Mapping cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let pV, dVq and pW, dWq be two complexes and L: V‚ Ñ W‚ a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The mapping cone is the complex pC, Bq whose degree i is given by Vi´1 ‘ Wi and whose differential is defined as B “ ˜ ´dV 0 ´L dW ¸ APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 173 That is, the differential is given on elements pv, wq P V ‘ W by Bpv, wq “ p´dVpvq, dWpwq ´ Lpvqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The mapping cone pC, Bq is exact if and only if L: V‚ Ñ W‚ is a quasi-isomorphism, [Cha14], Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 Resolutions of a module Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' An O-module V is said to be projective if it fulfills the following: given a O-linear map L: V : Ñ Z, every surjective O-linear map J : W Ñ Z admits a O-linear map rL: V Ñ W such that the following diagram commutes V � � W � Z � 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11) Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Free modules are projective modules (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2 of [Mic07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let A be a O-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A free/projective resolution (or resolution by free/projective modules) of A is an exact complex pV‚, dq ¨ ¨ ¨ ÝÑV´i´1 d ÝÑ V´i d ÝÑ V´i`1 d ÝÑ ¨ ¨ ¨ d ÝÑ V´2 d ÝÑ V´1 π ÝÑ A ÝÑ 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) such that the V´i’s are free/projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The complex (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12) may be of infinite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When the length is finite, we say that we have a finite free/projective resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In that case, the sequence 0ÝÑV´n d ÝÑ V´n`1 d ÝÑ V´n`2 d ÝÑ ¨ ¨ ¨ d ÝÑ V´2 d ÝÑ V´1 π ÝÑ A ÝÑ 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13) is exact in every degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, the sequence 0 Ñ V´n dÑ Vn´1 is exact at ´n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For instance, the map V´n dÑ Vn´1 injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, V´n » ImpV´n dÑ Vn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By exactness, we have that V´n » ImpV´n dÑ Vn´1q “ kerpV´n`1 dÑ V´nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Every O-module A admits a free/projective resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Firstly, notice that every module is isomorphic to a quotient of a free module: to see this, choose a set of generators tvi P A | i P Iu of the module A so that V “ ÿ iPI Ovi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The O-linear map π: à iPI O Ñ A, pfjqjPJĎI ÞÑ ÿ jPJ fjvj, J is finite is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By the first isomorphism theorem, it follows that A » pÀ iPI Oq{ ker π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Now put À iPI O “: V´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This yields an exact sequence 0 ÝÑ ker π ãÝÑ V´1 π ÝÑ AÝÑ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But ker π does not need to be a free O-module, but there is a free O-module V´2 together with a surjective O-linear map V´2 π1 � � ker π such that V´2{ ker π1 » ker π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This is added to the previous sequence as follows APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 174 0 � ker d1 � � � V´2 π1 � � d1 � � ker π � � � V´1 π � � A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Once again, ker d1 does not need to be a free O-module, therefore we can continue the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Inductively, assume that we have constructed a O-linear map, dn : V´n´1 Ñ V´n, for n ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, there exists a free O-module V´n´2 together with a surjective O-linear map πn`1 : V´n´2 Ñ V´n´1 such that V´n´2{ ker πn`1 » ker dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Joining this to the previous sequence, one get 0 � ker dn`1 � � � V´n´2 πn`1� � dn`1 � � ker dn � � � V´n´1 dn � � V´n � ¨ ¨ ¨ � V´2 d1 � V´1 π � � A (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) By construction, we have kerpdnq “ Impdn`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Therefore, we have built an exact sequence up to length n ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One shall notice that the process (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14) may be continued forever without reaching a free kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is an important operation on complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following Proposition states the localization in the sense of item 2 of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1 preserves exactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More precisely, Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6 ([Sta22], Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9 or [And]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let A be an O-module and S Ă O a multiplicative subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any free resolution of A ¨ ¨ ¨ d � V´3 d � V´2 d � V´1 π � A , the complex ¨ ¨ ¨ S´1d� S´1V´3 S´1d � S´1V´2 S´1d � S´1V´1 S´1π � S´1A is a free resolution of S´1A by S´1O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is well-known that a submodule of a finitely generated module is not finitely generated in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following assertion guarantees this for finitely generated modules over Noetherian rings (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4 of [Eis95], Page 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If O is Noetherian as a ring, then all submodules of an O-module V are finitely generated if and only if V is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following theorem assures existence of free resolution of finite length in a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='8 (Hilbert Syzygy Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Pee, Eis04] Assume O “ Crx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any finitely generated (graded) O-module A admits a finite graded free resolution by finitely generated O-modules ¨ ¨ ¨ d � V´3 d � V´2 d � V´1 π � A of length N ď d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 175 Tor complex Let pV‚, dV, πq be a projective resolution of an O-module A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For any O-module B, we consider the complex ¨ ¨ ¨ � V´3 bO B dVbid� V´2 bO B dVbid� V´1 bO B dVbid � A bO B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We define for i ě 0 Tor´ipA, Bq :“ H´ipV‚ bO Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here are some properties of Tor [Las18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The functor Tor satisfies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Tor0pA, Bq » A bO B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If A is projective, then Tor´ipA, Bq “ 0 for every i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If B is flat, then Tor´ipA, Bq “ 0 for every i ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For all i, Tor´ipA, Bq “ Tor´ipB, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The construction of TorpA, Bq is independent of the choice of the resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' any other projective resolution of A or B yields the same Tor groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Minimal resolutions Detailed definitions on local rings can be found in [Mas62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A ring R is said to be a local ring if it has a "unique maximal ideal", i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', a proper ideal m Ă R such that m contains every other ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A local ring R with maximal ideal m is called regular if m can be generated by n elements, where n “ dim R is the krull dimension1 of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Assume now that R is a local ring with maximal ideal m, and let K :“ R{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Geometrically, local rings correspond to germs of functions on a manifolds or affine variety at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We assume that pR, mq is a local Noetherian commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here is an important lemma that uses definition of local rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='11 (Nakayama).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let V be a finitely generated R-module such that r “ dimpV{mVq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Then, any basis of the vector space V{mV lifts to a (minimal) generating set for V as a R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, V can be generated by r elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Eis04] Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Nakayama Lemma, a local ring R is regular if the dimension of the R{m-vector space m{m2 is dim R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1the Krull dimension of R is by definition the supremum of lengths of all chains of prime ideals in R [Hid89], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 30 APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 176 One can construct free resolutions of finitely generated modules over R like in the proof of Propo- sition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5 by taking a minimal set of homogeneous generators at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This can be formalized as follows, Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A projective resolution ¨ ¨ ¨ d � V´3 d � V´2 d � V´1 � π � A � 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15) of a finitely generated R-module A is said to be minimal if the differential map satisfies dpV´iq Ď mV´i`1 for all i ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Eis04] Every finitely generated R-module A admits a minimal free resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof goes just like in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5, one just need to take minimal set of generators in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By doing so, it suffices to check that dpV´2q Ă V´1: Let r “ dimpA{mAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By Nakayama Lemma, we can take V´1 “ Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One has a short exact sequence, 0 ÝÑ dpV´2q “ ker π ãÝÑ V´1 ÝÑ A ÝÑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It induces an exact sequence dpV´2q{m dpV´2q ãÝÑ V´1{mV´1 ÝÑ A{mA ÝÑ 0, by tensorizing with K “ R{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since by construction V´1{mV´1 » A{mA » Kr, the image of the map dpV´2q{m dpV´2q ãÝÑ V´1{mV´1 is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', dpV´2q Ď mV´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Likewise, by Noetheriality of R, ker π Ă V´1 is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' One can repeat the procedure by starting with a minimal set of generators of ker π and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The proof continues by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Minimal resolutions and the Tor complex: given a minimal projective resolution as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='15), its quotient by the maximal ideal m corresponds to the complex ¨ ¨ ¨ � V´3 bO K dbid � V´2 bO K dbid � V´1 bO K πbid � A bO K � 0 whose cohomology compute Tor‚pA, Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' By minimality of pV‚, d, πq one has d b id ” 0, therefore Tor´ipA, Kq » V´ibO K for every i ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In particular, the ranks of the V´i’s for i ě 2 are independent of the choices made in the construction of the minimal projective resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Koszul complex Assume that V is a free O-module of finite rank n, which is concentrated in degree ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here, we denote by Ź‚ V the graded symmetric algebra of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F : V Ñ O be a O-linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Given a free basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , en of V, F is completely determined by n-uplet pf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fnq Ă O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Eis95, Hid89] The Koszul complex associated to pf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fnq is the complex 0 ÝÑ n ľ V d ÝÑ n´1 ľ V d ÝÑ ¨ ¨ ¨ ÝÑ 2 ľ V d ÝÑ V F ÝÑ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) whose differential d: Ź‚ V d ÝÑ Ź‚´1 V is the unique derivation of Ź‚ V such that, d|V “ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' In other words, dpei1 ^ ¨ ¨ ¨ ^ eikq “ kÿ j“1 p´1qj´1Fpeijqei1 ^ ¨ ¨ ¨ ˆeij ¨ ¨ ¨ ^ eik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17) APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 177 It is easily checked that this gives a well-defined complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' An ordered sequence of elements f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fr P O is called a regular sequence on a module V if the following conditions are satisfied 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' fi is not a zero divisor on the quotient V{xf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fi´1yV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' xf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fryV ‰ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' If pf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , fkq is regular sequence on O, then the Koszul complex (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='16) has no cohomology in degree less equal to ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' of [Hid89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For O “ Krx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xds be the polynomial ring in d indeterminates, the Koszul complex which is associated to px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xdq induces a free resolution of K » O{xx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3 Geometric resolutions of a singular foliation Let us start with this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F be a singular foliation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A complex of vector bundles over F consists of a triple pE‚, d‚, ρq, where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' E‚ “ pE´iqiě1 is a family of vector bundles over M, indexed by negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' dpi`1q P HompE´i´1, E´iq is a vector bundle morphism over the identity of M called the differ- ential map 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' ρ: E´1 ÝÑ TM is a vector bundle morphism over the identity of M called the anchor map with ρpΓpE´1qq “ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' such that ¨ ¨ ¨ � E´i´1 dpi`1q � � E´i dpiq � � Ei´1 � � dp2q� E´1 ρ � � TM � M M M M M (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18) which form a (chain) complex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' dpiq ˝ dpi`1q “ 0 and ρ ˝ dp2q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Two main cohomology groups can be associated to a complex of vector bundles over F: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Cohomology at the level of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The complex of vector bundles (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18) induces a complex of sheaves of modules over functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More explicitly, for every open subset U Ă M, there is a complex: ¨ ¨ ¨ ÝÑΓUpE´i´1q dpi`1q ÝÑ ΓUpE´iq dpiq ÝÑ ΓUpE´i`1qÝÑ ¨ ¨ ¨ dp2q ÝÑ ΓUpE´1q ρ ÝÑ FU Ă XpUq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 178 Since Impdpi`1qq Ď ker dpiq for every i P N, we are allowed to define the quotient spaces, H´ipE‚, Uq “ $ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ % ker ´ ΓUpE´1q ρ ÝÑF ¯ Im ˆ ΓUpE´2q dp2q ÝÑΓUpE´1q ˙ for i “ 1 ker ˜ ΓUpE´iq dpiq ÝÑΓUpEi`1q ¸ Im ˆ ΓUpE´i´1q dpi`1q ÝÑ ΓUpE´iq ˙ if i ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' They are modules over functions on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For each i ě 1, H´ipE‚, Uq is called the i-th cohomology of pE‚, d‚, ρq at the level of sections on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (a) We way then pE‚, d‚, ρq is a geometric resolution of F if for every open set U Ă M and i ě 1, if it induces an exact complex on the level of sections, that is H´ipE‚, U1q “ t0u for every open subset U1 Ă U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (b) A geometric resolution pE‚, d‚, ρq of F is said to be minimal at a point m P M if for each i ě 2 the linear map dpiq |m : E´i ÝÑ E´i`1|m vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Cohomology at an arbitrary point m P M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Also, the complex of vector bundles (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='18), at an arbitrary point m P M, restricts to a complex of vector spaces ¨ ¨ ¨ ÝÑE´i´1|m dpi`1q |m ÝÑ E´i|m dpiq |m ÝÑ E´i`1|mÝÑ ¨ ¨ ¨ dp2q |m ÝÑ E´1 ρ|m ÝÑ TmM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We can look at the quotient vector spaces: H´ipE‚, mq “ $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % ker ˆ E´1|m ρ|m ÝÑTmM ˙ Im ˜ E´2|m dp2q|m ÝÑ E´1|m ¸ for i “ 1 ker ˜ E´i|m dpiq|m ÝÑ Ei`1|m ¸ Im ˜ E´i´1|m dpi`1q|m ÝÑ E´i|m ¸ if i ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here we call H´ipE‚, mq the i-th cohomology of pE‚, d‚, ρq at the point m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' It is important to notice the following: even if pE‚, d‚, ρq is a geometric resolution of F, there are no reasons for pE‚, d‚, ρq to be exact at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For example, if the resolution is minimal at some point m P M, then H´ipE‚, mq » E´i|m for each i ě 2 and H´1pE‚, mq » kerpρ|mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Most of the definitions and operations on chain complexes of modules are adapted in a obvious manner to complexes of vector bundles over M, both on the level of the sections or at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' See also [LLS20, Lav17] for more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 179 On existence of geometric resolutions Geometric resolutions of a singular foliation F as defined in Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2(a) are not guaranteed in all contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' However, there always exists a projective resolution of F as a O-module (see B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' But these resolutions do not induce always a geometric resolution of F, since the projective modules of a projective resolution may not correspond to vector bundles because they may not be locally finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' When the latter condition is satisfied, the Serre-Swan theorem [Swa62, Mor13] states that there is a one-to-one correspondence between locally finitely generated projective modules and sec- tions of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Under the assumptions of the latter theorem, geometric resolutions of singular foliations are exactly projective resolutions at the sections level in the category of chain complexes by O-modules, since sections of vector bundles over M are projective O-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following proposition summarizes some contexts where geometric resolutions exist, see [LLS20] for their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Any algebraic singular foliation2 on Kd admits geometric resolutions by trivial vector bundles and of length ď d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The same holds for a real analytic of holomorphic singular foliation, but only in a neighborhood of a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A locally real analytic singular foliation on a manifold of dimension d admits a geometric reso- lution of length ď d ` 1 on any relatively compact open subset of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here we have some examples of geometric resolutions of singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F0 “ tX P XpV q | Xp0q “ 0u be the singular foliation made of all vector fields vanishing at the origin of a vector space V (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' think of CN or RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The contraction by the Euler vector field ÝÑ E “ N ÿ i“1 xi B Bxi gives rise to a complex of trivial vector bundles ¨ ¨ ¨ ÝÑ ^3 T ˚V ιÝÑ E ÝÑ ^2T ˚V ιÝÑ E ÝÑ T ˚V ιÝÑ E ÝÑ C ˆ V “: C, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='19) whose complex on the sections level is pΩ‚pV q, ιÝÑ E q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Here px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xNq are the canonical coordinates on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter is the Kozul complex associated to the coordinate functions x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , xN of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Since the xi’s form a regular sequence, it is well known that pΩ‚pV q, ιÝÑ E q is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The following complex of vector bundles over V ¨ ¨ ¨ ÝÑ ^3 T ˚V b TV ιÝÑ E bid ÝÑ ^2T ˚V b TV ιÝÑ E bid ÝÑ T ˚V b TV ιÝÑ E bid ÝÑ C b TV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20) is a geometric resolution of F0 since ´ Ω‚pV q b XpV q, ιÝÑ E b id ¯ is also exact (here ΩipV q :“ Γp^iT ˚V q stands for the sheaf of i-forms on V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2A singular foliation which is generated by polynomial vector fields on Kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' HOMOLOGICAL ALGEBRA 180 More generally, the construction we have made in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='20) is still possible by contracting with any vector field X “ N ÿ i“1 Xi B Bxi P XpV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The latter yields a complex of vector bundles that covers the singular foliation FX generated by the Xi B Bxj ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' For instance, if X is a polynomial vector field and pX1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' , XNq form a regular sequence, we get a geometric resolution of FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Let F2 “ I2 0XpK2q Ă F0 be the sub-singular singular foliation made of vector fields vanishing at order 2 at the origin of K2, where I2 0 Ă OpK2q is the ideal generated by the monomials x2, xy, y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that the ideal I2 0 admits a free resolution of the form 0 ÝÑ OpK2q ‘ OpK2q δ1 ÝÑ OpK2q ‘ OpK2q ‘ OpK2q δ0 ÝÑ I2 0 ÝÑ 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21) where for all f, g, h P OpK2q, δ0pf, g, hq “ x2f ` xyg ` y2h and δ1pf, gq “ pxf, xf ´ yg, xgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The free resolution (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='21) has to take the form 0 ÝÑ ΓpI´2q δ1 ÝÑ ΓpI´1q δ0 ÝÑ I2 0 ÝÑ 0, for sum trivial vector bundles I´1, I´2 on K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Thus, the following complex 0 ÝÑ I´2 b TK2 δ1bid ÝÑ I´1 b TK2 δ0bid ÝÑ I2 0 b TK2 ÝÑ 0 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='22) is a geometric resolution of F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Note that I´1 can be identified with the tivial vector bundle S2ppK2q˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' More generally, let Fk be the singular foliation made of vector fields vanishing at order k at the origin of a vector space V of dimension N over R or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The Hilbert’s syzygy theorem assures the existence of a free resolution of length N `1 of the ideal Ik 0 made of functions on V vanishing to order k at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' This resolution is of the form ¨ ¨ ¨ ÝÑ ΓpI´2q δ1 ÝÑ ΓpI´1q δ0 ÝÑ I2 0 ÝÑ 0, for some family of trivial vector bundles pI´iqiě1 over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' We obtain a geometric resolution of Fk of the form ¨ ¨ ¨ ÝÑ ΓpI´2 b TV q δ1bid ÝÑ ΓpI´1 b TV q δ0bid ÝÑ I2 0 b XpV q “ Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Bibliography [And] Gathmann Andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Commutative Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Class Notes TU Kaiserslautern 2013/14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='uni-kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='de/~gathmann/en/commalg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='php.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [AS09] Iakovos Androulidakis and Georges Skandalis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' The holonomy groupoid of a singu- lar foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Journal für die reine und angewandte Mathematik, 2009(626):1–37, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='5167/uzh-23589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [AS11] Iakovos Androulidakis and Georges Skandalis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Pseudodifferential calculus on a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Journal of noncommutative geometry, 5(1):125–152, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='4171/jncg/72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [AS19] Iakovos Androulidakis and Georges Skandalis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' A Baum–Connes conjecture for singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Annals of K-Theory, 4(4):561 – 620, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='2140/akt.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Advances in mathematics (New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1965), 256:348–397, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [DHH86] Klas Diederich, Gilbert Hector, and Ulrich Hirsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Introduction to the Geometry of Folia- tions, Part A: Foliations on Compact Surfaces, Fundamentals for Arbitrary Codimension, and Holonomy, volume 1 of Aspects of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Springer Vieweg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' in Springer Fachme- dien Wiesbaden GmbH, Wiesbaden, second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' edition, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [dJP00] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' de Jong and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Pfister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Local Analytic Geometry: Basic Theory and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Advanced Lectures in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Vieweg+Teubner Verlag, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='com/book/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1007/978-3-322-90159-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [dS01] Ana Cannas da Silva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lectures on symplectic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lecture notes in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Springer, Berlin Paris [etc, C 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='1007/b80865.' metadata={'source': 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algebras of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Manuscripta mathematica, 80(1):309–337, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Hue98] Johannes Huebschmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie-Rinehart algebras, Gerstenhaber algebras and Batalin- Vilkovisky algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Fourier (Grenoble), 48(2):425–440, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Hue03] Johannes Huebschmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Higher homotopies and Maurer-Cartan algebras: Quasi-Lie- Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Hue04] Johannes Huebschmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Lie-Rinehart algebras, descent, and quantization, volume 43 of Fields Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=', Providence, RI, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' 1090/memo/0814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Iva93] Kolář Ivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Natural operations in differential geometry / Ivan Kolář, Peter W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Michor, Jan Slovák.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Springer-Verlag, Berlin New York Heidelberg, C 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [JLL] Bruno Vallette 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Singular subalgebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' arXiv, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/abs/1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='02480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Zam18] Marco Zambon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Holonomy transformations for Lie subalgebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' arXiv, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/abs/2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='10409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' [Še01] Pavol Ševera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Some title containing the words “homotopy” and “symplectic”, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' arXiv, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content='org/abs/math/0105080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE_T4oBgHgl3EQf1hxM/content/2301.08335v1.pdf'} +page_content=' Université de Lorraine, CNRS, IECL, F-57000 Metz, France.' metadata={'source': 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To achieve +better fitting capability, most GNNs are with a large +number of parameters, which makes these GNNs compu- +tationally expensive. Therefore, it is difficult to deploy +them onto edge devices with scarce computational re- +sources, e.g., mobile phones and wearable smart devices. +Knowledge Distillation (KD) is a common solution to +compress GNNs, where a light-weighted model (i.e., the +student model) is encouraged to mimic the behavior of a +computationally expensive GNN (i.e., the teacher GNN +model). Nevertheless, most existing GNN-based KD +methods lack fairness consideration. As a consequence, +the student model usually inherits and even exagger- +ates the bias from the teacher GNN. To handle such a +problem, we take initial steps towards fair knowledge +distillation for GNNs. Specifically, we first formulate a +novel problem of fair knowledge distillation for GNN- +based teacher-student frameworks. Then we propose a +principled framework named RELIANT to mitigate the +bias exhibited by the student model. Notably, the de- +sign of RELIANT is decoupled from any specific teacher +and student model structures, and thus can be easily +adapted to various GNN-based KD frameworks. We +perform extensive experiments on multiple real-world +datasets, which corroborates that RELIANT achieves +less biased GNN knowledge distillation while maintain- +ing high prediction utility. Open-source code can be +found at https://github.com/yushundong/RELIANT. +Keywords: Graph Neural Networks, Algorithmic Fair- +ness, Knowledge Distillation +1 +Introduction +In recent years, Graph Neural Networks (GNNs) have +shown satisfying performance in a plethora of real-world +applications, e.g., medical diagnosis [27] and credit +risk scoring [30], to name a few. +In practice, the +depth and the number of parameters of GNNs largely +∗University +of +Virginia, +Email: +{yd6eb, +epb6gw, +jun- +dong}@virginia.edu +†Beijing University of Posts and Telecommunications, Email: +yuanyiling@bupt.edu.cn +‡Texas A&M University, Email: nzou1@tamu.edu +§Northeastern University, Email: q.wang@northeastern.edu +Recidivism +Credit +4 +8 +12 +16 +SP +T-GCN +S-GCN +T-GAT +S-GAT ++13.8% ++21.6% ++15.8% ++28.3% +(a) Bias under ∆SP on CPF. +Recidivism +Credit +2 +4 +6 +8 +10 +EO +T-GCN +S-GCN +T-GAT +S-GAT ++34.8% ++9.0% ++22.5% ++18.3% +(b) Bias under ∆EO on AKD. +Figure 1: +A comparison of exhibited bias between +teacher and student models based on two representative +GNN knowledge distillation frameworks (CPF and +GraphAKD). "T" and "S" represent the teacher and +the student model, respectively. The names of GNN +mark out the corresponding teacher models. +determine their expressive power [16], which directly +influence their performances in various graph learning +tasks [2]. +Typically, deeper GNN layers enable the +model to capture information that is multiple hops away +from any node [21], while a larger number of learnable +parameters enable GNN to fit more complex underlying +data patterns [2]. However, in most cases, the inference +efficiency of GNNs is inevitably degraded by the deep +layers or the large number of parameters. Such efficiency +degradation naturally makes these GNNs inapplicable to +be deployed on edge devices (e.g., mobile phones) with +limited computational resources [16, 19]. +Due to the problem above, it is necessary to compress +those computationally expensive GNNs for deployment +on edge devices. Knowledge Distillation (KD) is a com- +mon approach to compress GNNs but still maintains +a similar level of prediction performance [34, 16, 19]. +Here, the basic idea of KD is to let a light-weighted +student model (as the compressed GNN) learn to mimic +the behavior (e.g., output logits) of the teacher model +(usually a computationally expensive GNN). However, +most existing KD approaches do not have any fairness +consideration over different demographic subgroups, and +the optimized student model often preserves and even +exaggerates the exhibited bias from the teacher GNN. +Consequently, when the compressed model is deployed +in real-world application scenarios, there could exist dis- +crimination toward specific populations. Here we provide +preliminary analysis based on two representative GNN +knowledge distillation frameworks, namely CPF [34] and +GraphAKD [16]. Specifically, we measure the exhibited +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited +arXiv:2301.01150v1 [cs.LG] 3 Jan 2023 + +bias in the widely-studied node classification task on +two real-world datasets. Here Recidivism is a network +of defendants [18, 1], while Credit is a network between +bank clients [35, 1]. We adopt two traditional metrics, +i.e., ∆SP (measuring the level of bias under Statistical +Parity [10]) and ∆EO (measuring the level of bias under +Equal Opportunity [15]), to measure the exhibited bias +of GNN predictions. We present a comparison of the +exhibited bias between teacher and student models in +Fig. 1. Empirical results show that student models tend +to yield more biased results compared with the teacher +GNN model, which could be attributed to the biased +guidance from the teacher GNN during training. It is +worth noting that in most cases, directly retraining the +teacher GNN for debiasing is infeasible, since retraining +the teacher GNN with a large number of parameters is +computationally expensive. Hence, mitigating the bias +for the student model is an urgent need. +Despite the necessity of mitigating bias for the stu- +dent model, existing exploration remains scarce. In this +paper, we aim to make an initial step towards develop- +ing a debiasing framework that can be easily adapted to +various existing GNN-based KD methods. However, this +task is non-trivial mainly due to the following three chal- +lenges: (1) Gap towards Fair Knowledge: For most +KD frameworks designed for compressing GNNs, the +teacher GNN model usually serves as the sole source of +supervision signal for the training of the student model. +Therefore, if the teacher GNN exhibits any bias, such +biased knowledge tends to be inherited by the student +model. Hence, learning a fair student model with biased +supervision from the teacher GNN is our first challenge. +(2) Gap towards End-to-End Learning: A critical +advantage of existing KD models is the end-to-end learn- +ing paradigm, which enables the distilled knowledge to +be tailored to specific downstream tasks. In such an end- +to-end learning process, highly efficient gradient-based +optimization techniques are widely adopted. However, +widely-used fairness notions (e.g., Statistical Parity and +Equal Opportunity) are defined on the predicted la- +bels. Hence the corresponding bias metrics are naturally +non-differentiable w.r.t. the student model parameters. +Developing a debiasing framework suitable for gradient- +based optimization techniques in the end-to-end learning +paradigm is our second challenge. (3) Gap towards +Generalization: Various KD models have been pro- +posed for compressing GNNs to satisfy different applica- +tion scenarios. In fact, most KD models are developed +based on certain designs of student models. Developing +a framework that is student-agnostic and easily adapted +to different KD models is our third challenge. +To tackle the above challenges, in this paper, we +propose a novel framework named RELIANT (faiR +knowlEdge distiLlatIon for grAph Neural neTworks) +to mitigate the bias learned by the student model. +Specifically, we first formulate a novel research problem +of Fair Knowledge Distillation for GNN-based Teacher- +Student Frameworks. To tackle the first challenge, we +incorporate a learnable proxy of the exhibited bias for +the student model. In this way, despite the knowledge +(from the teacher GNN) being biased, the student +model still makes less biased predictions under proper +manipulations on the proxy. +To tackle the second +challenge, we propose to approximate the bias level +of the student model, where the approximation is +differentiable (w.r.t. +the student model parameters) +manner. +In this way, the highly efficient end-to-end +learning paradigm is preserved, and the gradient-based +optimization techniques are still applicable. To tackle +the third challenge, we design the proposed framework +RELIANT in a student-agnostic manner. In other words, +the debiasing for the student model does not rely on any +specific design tailored for the student model structure. +Therefore, RELIANT can be easily adapted to different +GNN-based knowledge distillation approaches. The main +contributions of this paper are summarized as follows. +• Problem Formulation. We formulate and make +an initial investigation on a novel research problem +of fair knowledge distillation for GNN-Based teacher- +student frameworks. +• Algorithmic Design. We propose a principled +framework named RELIANT that learns the proxy +of bias for the student model during KD. RELIANT +achieves student-agnostic debiasing via manipulat- +ing the proxy during inference. +• Experimental Evaluation. We conduct compre- +hensive experiments on multiple real-world datasets +to verify the effectiveness of the proposed framework +RELIANT in learning less biased student models. +2 +Problem Definition +Notations. We denote matrices, vectors, and scalars +by bold uppercase letters (e.g., X), bold lowercase +letters (e.g., x), and regular lowercase letters (e.g., x), +respectively. For any matrix, e.g., X, we use Xi,j to +indicate the element at the i-th row and j-th column. +Preliminaries. We utilize G = {V, E, X} to denote an +attributed network (graph). Here, V = {v1, ..., vn} is +the set of nodes, E ⊆ V × V is the set of edges, and +X = {x1, ..., xn} (xi ∈ Rd, 1 ≤ i ≤ n) is the set +of node attribute vectors. We use A ∈ {0, 1}n×n to +denote the adjacency matrix of the graph. If there is an +edge from the i-th node to the j-th node, Ai,j = 1; +otherwise Ai,j = 0. +Moreover, we denote the pre- +trained teacher GNN model in a knowledge distillation +framework as f ˆθ parameterized by ˆθ. Here ˆθ denotes the +optimized θ of the pre-trained teacher model. Similarly, +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +… +Teacher GNN Model +Student Model (GNN as an example) +Input Attributed Graph +Attributed Graph with Proxy of Bias +Node Embeddings from Teacher Model +Node Embeddings from Student Model +Teacher Logits +Student Logits +Training Stage +Inference Stage +Input Attributed Graph +Attributed Graph with +Manipulated Proxy of Bias +Adding Expectation +of Bias Proxy +Less Biased Node +Embeddings +Less Biased Predictions +… +Maximizing Utility: 111 +Learning Proxy of Bias: +111 +Enforcing the Attribution +of Bias to Proxy: 111 +… +Approximation with +Differentiable +Polynomials +Quantitative +Level of Bias +ℒUtility +ℒAttr +ℒProxy +Figure 2: An overview of the proposed framework RELIANT including the training and inference stage. +we denote the student model as gφ parameterized by +φ. We represent the optimized φ after the training of +the student model as ˆφ. Without loss of generality, we +consider the most widely studied node classification as +the downstream task. For the teacher model f ˆθ(v), we +denote the set of outcome logits, i.e., the continuous +output vector corresponding to each node, as ˆY(t) = +{ˆy(t) +1 , ˆy(t) +2 , ..., ˆy(t) +n }, where ˆy(t) +i +∈ Rc. Here c is the total +number of node classes. Correspondingly, we represent +the set of outcome logits of the student model gφ(v) +as ˆY(s) = {ˆy(s) +1 , ˆy(s) +2 , ..., ˆy(s) +n }. +For any node vi, the +predicted label given by the student model (denoted +as ˆY (s) +i +for the i-th node) is determined by the largest +value across all c dimensions in ˆy(s) +i . +Based on the definitions above, we formulate the +problem of Fair Knowledge Distillation for GNN-based +Teacher-Student Frameworks as follows. +Problem 1. Fair +Knowledge +Distillation +for +GNN-Based Teacher-Student Frameworks. Given +an attributed network G and a GNN-based teacher- +student framework including a trained teacher GNN f ˆθ +and a student model gφ to be trained, our goal is to +achieve a more fair student model with similar prediction +utility compared with f ˆθ for the node classification task. +3 +Methodology +In this section, we first present an overview of the pro- +posed framework RELIANT, followed by the objective +function formulation and optimization strategy. +3.1 +Workflow of RELIANT Here we first introduce +the workflow of the proposed framework RELIANT. +In general, we introduce the three main functionalities +involved in the proposed framework RELIANT, namely +maximizing the utility, learning proxy of bias, and +enforcing the attribution of bias to the proxy. We present +an overview of RELIANT in Fig. 2. +Specifically, to +tackle the first challenge (gap towards fair knowledge), +we propose to learn the proxy of bias as extra input +attributes for the student model to account for the +exhibited bias, and wipe out such bias during inference +via proper manipulations on the proxy. To tackle the +second challenge (gap towards end-to-end learning), we +formulate our debiasing objectives in a differentiable +(w.r.t. the parameters of the student model) manner. To +tackle the third challenge (gap towards generalization), +we achieve debiasing in a student-agnostic manner. In +other words, the proposed framework RELIANT does not +rely on any specific student model structure to achieve +debiasing. We elaborate more details as follows. +Maximizing Utility. In general, existing GNN-based +KD frameworks consider the GNNs with high compu- +tational costs as the teacher model, and the goal is to +learn a student model with limited computational costs +but similar prediction utility (e.g., accuracy in node clas- +sification tasks). To maintain the utility of the teacher +model, it is necessary to utilize the knowledge from the +teacher model as the supervision signal for the training +of the student. In particular, a common approach is to +utilize the output classification logits from the teacher +model as the supervision signal, which we take as an +example here. Specifically, we minimize the distance be- +tween the logits from the student model and the teacher +model. We formally formulate the optimization goal as +min +φ +� +vi∈V +γd +� +ˆy(t) +i , ˆy(s) +i +� +, +(3.1) +where ˆy(s) +i +and ˆy(t) +i +are the output logits from the student +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +Layer +OLayer 2 +OLayer. +OLayerL +Omodel gφ(vi) and teacher model f ˆθ(vi), respectively. The +function γd(., .) measures the distance between two logit +vectors. Different choices can be adopted to measure the +distance, e.g., cosine distance and Euclidean distance. +Correspondingly, to maximize the prediction utility, we +minimize the objective function +LUtility(φ) = +� +vi∈V +γd +� +ˆy(t) +i , ˆy(s) +i +� +. +(3.2) +Learning Proxy of Bias. +It is worth noting that +even if the sensitive attributes are removed from the +input data, the student model could still exhibit bias +in its predictions. The main reason is that there could +exist dependencies between those sensitive attributes +and non-sensitive ones. Moreover, the information about +sensitive attributes could also be encoded in the input +network structure [6]. As a consequence, it is difficult to +prevent the student model from leveraging information +about sensitive attributes. To handle such a problem, +we propose to learn the proxy of bias x(p) +i +as extra input +attributes for each node vi. Here, the rationale is that +if much information about bias comes from the learned +proxy instead of those encoded in the non-sensitive +attributes or the network structures, then we are able +to achieve less biased inference results by removing such +a proxy. As a consequence, such a proxy of bias should +account for the exhibited bias of the student model +as much as possible. +In other words, the exhibited +bias should be largely attributed to the proxy of bias +rather than the sensitive information encoded in the +network data. More specifically, to enforce the proxy of +bias contributing to the exhibited bias in the student +model, we propose to maximize the exhibited bias when +these proxies are taken as input into the student model +together with other attributes and the network structure. +We formally formulate our goal as +max +X(p) JBias({gφ(γ(vi, X(p))) : i ∈ V}), +(3.3) +where γ(., .) is a function that takes a node and the +proxy of bias matrix as input, and outputs the node +with a concatenated node attribute vector [xi, x(p) +i ]. +Here x(p) +i +is the i-th row of X(p). JBias(.) is a function +that takes the set of logits from the student model +as input and outputs a value indicating the level of +exhibited bias. Nevertheless, the computation is non- +differentiable under traditional fairness notions such +as Statistical Parity and Equal Opportunity. Here we +propose to utilize orthogonal polynomials (e.g., Legendre +polynomials [4]) that are differentiable w.r.t. the output +logits to approximate the level of bias under traditional +fairness notions. This makes JBias differentiable w.r.t. +the learnable parameter φ. Correspondingly, we formally +give the objective function towards the goal above as +LProxy(X(p)) = −JBias({gφ(γ(vi, X(p))) : i ∈ V}). +(3.4) +Enforcing the Attribution of Bias to the Proxy. +Only achieving Eq. (3.3) is not enough to enforce the +proxy of bias largely accounting for the exhibited bias +of the student model. This is because the exhibited bias +may come from the vanilla node attributes instead of +the learned proxy. More specifically, we denote P( ˆY (s)) +as the probability of the positive prediction given by +the student model for any specific node1. We assume +that there are underlying unbiased and biased node +attributes X(u) and X(b), respectively. When Eq. (3.3) +is achieved, it is clear that P( ˆY (s)|X(u), X(b), X(p)), i.e., +P( ˆY (s)|X, X(p)), is biased. However, both X(b) and X(p) +could be the source of the exhibited bias. It is worth +noting that our goal is to learn proxy X(p) to account for +as much of the exhibited bias as possible. Therefore, to +enforce the effectiveness of the proxy, it is necessary to +ensure that the exhibited bias is attributed to the biased +information from X(p) instead of X(b). In other words, +we need to enforce P( ˆY (s)|X(u), X(b)) being less biased, +which ensures that X(p) accounts for the exhibited bias +as much as possible. Nevertheless, P( ˆY (s)|X(u), X(b)) +is intractable considering that the number of the input +dimension number for the student model is fixed. Hence +we propose an alternative approach. Denote the learned +proxy of bias and the underlying sensitive attribute +vector of any node as x(p) and s, respectively. +We +propose to utilize a vector E[x(p)] to replace each row +in X(p) as ˜X(p). +In this way, the rows in ˜X(p) are +independent from s, i.e., the information about sensitive +attributes encoded in X(p) is wiped out. To enforce the +attribution of bias to the proxy X(p), the predictions +should be as fair as possible when the information +about X(p) is removed. Therefore, we formulate our +last optimization goal as +min +φ JBias({gφ(˜γ(vi, ˜X(p))) : i ∈ V}), +(3.5) +where ˜γ(., .) is a function that takes a node and the +matrix ˜X(p) as input, and returns the input node with a +concatenated node attribute vector [xi, ˜x(p) +i +]. Here ˜x(p) +i +is the i-th row of matrix ˜X(p). We formally present the +corresponding objective function as +LAttr(φ) = JBias({gφ(˜γ(vi, ˜X(p))) : i ∈ V}). +(3.6) +Inference with Student Model. To achieve less bi- +ased inference, an ideal case is to make predictions +1Here we consider binary classification task for simplicity. +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +with P( ˆY (s)|X(u)). +However, it is difficult to explic- +itly extract X(u) from X. +Instead, we argue that +P( ˆY (s)|X(u), X(b), ˜X(p)) exhibits similar level of bias +compared with P( ˆY (s)|X(u)). This is because (1) the +bias exhibited by P( ˆY (s)|X(u), X(b), ˜X(p)) minimally re- +lies on X(b) after enforcing Eq. (3.3) and Eq. (3.5); and +(2) there is no further information about sensitive at- +tributes encoded in ˜X(p) (as discussed above). Conse- +quently, we propose to utilize gφ(˜γ(vi, ˜X(p))) to achieve +less biased prediction for node vi in the inference stage. +4 +Optimization Objectives & Strategy +We present the optimization objectives of RELIANT +followed by the training strategy in this section. +Optimization Objectives. Based on our discussions +above, here we present a summary of the optimization +objectives for the proposed RELIANT. First, to optimize +the parameter φ, we formally formulate a unified +objective function as +Lφ = LUtility(φ) + λ · LAttr(φ). +(4.7) +Here λ serves as a hyper-parameter controlling the effect +of debiasing the student model. Second, to optimize the +learnable proxy of bias X(p), we formally present the +objective function as +LX(P) = LProxy(X(p)). +(4.8) +Optimization Strategy. To train the proposed frame- +work RELIANT, we propose to optimize the parameter +φ and learnable proxy of bias X(p) in an alternative man- +ner. We present the algorithmic routine of RELIANT +in Algorithm 1. +Algorithm 1 Fair Knowledge Distillation for GNNs +Input: G: the graph data; f ˆθ: the trained teacher GNN model; +gφ: the student model; +Output: g ˆ +φ: the optimized student model; X(p): the proxy of +bias matrix; +1: Randomly initialize X(p); +2: while stop training condition not satisfied do +3: +Compute Lφ according to Eq. (4.7); +4: +Update φ with +∂Lφ +∂φ ; +5: +Compute LX(p) according to Eq. (4.8); +6: +Update X(p) with +∂LX(p) +∂X(p) ; +7: end while +8: return g ˆ +φ and X(p); +5 +Experimental Evaluations +In this section, we will first introduce the downstream +learning task and adopted real-world datasets, followed +by the backbone models, baseline methods, and eval- +uation metrics. Next, we present the implementation +details of the models. Finally, we discuss the evaluation +results of the proposed RELIANT. In particular, we +aim to answer the following research questions through +experiments: RQ1: How well can RELIANT balance +the utility and fairness of the student model compared +with other baselines? RQ2: To what extent each com- +ponent of RELIANT contributes to the overall debiasing +performance? RQ3: How will the choice of the hyper- +parameter λ affect the performance of RELIANT? +5.1 +Experimental Settings Here we introduce the +settings for our experimental evaluation. +Downstream Task & Real-world Datasets. +We +adopt the widely studied node classification as the +downstream task in this paper. We adopt four real-world +datasets for the experimental evaluations, including two +widely used network datasets (Recidivism [18, 1] and +Credit Defaulter [35, 1]) and two newly constructed +ones based on real-world data (DBLP and DBLP-L). +In Recidivism, nodes are defendants released on bail, +and edges denote the connections between defendants +computed from their past criminal records. Here the +sensitive feature is race, and we aim to classify if a +certain defendant is unlikely to commit a crime after +bail. In Credit Defaulter, nodes are credit card users, +and edges are the connections between these users. +Here we consider the age period of these users as their +sensitive feature, and we aim to predict the future +default of credit card payments. +Additionally, we +also construct two co-author networks, namely DBLP +and DBLP-L based on AMiner network [29], which is +a co-author network collected from computer science +bibliography. Specifically, we first filter out the nodes in +AMiner network with incomplete information. Then we +adopt two different approaches to sample a connected +network from the filtered dataset: DBLP is a subgraph +sampled with random walk, while DBLP-L is the largest +connected component of the filtered AMiner network. In +both datasets, nodes represent the researchers in different +fields, and edges denote the co-authorship between +researchers. The sensitive attribute is the continent of +the affiliation each researcher belongs to, and we aim to +predict the primary research field of each researcher. The +detailed statistics of these four datasets are in Table 1. +KD Framework Backbones & Teacher GNNs. To +evaluate the capability of RELIANT in generalizing to +different GNN-based KD backbones, here we adopt two +representative KD frameworks designed for compressing +GNNs, namely CPF [34] and GraphAKD [16]. In general, +CPF minimizes the distribution distance between the +logits from teacher and student to provide supervision +information for the student, while GraphAKD utilizes +adversarial training to achieve knowledge distillation for +the student. The student model of CPF and GraphAKD +is PLP [34] and SGC [32], respectively. For each KD +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +Table 1: The basic information about the real-world datasets adopted for experimental evaluation. Sens. denotes +the semantic meaning of sensitive attribute. +Dataset +Recidivism +Credit Defaulter +DBLP +DBLP-L +# Nodes +18,876 +30,000 +39,424 +129,726 +# Edges +321,308 +1,436,858 +52,460 +591,039 +# Attributes +18 +13 +5,693 +5,693 +Avg. degree +34.0 +95.8 +1.3 +4.6 +Sens. +Race +Age +Continent of Affiliation +Continent of Affiliation +Label +Bail Decision +Future Default +Research Field +Research Field +Table 2: The experimental results based on node classification accuracy and ∆SP. We use "(T)" and "(S)" suffixes +to represent the teacher model and the student model, respectively. Here Vanilla(S) denotes the student model +trained with the vanilla KD framework; One-Hot(S) represents the student model trained with the one-hot bias +proxy; RELIANT(S) is the student model trained with our proposed model. ↑ denotes the larger, the better; while +↓ denotes the opposite. All quantitative results are presented in percentages. The best results are in Bold. +DBLP +DBLP-L +Credit +Recidivism +CPF ++GCN +Accuracy (↑) +GCN(T) +92.37 ± 0.06 +94.20 ± 0.09 +76.39 ± 0.48 +93.68 ± 0.21 +Vanilla(S) +93.14 ± 0.10 +94.30 ± 0.04 +77.85 ± 0.10 +89.41 ± 0.12 +One-Hot(S)) +93.04 ± 0.34 +94.16 ± 0.02 +77.65 ± 0.10 +89.15 ± 0.37 +RELIANT(S) +92.70 ± 0.40 +94.07 ± 0.18 +77.82 ± 0.45 +88.88 ± 0.57 +∆SP (↓) +GCN(T) +7.66 ± 0.26 +7.33 ±0.44 +15.81 ±0.40 +6.10 ±0.05 +Vanilla(S) +8.55 ± 0.50 +7.16 ± 0.16 +14.90 ± 0.89 +6.85 ± 0.05 +One-Hot(S) +7.97 ± 0.63 +7.46 ± 0.24 +13.80 ± 0.32 +6.78 ± 0.51 +RELIANT(S) +2.27 ± 1.00 +3.09 ± 0.36 +10.28 ± 1.86 +4.06 ± 0.64 +CPF ++SAGE +Accuracy (↑) +SAGE(T) +92.57 ± 0.28 +94.10 ± 0.25 +77.88 ± 0.06 +89.71 ± 0.14 +Vanilla(S) +93.25 ± 0.15 +94.97 ± 0.10 +77.97 ± 0.26 +89.20 ± 0.11 +One-Hot(S) +93.07 ± 0.10 +94.32 ± 0.07 +78.01 ± 0.23 +89.11 ± 0.29 +RELIANT(S) +92.91 ± 0.51 +94.17 ± 0.93 +78.28 ± 0.36 +88.85 ± 0.27 +∆SP (↓) +SAGE(T) +8.32 ±0.24 +7.81 ±0.08 +14.08 ± 1.37 +6.50 ±0.39 +Vanilla(S) +8.29 ± 0.85 +7.02 ± 0.13 +13.44 ± 5.23 +4.41 ± 0.43 +One-Hot(S) +8.01 ± 0.25 +7.52 ± 0.32 +16.86 ± 3.86 +6.62 ± 0.38 +RELIANT(S) +2.01 ± 1.21 +2.97 ± 0.61 +10.06 ± 1.70 +3.94 ± 0.60 +AKD ++GCN +Accuracy (↑) +GCN(T) +92.37 ± 0.06 +94.20 ± 0.09 +76.39 ± 0.48 +93.68 ± 0.21 +Vanilla(S) +92.06 ± 0.16 +94.07 ± 0.11 +76.35 ± 0.31 +92.08 ± 0.29 +One-Hot(S) +91.55 ± 0.40 +94.07 ± 0.04 +75.65 ± 0.75 +92.07 ± 0.03 +RELIANT(S) +91.39 ± 0.24 +93.98 ± 0.08 +75.64 ± 0.06 +91.21 ± 0.14 +∆SP (↓) +GCN(T) +7.66 ±0.26 +7.33 ±0.44 +15.81±0.40 +6.10 ±0.05 +Vanilla(S) +7.87 ± 0.25 +6.79 ± 0.10 +13.61 ± 2.00 +6.54 ± 0.17 +One-Hot(S) +7.39 ± 0.35 +6.72 ± 0.19 +14.30 ± 0.24 +6.44 ± 0.32 +RELIANT(S) +3.66 ± 1.09 +5.18 ± 0.16 +8.47 ± 1.92 +5.70 ± 0.18 +AKD ++SAGE +Accuracy (↑) +SAGE(T) +92.57 ± 0.28 +94.10 ± 0.25 +77.88 ± 0.06 +89.71 ± 0.14 +Vanilla(S) +92.23 ± 0.07 +94.45 ± 0.03 +78.10 ± 0.24 +89.67 ± 0.07 +One-Hot(S) +92.31 ± 0.06 +94.52 ± 0.11 +78.24 ± 0.45 +89.60 ± 0.12 +RELIANT(S) +92.07 ± 0.07 +94.28 ± 0.06 +78.60 ± 0.33 +88.87 ± 0.31 +∆SP (↓) +SAGE(T) +8.32 ±0.24 +7.81 ±0.08 +14.08 ± 1.37 +6.50 ±0.39 +Vanilla(S) +7.53 ± 0.29 +7.34 ± 0.41 +14.41 ± 0.15 +6.24 ± 0.20 +One-Hot(S) +7.72 ± 0.44 +7.26 ± 0.36 +11.69 ± 0.93 +6.18 ± 0.30 +RELIANT(S) +4.91 ± 0.64 +4.05 ± 0.14 +5.00 ± 1.63 +6.06 ± 0.26 +framework, we adopt two types of GNNs (including +GCN [21] and GraphSAGE [14]) as the teacher GNN. +Baselines. To the best of our knowledge, this is the +first study on how to mitigate the bias exhibited in GNN- +based KD frameworks. In experiments, we adopt the +student model yielded by the vanilla KD framework as +our first baseline. For our second baseline, we replace +the learnable proxy of bias with a naive proxy for the +input of the KD framework. Specifically, we utilize one- +hot vectors as the naive proxy for different demographic +subgroups during training, where the one-hot vector +flags the membership of different nodes. We replace +all proxy vectors during inference with an averaged +proxy vector across all instances. Here, the rationale +is that more distinguishable attributes are easier for +deep learning models to learn during training, and these +one-hot vectors serve as an "easier" indicator of biased +information. In this way, if these one-hot proxy accounts +for the exhibited bias of the student model after training, +then the exhibited bias could also be mitigated during +inference, where such information is wiped out. +Evaluation Metrics. We evaluate the performance of +the compressed GNN models (i.e., the output student +model of KD frameworks) from two perspectives, namely +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +utility and fairness. Specifically, in terms of utility, we +adopt the node classification accuracy as the correspond- +ing metric; in terms of fairness, we adopt two traditional +metrics ∆SP and ∆EO. Here ∆SP measures the bias level +(of predictions) under the fairness notion of Statistical +Parity, while ∆EO measures the bias level under the no- +tion of Equal Opportunity. We present the quantitative +results of ∆EO in Appendix B.1 due to space limit. +Implementation Details. RELIANT is implemented +in PyTorch [25] and optimized with Adam optimizer [20]. +In our experiments, the learning rate is chosen in +{10−2, 10−3, 10−4} and the training epoch number is set +as 1,000 for CPF and 600 for GraphAKD. Experiments +are carried out on an Nvidia RTX A6000, and the +reported numerical results are averaged across three +different runs. We introduce more details in Appendix A. +5.2 +Effectiveness of RELIANT Here we aim to +answer RQ1. Specifically, we evaluate our proposed +framework RELIANT on two KD backbones, namely +CPF and GraphAKD. For each KD backbone, we adopt +two different GNNs (GCN and GraphSAGE) to evaluate +the capability of our proposed framework in generalizing +to different GNNs. +We compare the corresponding +performances between the teacher GNN model and the +student models trained with three different frameworks +(i.e., the vanilla KD framework, the KD framework with +the one-hot proxy of bias, and our proposed RELIANT). +We present quantitative results on node classification +accuracy and ∆SP in Table 2. +In addition, we also +perform experiments based on Equal Opportunity (see +Appendix B.1), where we have consistent observations. +We make the following observations from Table 2. +• From the perspective of prediction utility, stu- +dent models trained with all three KD frameworks +achieve comparable performances with the teacher +model. This implies that effective knowledge distil- +lation can be achieved by all three KD frameworks. +• From the perspective of bias mitigation, the student +models trained with the vanilla KD frameworks +inherit and even exaggerate the exhibited bias from +the teacher GNN model in all cases. Training the +student models with the one-hot proxy can mitigate +bias in most cases. +Compared with the student +models trained with the vanilla KD framework and +the one-hot proxy, RELIANT consistently exhibits +less bias w.r.t. Statistical Parity. +• Based on the performance of RELIANT in both +perspectives, RELIANT achieves effective debiasing +for the student model but still maintains comparable +model utility with the teacher model. Therefore, we +argue that RELIANT achieves a satisfying balance +between debiasing and maintaining utility. +0 +1 +2 +3 +EO +92.0 +92.5 +93.0 +93.5 +Accuracy +Vanilla +V. w/ Proxy +RELIANT +(a) GraphAKD on Credit. +0 +5 +10 +15 +20 +SP +75 +76 +77 +78 +79 +80 +Accuracy +Vanilla +V. w/ Proxy +RELIANT +(b) CPF on DBLP. +Figure 3: +Ablation study of RELIANT. "Vanilla" +denotes the student model trained with the original +KD framework, while "V. w/ Proxy" represents the +student model trained under the KD framework with +only learning the proxy of bias. +100 +101 +102 +103 +104 +92.7 +93.6 +94.5 +95.4 +96.3 +Accuracy +1.5 +3.0 +4.5 +6.0 +7.5 +SP +Accuracy +SP +(a) CPF on DBLP-L. +100 +101 +102 +103 +104 +81.6 +84.0 +86.4 +88.8 +91.2 +93.6 +Accuracy +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +EO +Accuracy +EO +(b) GraphAKD on Recidivism. +Figure 4: +Parameter sensitivity of λ based on two +different KD backbones on two real-world datasets. We +also have similar observations on other datasets. +5.3 +Ablation Study We aim to answer RQ2 in this +subsection. Specifically, for each framework, we evaluate +to what extent the two modules of RELIANT (including +learning proxy of bias and enforcing the attribution of +bias to the proxy) contribute to the performance of the +student model. We present the results in Fig. 3. Here, +Fig. 3a is the performance of accuracy vs. ∆SP from +CPF based on the DBLP-L dataset, while Fig. 3b is +the performance of accuracy vs. ∆EO from GraphAKD +based on the Recidivism dataset. Notably, we also have +similar observations under other settings. We make the +following observations. +• From the perspective of prediction utility, we +observe that the prediction utility is comparable +among all three cases. +This corroborates that +both modules exert limited influence on the node +classification accuracy. +• From the perspective of bias mitigation, adding the +module of learning proxy of bias to the vanilla KD +framework brings limited bias mitigation. This is be- +cause the bias could also come from the non-sensitive +node attributes (as discussed in Section 3.1). After +the module of enforcing the attribution of bias to +the proxy is added together with learning proxy of +bias, RELIANT is then able to achieve satisfying +performance on bias mitigation. +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +5.4 +Parameter Sensitivity We answer RQ3 by +studying the tendency of model utility and exhibited +bias w.r.t. +the change of hyper-parameter λ. +Here +λ controls the effect of LAttr. +More specifically, we +vary λ in {100, 101, 102, 103, 104}, and we present the +corresponding tendency of node classification accuracy +and the exhibited bias of the trained student model with +RELIANT in Fig. 4. Here, Fig. 4a is based on the Credit +dataset under GraphAKD, while Fig. 4b is based on +the DBLP dataset under CPF. We also have similar +observations on other datasets. We make the following +observations from Fig. 4. +• From the perspective of prediction utility, the node +classification accuracies on both datasets and KD +backbones do not exhibit apparent reduction when +the value of λ increases from 1 to 104. This verifies +that the prediction utility is not sensitive to λ. +• From the perspective of bias mitigation, the student +model exhibits less bias when λ increases from 1 +to 104. +Specifically, when λ is relatively small +(e.g., 1), the learned proxy of bias only partially +accounts for the exhibited bias; when the value of +λ increases, more bias is then attributed to the +learned proxy. Considering the balance between +model utility and bias mitigation, a recommended +range of λ is between 102 and 103. +6 +Related Works +Algorithmic Fairness in GNNs. Most existing works +promoting the algorithmic fairness of GNNs focus either +on Group Fairness [10] or Individual Fairness [36]. +Specifically, group fairness is defined based on a set +of pre-defined sensitive attributes (e.g., gender and +race). +These sensitive attributes divide the whole +population into different demographic subgroups. Group +fairness requires that each subgroup should receive their +fair share of interest according to the output GNN +predictions [23]. Various explorations have been made +towards achieving a higher level of group fairness for +GNNs [7]. +Decoupling the output predictions from +sensitive attributes via adversarial learning is one of the +most popular approaches among existing works [31, 3]. +Other common strategies include reformulating the +objective function with fairness regularization [11, 24], +rebalancing the number of intra-group edges between two +demographic subgroups [6, 22], deleting nodes or edges +that contribute the most to the exhibited bias [8, 9], +etc. On the other hand, individual fairness does not +rely on any sensitive attributes. +Instead, individual +fairness requires that similar nodes (in the input space) +should be treated similarly (in the output space) [10]. To +fulfill individual fairness in GNNs, adding fairness-aware +regularization terms to the optimization objective is the +most widely adopted approach [5, 28]. +Knowledge Distillation. In recent years, knowledge +distillation has been proven to be effective in compressing +the model but still maintaining similar model prediction +performance [12]. Correspondingly, it has been widely +adopted in a plethora of applications, including visual +recognition [33], natural language processing [13, 17], +etc. +The main idea of knowledge distillation is to +transfer the knowledge of a computationally expensive +teacher model to a light student model, and thus the +student model is able to fit in platforms with limited +computing resources [16, 19]. It is worth noting that such +a strategy is also proved to be effective in compressing +GNNs [34, 16, 19]. +Consequently, there is growing +research attention on utilizing knowledge distillation +to compress GNNs for more efficient inference. +For +example, encouraging the student model to yield similar +output to the teacher GNN via regularization is proved +to be effective [16]. In addition, adversarial learning is +also a popular technique to obtain light-weighted but +accurate student models [16]. However, most of these +frameworks for GNNs do not have fairness consideration. +Hence the student model tends to be influenced by biased +knowledge from the teacher GNN. Different from existing +works, we develop a generalizable knowledge distillation +framework that explicitly considers fairness in GNNs but +still maintains the utility of GNN predictions. +7 +Conclusion +Despite the success of Knowledge Distillation (KD) in +compressing GNNs, most existing works do not consider +fairness. +Hence the student model trained with the +KD framework tends to inherit and even exaggerate +the bias from the teacher GNN. In this paper, we +take initial steps towards learning less biased student +models for GNN-based KD frameworks. +Specifically, +we first formulate a novel problem of fair knowledge +distillation for GNN-based teacher-student frameworks, +then propose a framework named RELIANT to achieve +a less biased student model. +Notably, the design of +RELIANT is agnostic to the specific structures of teacher +and student models. Therefore, it can be easily adapted +to different KD approaches for debiasing. +Extensive +experiments demonstrate the effectiveness of RELIANT +in fulfilling fairness for GNN compression with KD. +8 +Acknowledgments +This work is supported by the National Science Foun- +dation under grants IIS-2006844, IIS-2144209, IIS- +2223768, IIS-2223769, CNS-2154962, CMMI-2125326, +BCS-2228533, and BCS-2228534, the JP Morgan Chase +Faculty Research Award, and the Cisco Faculty Research +Award. We would like to thank the anonymous reviewers +for their constructive feedback. +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +References +[1] Agarwal, C., Lakkaraju, H., and Zitnik, M. +Towards a unified framework for fair and stable graph +representation learning. In UAI (2021). +[2] Chen, M., Wei, Z., Huang, Z., Ding, B., and Li, +Y. Simple and deep graph convolutional networks. In +ICML (2020). +[3] Dai, E., and Wang, S. Say no to the discrimination: +Learning fair graph neural networks with limited +sensitive attribute information. In WSDM (2021). +[4] Dattoli, G., Ricci, P. E., and Cesarano, C. A +note on legendre polynomials. International Journal +of Nonlinear Sciences and Numerical Simulation 2, 4 +(2001), 365–370. +[5] Dong, +Y., +Kang, +J., +Tong, +H., +and Li, +J. +Individual fairness for graph neural networks: A ranking +based approach. In SIGKDD (2021). +[6] Dong, Y., Liu, N., Jalaian, B., and Li, J. 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In ICML +(2013). +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +A +Reproducibility +In this section, we introduce the reproducibility of the +presented experiments as a supplement of Section 5. +More specifically, we first introduce the experimental +settings in detail, followed by the implementation +details of our proposed framework RELIANT, GNNs, +KD backbones, and the baseline debiasing framework. +We finally introduce several key packages and their +corresponding versions used in our implementations. The +code will be released upon acceptance. +A.1 +Experimental Settings. The implementation +of our experiments could mainly be divided into three +parts, well-trained teacher GNNs, knowledge distillation +backbones, and our RELIANT module. +Generally, +our experiments are implemented on an Nvidia RTX +A6000. We write our experiment code in PyTorch [26] +framework, and use Adam [20] optimizer to learn the +model parameters. We run all experiments three times +and record the average and the standard deviation where +the random seeds are chosen in {0,10,100,1000,10000}. +A.2 +Implementation of RELIANT. We imple- +ment RELIANT in PyTorch and use Adam as the op- +timizer of the learnable proxy. +For results shown in +Table 2 and Table 4, we set the proxy learning rate +as 10−2 and weight decay as 10−2 for CPF and set the +proxy learning rate as 10−2 and weight decay as 5×10−4 +for GraphAKD. For all the results, we search for the +optimal coefficient λ in the set {1,10,100,1000,10000}. +A.3 +Implementation +of +Graph +Neural +Net- +works. For the training of the teacher GNN models, +we use the code in the CPF framework where the details +of implementation setting are shown in Table 3. +A.4 +Implementation of KD Frameworks. In our +experimental setting, the teacher model is fixed during +the training stage. Hence we first train a teacher GNN +with the parameters shown in Table 3. After obtaining +a well-trained teacher GNN, we use it as supervision to +train the student model. Details of student training are +shown as follows. +• For CPF, we follow the implementation in [34], +where we use PLP as the student model. +For +the training settings of the student model, we +set the maximum epoch as 1,000, early stopping +as 500, layer number as 5, feature dropout as +0.8, edge weight dropout as 0.2, learning rate in +{10−2, 10−3, 10−4}, and weight decay as 10−2. +• For GraphAKD, we follow the implementation in +[16], where we utilize SGC [32] as the student model. +For detailed training settings, we set the maximum +Table 3: Experimental settings of teacher GNN models. +Hyperparameter +GCN +GraphSAGE +Layer +3 +3 +Hidden Dimension +64 +128 +Epoch +1000 +1000 +Early Stopping +500 +500 +Learning Rate +10−2 +10−2 +Weight Decay +10−3 +5 × 10−4 +Dropout +0.8 +0.5 +epoch as 600, layer number as 3, learning rate in +{10−2, 10−3, 10−4}, weight decay as 5 × 10−4, and +dropout as 0.5. +A.5 +Implementation of the Baseline Debiasing +Framework. For the vanilla KD frameworks, we follow +the settings in Appendix A.4 to directly implement the +KD framework. For the baselines with the one-hot proxy +for bias, we use the one-hot vector of sensitive attributes +as the constant proxy. It is added to the input data +before training and is fixed during the training stage. +Hence there is no loss term (as Eq. 3.4 shows) for the +constant proxy. The training settings of the one-hot +baseline are the same as Vanilla since the constant proxy +vectors do not contain any extra parameters. +A.6 +Packages Required for Implementations. +We list the key packages and corresponding versions +in our implementations as below. +• Python == 3.7.10 +• torch == 1.8.1 +• torch-cluster == 1.5.9 +• torch-geometric == 1.4.1 +• torch-sparse == 0.6.9 +• numpy == 1.20.0 +• networkx == 2.5.1 +• scikit-learn == 0.24.1 +• pandas==1.2.3 +• scipy==1.4.1 +B +Complementary Experiments +In this section, we perform experiments as a supplement +of Section 5. Specifically, we first present the perfor- +mance of RELIANT over prediction utility and fairness +under another widely studied fairness notion – Equal +Opportunity. Then, we perform experiments to com- +pare the performance of RELIANT on balancing the +GNN prediction utility and fairness with state-of-the-art +methods that directly debias GNNs. +B.1 +Effectiveness of RELIANT Here we present +complementary experiments to answer RQ1, where the +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +Recidivism +Credit +40 +60 +80 +100 +Accuracy +(a) Comparison on the node classification accuracy. +Recidivism +Credit +0 +10 +20 +SP +Teacher +RELIANT +EDITS +NIFTY +(b) Comparison on ∆SP. +Figure 5: Performance comparison on balancing predic- +tion utility and bias mitigation between the proposed +framework RELIANT and other state-of-the-art meth- +ods that directly debias GNNs. All numerical results are +in percentage. +fairness notion is instantiated with Equal Opportunity +(measured with ∆EO). Here we adopt the same settings +as those in Section 5.2. We compare the performances +between the teacher GNN model and the student +models trained with three different framework variants, +including the vanilla KD framework, the KD framework +with the one-hot proxy of bias, and our proposed +RELIANT. We present quantitative results on node +classification accuracy and ∆EO in Table 4. We make +the following observations from Table 4. +• From the perspective of prediction utility, all stu- +dent models trained with the adopted three KD +frameworks are able to achieve comparable perfor- +mances with the teacher model. This corroborates +that all three KD frameworks are capable of achiev- +ing effective knowledge distillation. +• From the perspective of bias mitigation, the student +models trained with the vanilla KD frameworks +inherit or exaggerate the bias from the teacher GNN +in all cases. Compared with the student models +trained with the vanilla KD framework and the +one-hot proxy, RELIANT consistently exhibits less +bias under the fairness notion of Equal Opportunity. +The student model trained with RELIANT even +exhibits less bias than the teacher model in certain +cases, which further corroborates the effectiveness +of RELIANT in training less biased student models. +• According to the performance of RELIANT in +both perspectives above, RELIANT is proved to +achieve effective debiasing for the student model but +maintains comparable prediction utility compared +with the teacher model. Therefore, we argue that +RELIANT achieves a satisfying balance between +debiasing and maintaining the prediction utility. +B.2 +RELIANT vs. +GNN-Debiasing Methods +In this subsection, we perform experiments and compare +the performance of RELIANT with other state-of-the- +art GNN debiasing methods on balancing the prediction +utility and bias mitigation. +Baselines. Here we choose two state-of-the-art GNN de- +biasing methods as our baselines, namely EDITS [6] and +NIFTY [1]. EDITS is a recent GNN debiasing method +that learns less biased network data in the pre-processing +stage. After debiasing, the network data will be fed into +the GNN model for evaluation. NIFTY is another recent +GNN-based debiasing framework that achieves bias mit- +igation in the processing stage. During training, node +representations are learned to be invariant to the sensi- +tive attributes after counterfactual perturbations. +Backbones. +Here we choose the most widely used +GCN as the backbone GNN model for all methods. We +choose GraphAKD as the KD backbone of the proposed +RELIANT. It is also worth noting that we also have +similar observations with other GNN backbones. +Discussion. We present the performance comparison +results on node classification accuracy and ∆SP in Fig. 5. +We make the following observations. +• From the perspective of prediction utility, RE- +LIANT keeps comparable to the teacher GCN +model, while other debiasing methods bear different +levels of prediction utility corruption. Therefore, +RELIANT achieves the best performance in main- +taining the prediction utility among all methods. +• From the perspective of bias mitigation, RELIANT +is able to achieve comparable performance with +other baselines when all models bear similar predic- +tion utility (e.g., on Credit dataset); when baselines +outperform RELIANT on bias mitigation, there is +also much more prediction utility sacrifice (e.g., on +Recidivism dataset). Considering that debiasing +the student model with biased supervision is much +more difficult than directly debiasing GNNs, we +argue that the performances of RELIANT in both +cases should be considered satisfying. +• According to the performance of RELIANT in +both perspectives above, we argue that RELIANT +achieves comparable performance with other state- +of-the-art GNN debiasing approaches, which further +corroborates its satisfying performance on balancing +the prediction utility and bias mitigation. +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + +Table 4: The experimental results based on node classification accuracy and ∆EO. We use "(T)" and "(S)" suffixes +to represent the teacher model and the student model, respectively. Here Vanilla(S) denotes the student model +trained with the vanilla KD framework; One-Hot(S) represents the student model trained with the one-hot bias +proxy; RELIANT(S) is the student model trained with our proposed model. ↑ denotes the larger, the better; while +↓ denotes the opposite. All quantitative results are presented in percentages. The best results are in Bold. +DBLP +DBLP-L +Credit +Recidivism +CPF ++GCN +Accuracy (↑) +GCN(T) +92.37 ± 0.06 +94.20 ± 0.09 +76.39 ± 0.48 +93.68 ± 0.21 +Vanilla(S) +93.24 ± 0.18 +94.15 ± 0.04 +77.58 ± 0.20 +89.36 ± 0.16 +One-Hot(S)) +93.07 ± 0.35 +94.15 ± 0.07 +77.65 ± 0.10 +89.38 ± 0.15 +RELIANT(S) +93.20 ± 0.12 +94.15 ± 0.08 +77.00 ± 1.57 +89.30 ± 0.19 +∆EO (↓) +GCN(T) +2.31 ± 0.13 +2.29 ± 0.34 +12.63 ± 0.24 +0.52 ± 0.06 +Vanilla(S) +2.56 ± 0.11 +2.34 ± 0.29 +11.56 ± 0.38 +1.25 ± 0.19 +One-Hot(S) +2.21 ± 0.32 +2.17 ± 0.26 +10.69 ± 0.31 +0.96 ± 0.28 +RELIANT(S) +0.42 ± 0.21 +1.08 ± 0.10 +6.02 ± 4.78 +0.35 ± 0.12 +CPF ++SAGE +Accuracy (↑) +SAGE(T) +92.57 ± 0.28 +94.10 ± 0.25 +77.88 ± 0.06 +89.71 ± 0.14 +Vanilla(S) +93.22 ± 0.03 +94.37 ± 0.08 +78.30 ± 0.23 +89.15 ± 0.27 +One-Hot(S) +93.13 ± 0.11 +94.36 ± 0.06 +78.01 ± 0.23 +88.98 ± 0.55 +RELIANT(S) +93.24 ± 0.09 +94.32 ± 0.06 +78.11 ± 0.40 +89.01 ± 0.26 +∆EO (↓) +SAGE(T) +2.51 ± 0.33 +2.67 ± 0.19 +11.05 ± 0.71 +0.86 ± 0.03 +Vanilla(S) +2.83 ± 0.34 +2.00 ± 0.18 +11.07 ± 4.61 +1.17 ± 0.11 +One-Hot(S) +2.16 ± 0.27 +2.05 ± 0.21 +12.73 ± 2.29 +1.23 ± 0.08 +RELIANT(S) +0.63 ± 0.42 +0.86 ± 0.18 +6.72 ± 4.49 +0.51 ± 0.25 +AKD ++GCN +Accuracy (↑) +GCN(T) +92.37 ± 0.06 +94.20 ± 0.09 +76.39 ± 0.48 +93.68 ± 0.21 +Vanilla(S) +92.12 ± 0.09 +94.06 ± 0.06 +78.12 ± 0.65 +92.29 ± 0.06 +One-Hot(S) +91.68 ± 0.28 +93.98 ± 0.13 +77.87 ± 0.48 +92.28 ± 0.13 +RELIANT(S) +91.69 ± 0.19 +94.09 ± 0.12 +77.88 ± 0.82 +92.46 ± 0.09 +∆EO (↓) +GCN(T) +2.31 ± 0.13 +2.29 ± 0.34 +12.63 ± 0.24 +0.52 ± 0.06 +Vanilla(S) +2.76 ± 0.33 +1.88 ± 0.08 +8.26 ± 3.41 +0.82 ± 0.17 +One-Hot(S) +2.69 ± 0.28 +1.87 ± 0.17 +8.43 ± 5.08 +0.97 ± 0.45 +RELIANT(S) +1.79 ± 0.31 +1.43 ± 0.09 +4.96 ± 3.77 +0.66 ± 0.21 +AKD ++SAGE +Accuracy (↑) +SAGE(T) +92.57 ± 0.28 +94.10 ± 0.25 +77.88 ± 0.06 +89.71 ± 0.14 +Vanilla(S) +92.23 ± 0.07 +94.45 ± 0.03 +78.10 ± 0.24 +90.56 ± 0.14 +One-Hot(S) +92.31 ± 0.06 +94.52 ± 0.11 +78.24 ± 0.45 +90.85 ± 0.20 +RELIANT(S) +92.15 ± 0.16 +94.42 ± 0.05 +79.08 ± 0.29 +90.00 ± 0.64 +∆EO (↓) +SAGE(T) +2.51 ± 0.33 +2.67 ± 0.19 +11.05 ± 0.71 +0.86 ± 0.03 +Vanilla(S) +2.06 ± 0.06 +2.23 ± 0.23 +10.56 ± 0.43 +1.61 ± 0.39 +One-Hot(S) +2.21 ± 0.39 +2.11 ± 0.21 +8.38 ± 0.73 +1.10 ± 0.37 +RELIANT(S) +1.60 ± 0.45 +1.89 ± 0.21 +2.33 ± 0.80 +0.91 ± 0.22 +Copyright © 2023 by SIAM +Unauthorized reproduction of this article is prohibited + diff --git a/ktAzT4oBgHgl3EQfNfuY/content/tmp_files/load_file.txt b/ktAzT4oBgHgl3EQfNfuY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..842aa908e9f4757589000a501a2e4bbd2c41aa2c --- /dev/null +++ b/ktAzT4oBgHgl3EQfNfuY/content/tmp_files/load_file.txt @@ -0,0 +1,1410 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf,len=1409 +page_content='RELIANT: Fair Knowledge Distillation for Graph Neural Networks Yushun Dong∗ Binchi Zhang∗ Yiling Yuan† Na Zou‡ Qi Wang§ Jundong Li∗ Abstract Graph Neural Networks (GNNs) have shown satisfying performance on various graph learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To achieve better fitting capability, most GNNs are with a large number of parameters, which makes these GNNs compu- tationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, it is difficult to deploy them onto edge devices with scarce computational re- sources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', mobile phones and wearable smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Knowledge Distillation (KD) is a common solution to compress GNNs, where a light-weighted model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', the student model) is encouraged to mimic the behavior of a computationally expensive GNN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', the teacher GNN model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Nevertheless, most existing GNN-based KD methods lack fairness consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' As a consequence, the student model usually inherits and even exagger- ates the bias from the teacher GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To handle such a problem, we take initial steps towards fair knowledge distillation for GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we first formulate a novel problem of fair knowledge distillation for GNN- based teacher-student frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Then we propose a principled framework named RELIANT to mitigate the bias exhibited by the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Notably, the de- sign of RELIANT is decoupled from any specific teacher and student model structures, and thus can be easily adapted to various GNN-based KD frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We perform extensive experiments on multiple real-world datasets, which corroborates that RELIANT achieves less biased GNN knowledge distillation while maintain- ing high prediction utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Open-source code can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='com/yushundong/RELIANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Keywords: Graph Neural Networks, Algorithmic Fair- ness, Knowledge Distillation 1 Introduction In recent years, Graph Neural Networks (GNNs) have shown satisfying performance in a plethora of real-world applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', medical diagnosis [27] and credit risk scoring [30], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In practice, the depth and the number of parameters of GNNs largely ∗University of Virginia, Email: {yd6eb, epb6gw, jun- dong}@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='edu †Beijing University of Posts and Telecommunications, Email: yuanyiling@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='cn ‡Texas A&M University, Email: nzou1@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='edu §Northeastern University, Email: q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='wang@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='edu Recidivism Credit 4 8 12 16 SP T-GCN S-GCN T-GAT S-GAT +13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8% +21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6% +15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8% +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3% (a) Bias under ∆SP on CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Recidivism Credit 2 4 6 8 10 EO T-GCN S-GCN T-GAT S-GAT +34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8% +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0% +22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5% +18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3% (b) Bias under ∆EO on AKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Figure 1: A comparison of exhibited bias between teacher and student models based on two representative GNN knowledge distillation frameworks (CPF and GraphAKD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' "T" and "S" represent the teacher and the student model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The names of GNN mark out the corresponding teacher models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' determine their expressive power [16], which directly influence their performances in various graph learning tasks [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Typically, deeper GNN layers enable the model to capture information that is multiple hops away from any node [21], while a larger number of learnable parameters enable GNN to fit more complex underlying data patterns [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' However, in most cases, the inference efficiency of GNNs is inevitably degraded by the deep layers or the large number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Such efficiency degradation naturally makes these GNNs inapplicable to be deployed on edge devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', mobile phones) with limited computational resources [16, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Due to the problem above, it is necessary to compress those computationally expensive GNNs for deployment on edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Knowledge Distillation (KD) is a com- mon approach to compress GNNs but still maintains a similar level of prediction performance [34, 16, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here, the basic idea of KD is to let a light-weighted student model (as the compressed GNN) learn to mimic the behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', output logits) of the teacher model (usually a computationally expensive GNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' However, most existing KD approaches do not have any fairness consideration over different demographic subgroups, and the optimized student model often preserves and even exaggerates the exhibited bias from the teacher GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Consequently, when the compressed model is deployed in real-world application scenarios, there could exist dis- crimination toward specific populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here we provide preliminary analysis based on two representative GNN knowledge distillation frameworks, namely CPF [34] and GraphAKD [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we measure the exhibited Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='01150v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='LG] 3 Jan 2023 bias in the widely-studied node classification task on two real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here Recidivism is a network of defendants [18, 1], while Credit is a network between bank clients [35, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We adopt two traditional metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', ∆SP (measuring the level of bias under Statistical Parity [10]) and ∆EO (measuring the level of bias under Equal Opportunity [15]), to measure the exhibited bias of GNN predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We present a comparison of the exhibited bias between teacher and student models in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Empirical results show that student models tend to yield more biased results compared with the teacher GNN model, which could be attributed to the biased guidance from the teacher GNN during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' It is worth noting that in most cases, directly retraining the teacher GNN for debiasing is infeasible, since retraining the teacher GNN with a large number of parameters is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hence, mitigating the bias for the student model is an urgent need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Despite the necessity of mitigating bias for the stu- dent model, existing exploration remains scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In this paper, we aim to make an initial step towards develop- ing a debiasing framework that can be easily adapted to various existing GNN-based KD methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' However, this task is non-trivial mainly due to the following three chal- lenges: (1) Gap towards Fair Knowledge: For most KD frameworks designed for compressing GNNs, the teacher GNN model usually serves as the sole source of supervision signal for the training of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, if the teacher GNN exhibits any bias, such biased knowledge tends to be inherited by the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hence, learning a fair student model with biased supervision from the teacher GNN is our first challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (2) Gap towards End-to-End Learning: A critical advantage of existing KD models is the end-to-end learn- ing paradigm, which enables the distilled knowledge to be tailored to specific downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In such an end- to-end learning process, highly efficient gradient-based optimization techniques are widely adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' However, widely-used fairness notions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Statistical Parity and Equal Opportunity) are defined on the predicted la- bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hence the corresponding bias metrics are naturally non-differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' the student model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Developing a debiasing framework suitable for gradient- based optimization techniques in the end-to-end learning paradigm is our second challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (3) Gap towards Generalization: Various KD models have been pro- posed for compressing GNNs to satisfy different applica- tion scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In fact, most KD models are developed based on certain designs of student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Developing a framework that is student-agnostic and easily adapted to different KD models is our third challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To tackle the above challenges, in this paper, we propose a novel framework named RELIANT (faiR knowlEdge distiLlatIon for grAph Neural neTworks) to mitigate the bias learned by the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we first formulate a novel research problem of Fair Knowledge Distillation for GNN-based Teacher- Student Frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To tackle the first challenge, we incorporate a learnable proxy of the exhibited bias for the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In this way, despite the knowledge (from the teacher GNN) being biased, the student model still makes less biased predictions under proper manipulations on the proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To tackle the second challenge, we propose to approximate the bias level of the student model, where the approximation is differentiable (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' the student model parameters) manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In this way, the highly efficient end-to-end learning paradigm is preserved, and the gradient-based optimization techniques are still applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To tackle the third challenge, we design the proposed framework RELIANT in a student-agnostic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In other words, the debiasing for the student model does not rely on any specific design tailored for the student model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, RELIANT can be easily adapted to different GNN-based knowledge distillation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Problem Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We formulate and make an initial investigation on a novel research problem of fair knowledge distillation for GNN-Based teacher- student frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Algorithmic Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We propose a principled framework named RELIANT that learns the proxy of bias for the student model during KD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' RELIANT achieves student-agnostic debiasing via manipulat- ing the proxy during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Experimental Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We conduct compre- hensive experiments on multiple real-world datasets to verify the effectiveness of the proposed framework RELIANT in learning less biased student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 2 Problem Definition Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We denote matrices, vectors, and scalars by bold uppercase letters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', X), bold lowercase letters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', x), and regular lowercase letters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For any matrix, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', X, we use Xi,j to indicate the element at the i-th row and j-th column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We utilize G = {V, E, X} to denote an attributed network (graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here, V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', vn} is the set of nodes, E ⊆ V × V is the set of edges, and X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', xn} (xi ∈ Rd, 1 ≤ i ≤ n) is the set of node attribute vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We use A ∈ {0, 1}n×n to denote the adjacency matrix of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' If there is an edge from the i-th node to the j-th node, Ai,j = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' otherwise Ai,j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Moreover, we denote the pre- trained teacher GNN model in a knowledge distillation framework as f ˆθ parameterized by ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here ˆθ denotes the optimized θ of the pre-trained teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Copyright © 2023 by SIAM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Unauthorized reproduction of this article is prohibited ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Teacher GNN Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Student Model (GNN as an example) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Input Attributed Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Attributed Graph with Proxy of Bias ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Node Embeddings from Teacher Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Node Embeddings from Student Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Teacher Logits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Student Logits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Training Stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Inference Stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Input Attributed Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Attributed Graph with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Manipulated Proxy of Bias ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Adding Expectation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='of Bias Proxy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Less Biased Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Less Biased Predictions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Maximizing Utility: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Learning Proxy of Bias: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Enforcing the Attribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='of Bias to Proxy: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Approximation with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Differentiable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Polynomials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Quantitative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Level of Bias ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='ℒUtility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='ℒAttr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='ℒProxy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='Figure 2: An overview of the proposed framework RELIANT including the training and inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' we denote the student model as gφ parameterized by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We represent the optimized φ after the training of the student model as ˆφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Without loss of generality, we consider the most widely studied node classification as the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For the teacher model f ˆθ(v), we denote the set of outcome logits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', the continuous output vector corresponding to each node, as ˆY(t) = {ˆy(t) 1 , ˆy(t) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', ˆy(t) n }, where ˆy(t) i ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here c is the total number of node classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Correspondingly, we represent the set of outcome logits of the student model gφ(v) as ˆY(s) = {ˆy(s) 1 , ˆy(s) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', ˆy(s) n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For any node vi, the predicted label given by the student model (denoted as ˆY (s) i for the i-th node) is determined by the largest value across all c dimensions in ˆy(s) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Based on the definitions above, we formulate the problem of Fair Knowledge Distillation for GNN-based Teacher-Student Frameworks as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Fair Knowledge Distillation for GNN-Based Teacher-Student Frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Given an attributed network G and a GNN-based teacher- student framework including a trained teacher GNN f ˆθ and a student model gφ to be trained, our goal is to achieve a more fair student model with similar prediction utility compared with f ˆθ for the node classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 3 Methodology In this section, we first present an overview of the pro- posed framework RELIANT, followed by the objective function formulation and optimization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 Workflow of RELIANT Here we first introduce the workflow of the proposed framework RELIANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In general, we introduce the three main functionalities involved in the proposed framework RELIANT, namely maximizing the utility, learning proxy of bias, and enforcing the attribution of bias to the proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We present an overview of RELIANT in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, to tackle the first challenge (gap towards fair knowledge), we propose to learn the proxy of bias as extra input attributes for the student model to account for the exhibited bias, and wipe out such bias during inference via proper manipulations on the proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To tackle the second challenge (gap towards end-to-end learning), we formulate our debiasing objectives in a differentiable (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' the parameters of the student model) manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To tackle the third challenge (gap towards generalization), we achieve debiasing in a student-agnostic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In other words, the proposed framework RELIANT does not rely on any specific student model structure to achieve debiasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We elaborate more details as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Maximizing Utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In general, existing GNN-based KD frameworks consider the GNNs with high compu- tational costs as the teacher model, and the goal is to learn a student model with limited computational costs but similar prediction utility (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', accuracy in node clas- sification tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To maintain the utility of the teacher model, it is necessary to utilize the knowledge from the teacher model as the supervision signal for the training of the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In particular, a common approach is to utilize the output classification logits from the teacher model as the supervision signal, which we take as an example here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we minimize the distance be- tween the logits from the student model and the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We formally formulate the optimization goal as min φ � vi∈V γd � ˆy(t) i , ˆy(s) i � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1) where ˆy(s) i and ˆy(t) i are the output logits from the student Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited Layer OLayer 2 OLayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' OLayerL Omodel gφ(vi) and teacher model f ˆθ(vi), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The function γd(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=') measures the distance between two logit vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Different choices can be adopted to measure the distance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', cosine distance and Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Correspondingly, to maximize the prediction utility, we minimize the objective function LUtility(φ) = � vi∈V γd � ˆy(t) i , ˆy(s) i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2) Learning Proxy of Bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' It is worth noting that even if the sensitive attributes are removed from the input data, the student model could still exhibit bias in its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The main reason is that there could exist dependencies between those sensitive attributes and non-sensitive ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Moreover, the information about sensitive attributes could also be encoded in the input network structure [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' As a consequence, it is difficult to prevent the student model from leveraging information about sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To handle such a problem, we propose to learn the proxy of bias x(p) i as extra input attributes for each node vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here, the rationale is that if much information about bias comes from the learned proxy instead of those encoded in the non-sensitive attributes or the network structures, then we are able to achieve less biased inference results by removing such a proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' As a consequence, such a proxy of bias should account for the exhibited bias of the student model as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In other words, the exhibited bias should be largely attributed to the proxy of bias rather than the sensitive information encoded in the network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' More specifically, to enforce the proxy of bias contributing to the exhibited bias in the student model, we propose to maximize the exhibited bias when these proxies are taken as input into the student model together with other attributes and the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We formally formulate our goal as max X(p) JBias({gφ(γ(vi, X(p))) : i ∈ V}), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3) where γ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=') is a function that takes a node and the proxy of bias matrix as input, and outputs the node with a concatenated node attribute vector [xi, x(p) i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here x(p) i is the i-th row of X(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' JBias(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=') is a function that takes the set of logits from the student model as input and outputs a value indicating the level of exhibited bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Nevertheless, the computation is non- differentiable under traditional fairness notions such as Statistical Parity and Equal Opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here we propose to utilize orthogonal polynomials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Legendre polynomials [4]) that are differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' the output logits to approximate the level of bias under traditional fairness notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' This makes JBias differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' the learnable parameter φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Correspondingly, we formally give the objective function towards the goal above as LProxy(X(p)) = −JBias({gφ(γ(vi, X(p))) : i ∈ V}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4) Enforcing the Attribution of Bias to the Proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Only achieving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3) is not enough to enforce the proxy of bias largely accounting for the exhibited bias of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' This is because the exhibited bias may come from the vanilla node attributes instead of the learned proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' More specifically, we denote P( ˆY (s)) as the probability of the positive prediction given by the student model for any specific node1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We assume that there are underlying unbiased and biased node attributes X(u) and X(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' When Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3) is achieved, it is clear that P( ˆY (s)|X(u), X(b), X(p)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', P( ˆY (s)|X, X(p)), is biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' However, both X(b) and X(p) could be the source of the exhibited bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' It is worth noting that our goal is to learn proxy X(p) to account for as much of the exhibited bias as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, to enforce the effectiveness of the proxy, it is necessary to ensure that the exhibited bias is attributed to the biased information from X(p) instead of X(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In other words, we need to enforce P( ˆY (s)|X(u), X(b)) being less biased, which ensures that X(p) accounts for the exhibited bias as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Nevertheless, P( ˆY (s)|X(u), X(b)) is intractable considering that the number of the input dimension number for the student model is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hence we propose an alternative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Denote the learned proxy of bias and the underlying sensitive attribute vector of any node as x(p) and s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We propose to utilize a vector E[x(p)] to replace each row in X(p) as ˜X(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In this way, the rows in ˜X(p) are independent from s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', the information about sensitive attributes encoded in X(p) is wiped out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To enforce the attribution of bias to the proxy X(p), the predictions should be as fair as possible when the information about X(p) is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, we formulate our last optimization goal as min φ JBias({gφ(˜γ(vi, ˜X(p))) : i ∈ V}), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5) where ˜γ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=') is a function that takes a node and the matrix ˜X(p) as input, and returns the input node with a concatenated node attribute vector [xi, ˜x(p) i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here ˜x(p) i is the i-th row of matrix ˜X(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We formally present the corresponding objective function as LAttr(φ) = JBias({gφ(˜γ(vi, ˜X(p))) : i ∈ V}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6) Inference with Student Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To achieve less bi- ased inference, an ideal case is to make predictions 1Here we consider binary classification task for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited with P( ˆY (s)|X(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' However, it is difficult to explic- itly extract X(u) from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Instead, we argue that P( ˆY (s)|X(u), X(b), ˜X(p)) exhibits similar level of bias compared with P( ˆY (s)|X(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' This is because (1) the bias exhibited by P( ˆY (s)|X(u), X(b), ˜X(p)) minimally re- lies on X(b) after enforcing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' and (2) there is no further information about sensitive at- tributes encoded in ˜X(p) (as discussed above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Conse- quently, we propose to utilize gφ(˜γ(vi, ˜X(p))) to achieve less biased prediction for node vi in the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 4 Optimization Objectives & Strategy We present the optimization objectives of RELIANT followed by the training strategy in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Optimization Objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Based on our discussions above, here we present a summary of the optimization objectives for the proposed RELIANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' First, to optimize the parameter φ, we formally formulate a unified objective function as Lφ = LUtility(φ) + λ · LAttr(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='7) Here λ serves as a hyper-parameter controlling the effect of debiasing the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Second, to optimize the learnable proxy of bias X(p), we formally present the objective function as LX(P) = LProxy(X(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8) Optimization Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To train the proposed frame- work RELIANT, we propose to optimize the parameter φ and learnable proxy of bias X(p) in an alternative man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We present the algorithmic routine of RELIANT in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Algorithm 1 Fair Knowledge Distillation for GNNs Input: G: the graph data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' f ˆθ: the trained teacher GNN model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' gφ: the student model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Output: g ˆ φ: the optimized student model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' X(p): the proxy of bias matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 1: Randomly initialize X(p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 2: while stop training condition not satisfied do 3: Compute Lφ according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 4: Update φ with ∂Lφ ∂φ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 5: Compute LX(p) according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 6: Update X(p) with ∂LX(p) ∂X(p) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 7: end while 8: return g ˆ φ and X(p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 5 Experimental Evaluations In this section, we will first introduce the downstream learning task and adopted real-world datasets, followed by the backbone models, baseline methods, and eval- uation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Next, we present the implementation details of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Finally, we discuss the evaluation results of the proposed RELIANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In particular, we aim to answer the following research questions through experiments: RQ1: How well can RELIANT balance the utility and fairness of the student model compared with other baselines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' RQ2: To what extent each com- ponent of RELIANT contributes to the overall debiasing performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' RQ3: How will the choice of the hyper- parameter λ affect the performance of RELIANT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 Experimental Settings Here we introduce the settings for our experimental evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Downstream Task & Real-world Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We adopt the widely studied node classification as the downstream task in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We adopt four real-world datasets for the experimental evaluations, including two widely used network datasets (Recidivism [18, 1] and Credit Defaulter [35, 1]) and two newly constructed ones based on real-world data (DBLP and DBLP-L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In Recidivism, nodes are defendants released on bail, and edges denote the connections between defendants computed from their past criminal records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here the sensitive feature is race, and we aim to classify if a certain defendant is unlikely to commit a crime after bail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In Credit Defaulter, nodes are credit card users, and edges are the connections between these users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here we consider the age period of these users as their sensitive feature, and we aim to predict the future default of credit card payments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Additionally, we also construct two co-author networks, namely DBLP and DBLP-L based on AMiner network [29], which is a co-author network collected from computer science bibliography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we first filter out the nodes in AMiner network with incomplete information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Then we adopt two different approaches to sample a connected network from the filtered dataset: DBLP is a subgraph sampled with random walk, while DBLP-L is the largest connected component of the filtered AMiner network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In both datasets, nodes represent the researchers in different fields, and edges denote the co-authorship between researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The sensitive attribute is the continent of the affiliation each researcher belongs to, and we aim to predict the primary research field of each researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The detailed statistics of these four datasets are in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' KD Framework Backbones & Teacher GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To evaluate the capability of RELIANT in generalizing to different GNN-based KD backbones, here we adopt two representative KD frameworks designed for compressing GNNs, namely CPF [34] and GraphAKD [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In general, CPF minimizes the distribution distance between the logits from teacher and student to provide supervision information for the student, while GraphAKD utilizes adversarial training to achieve knowledge distillation for the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The student model of CPF and GraphAKD is PLP [34] and SGC [32], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For each KD Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited Table 1: The basic information about the real-world datasets adopted for experimental evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' denotes the semantic meaning of sensitive attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Dataset Recidivism Credit Defaulter DBLP DBLP-L # Nodes 18,876 30,000 39,424 129,726 # Edges 321,308 1,436,858 52,460 591,039 # Attributes 18 13 5,693 5,693 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' degree 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6 Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Race Age Continent of Affiliation Continent of Affiliation Label Bail Decision Future Default Research Field Research Field Table 2: The experimental results based on node classification accuracy and ∆SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We use "(T)" and "(S)" suffixes to represent the teacher model and the student model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here Vanilla(S) denotes the student model trained with the vanilla KD framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' One-Hot(S) represents the student model trained with the one-hot bias proxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' RELIANT(S) is the student model trained with our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' ↑ denotes the larger, the better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' while ↓ denotes the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' All quantitative results are presented in percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The best results are in Bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' DBLP DBLP-L Credit Recidivism CPF +GCN Accuracy (↑) GCN(T) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='09 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='48 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='21 Vanilla(S) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='04 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='12 One-Hot(S)) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='34 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='02 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 RELIANT(S) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='40 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='18 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='45 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='57 ∆SP (↓) GCN(T) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='33 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='44 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='81 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='05 Vanilla(S) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='16 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='89 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='05 One-Hot(S) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='63 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='51 RELIANT(S) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='27 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='36 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='28 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='64 CPF +SAGE Accuracy (↑) SAGE(T) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='28 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='25 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='14 Vanilla(S) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='15 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='26 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='11 One-Hot(S) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='23 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='29 RELIANT(S) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='51 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='93 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='36 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='27 ∆SP (↓) SAGE(T) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='32 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='81 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='50 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='39 Vanilla(S) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='85 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='44 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='43 One-Hot(S) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='32 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='86 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='86 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='38 RELIANT(S) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='01 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='61 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='60 AKD +GCN Accuracy (↑) GCN(T) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='09 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} 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+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='31 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='29 One-Hot(S) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='40 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='04 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='75 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='03 RELIANT(S) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='14 ∆SP (↓) GCN(T) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='66 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='33 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='44 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='05 Vanilla(S) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='61 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='17 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='32 RELIANT(S) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='09 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='18 AKD +SAGE Accuracy (↑) SAGE(T) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='28 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='25 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='14 Vanilla(S) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='03 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 One-Hot(S) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='11 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='45 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='12 RELIANT(S) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='33 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='31 ∆SP (↓) SAGE(T) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='32 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='81 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='50 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='39 Vanilla(S) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='29 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='41 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20 One-Hot(S) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='44 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='36 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='93 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='30 RELIANT(S) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='63 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='26 framework, we adopt two types of GNNs (including GCN [21] and GraphSAGE [14]) as the teacher GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To the best of our knowledge, this is the first study on how to mitigate the bias exhibited in GNN- based KD frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In experiments, we adopt the student model yielded by the vanilla KD framework as our first baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For our second baseline, we replace the learnable proxy of bias with a naive proxy for the input of the KD framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we utilize one- hot vectors as the naive proxy for different demographic subgroups during training, where the one-hot vector flags the membership of different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We replace all proxy vectors during inference with an averaged proxy vector across all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here, the rationale is that more distinguishable attributes are easier for deep learning models to learn during training, and these one-hot vectors serve as an "easier" indicator of biased information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In this way, if these one-hot proxy accounts for the exhibited bias of the student model after training, then the exhibited bias could also be mitigated during inference, where such information is wiped out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We evaluate the performance of the compressed GNN models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', the output student model of KD frameworks) from two perspectives, namely Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited utility and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, in terms of utility, we adopt the node classification accuracy as the correspond- ing metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' in terms of fairness, we adopt two traditional metrics ∆SP and ∆EO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here ∆SP measures the bias level (of predictions) under the fairness notion of Statistical Parity, while ∆EO measures the bias level under the no- tion of Equal Opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We present the quantitative results of ∆EO in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 due to space limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' RELIANT is implemented in PyTorch [25] and optimized with Adam optimizer [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In our experiments, the learning rate is chosen in {10−2, 10−3, 10−4} and the training epoch number is set as 1,000 for CPF and 600 for GraphAKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Experiments are carried out on an Nvidia RTX A6000, and the reported numerical results are averaged across three different runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We introduce more details in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2 Effectiveness of RELIANT Here we aim to answer RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we evaluate our proposed framework RELIANT on two KD backbones, namely CPF and GraphAKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For each KD backbone, we adopt two different GNNs (GCN and GraphSAGE) to evaluate the capability of our proposed framework in generalizing to different GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We compare the corresponding performances between the teacher GNN model and the student models trained with three different frameworks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', the vanilla KD framework, the KD framework with the one-hot proxy of bias, and our proposed RELIANT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We present quantitative results on node classification accuracy and ∆SP in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In addition, we also perform experiments based on Equal Opportunity (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1), where we have consistent observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We make the following observations from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of prediction utility, stu- dent models trained with all three KD frameworks achieve comparable performances with the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' This implies that effective knowledge distil- lation can be achieved by all three KD frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of bias mitigation, the student models trained with the vanilla KD frameworks inherit and even exaggerate the exhibited bias from the teacher GNN model in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Training the student models with the one-hot proxy can mitigate bias in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Compared with the student models trained with the vanilla KD framework and the one-hot proxy, RELIANT consistently exhibits less bias w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Statistical Parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Based on the performance of RELIANT in both perspectives, RELIANT achieves effective debiasing for the student model but still maintains comparable model utility with the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, we argue that RELIANT achieves a satisfying balance between debiasing and maintaining utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 0 1 2 3 EO 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5 Accuracy Vanilla V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' w/ Proxy RELIANT (a) GraphAKD on Credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 0 5 10 15 20 SP 75 76 77 78 79 80 Accuracy Vanilla V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' w/ Proxy RELIANT (b) CPF on DBLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Figure 3: Ablation study of RELIANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' "Vanilla" denotes the student model trained with the original KD framework, while "V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' w/ Proxy" represents the student model trained under the KD framework with only learning the proxy of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 100 101 102 103 104 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3 Accuracy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5 SP Accuracy SP (a) CPF on DBLP-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 100 101 102 103 104 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6 EO Accuracy EO (b) GraphAKD on Recidivism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Figure 4: Parameter sensitivity of λ based on two different KD backbones on two real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We also have similar observations on other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3 Ablation Study We aim to answer RQ2 in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, for each framework, we evaluate to what extent the two modules of RELIANT (including learning proxy of bias and enforcing the attribution of bias to the proxy) contribute to the performance of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We present the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 3a is the performance of accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' ∆SP from CPF based on the DBLP-L dataset, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 3b is the performance of accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' ∆EO from GraphAKD based on the Recidivism dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Notably, we also have similar observations under other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We make the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of prediction utility, we observe that the prediction utility is comparable among all three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' This corroborates that both modules exert limited influence on the node classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of bias mitigation, adding the module of learning proxy of bias to the vanilla KD framework brings limited bias mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' This is be- cause the bias could also come from the non-sensitive node attributes (as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' After the module of enforcing the attribution of bias to the proxy is added together with learning proxy of bias, RELIANT is then able to achieve satisfying performance on bias mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4 Parameter Sensitivity We answer RQ3 by studying the tendency of model utility and exhibited bias w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' the change of hyper-parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here λ controls the effect of LAttr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' More specifically, we vary λ in {100, 101, 102, 103, 104}, and we present the corresponding tendency of node classification accuracy and the exhibited bias of the trained student model with RELIANT in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 4a is based on the Credit dataset under GraphAKD, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 4b is based on the DBLP dataset under CPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We also have similar observations on other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We make the following observations from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of prediction utility, the node classification accuracies on both datasets and KD backbones do not exhibit apparent reduction when the value of λ increases from 1 to 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' This verifies that the prediction utility is not sensitive to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of bias mitigation, the student model exhibits less bias when λ increases from 1 to 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, when λ is relatively small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', 1), the learned proxy of bias only partially accounts for the exhibited bias;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' when the value of λ increases, more bias is then attributed to the learned proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Considering the balance between model utility and bias mitigation, a recommended range of λ is between 102 and 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 6 Related Works Algorithmic Fairness in GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Most existing works promoting the algorithmic fairness of GNNs focus either on Group Fairness [10] or Individual Fairness [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, group fairness is defined based on a set of pre-defined sensitive attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', gender and race).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' These sensitive attributes divide the whole population into different demographic subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Group fairness requires that each subgroup should receive their fair share of interest according to the output GNN predictions [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Various explorations have been made towards achieving a higher level of group fairness for GNNs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Decoupling the output predictions from sensitive attributes via adversarial learning is one of the most popular approaches among existing works [31, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Other common strategies include reformulating the objective function with fairness regularization [11, 24], rebalancing the number of intra-group edges between two demographic subgroups [6, 22], deleting nodes or edges that contribute the most to the exhibited bias [8, 9], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' On the other hand, individual fairness does not rely on any sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Instead, individual fairness requires that similar nodes (in the input space) should be treated similarly (in the output space) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' To fulfill individual fairness in GNNs, adding fairness-aware regularization terms to the optimization objective is the most widely adopted approach [5, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Knowledge Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In recent years, knowledge distillation has been proven to be effective in compressing the model but still maintaining similar model prediction performance [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Correspondingly, it has been widely adopted in a plethora of applications, including visual recognition [33], natural language processing [13, 17], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The main idea of knowledge distillation is to transfer the knowledge of a computationally expensive teacher model to a light student model, and thus the student model is able to fit in platforms with limited computing resources [16, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' It is worth noting that such a strategy is also proved to be effective in compressing GNNs [34, 16, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Consequently, there is growing research attention on utilizing knowledge distillation to compress GNNs for more efficient inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For example, encouraging the student model to yield similar output to the teacher GNN via regularization is proved to be effective [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In addition, adversarial learning is also a popular technique to obtain light-weighted but accurate student models [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' However, most of these frameworks for GNNs do not have fairness consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hence the student model tends to be influenced by biased knowledge from the teacher GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Different from existing works, we develop a generalizable knowledge distillation framework that explicitly considers fairness in GNNs but still maintains the utility of GNN predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 7 Conclusion Despite the success of Knowledge Distillation (KD) in compressing GNNs, most existing works do not consider fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hence the student model trained with the KD framework tends to inherit and even exaggerate the bias from the teacher GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In this paper, we take initial steps towards learning less biased student models for GNN-based KD frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we first formulate a novel problem of fair knowledge distillation for GNN-based teacher-student frameworks, then propose a framework named RELIANT to achieve a less biased student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Notably, the design of RELIANT is agnostic to the specific structures of teacher and student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, it can be easily adapted to different KD approaches for debiasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Extensive experiments demonstrate the effectiveness of RELIANT in fulfilling fairness for GNN compression with KD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 8 Acknowledgments This work is supported by the National Science Foun- dation under grants IIS-2006844, IIS-2144209, IIS- 2223768, IIS-2223769, CNS-2154962, CMMI-2125326, BCS-2228533, and BCS-2228534, the JP Morgan Chase Faculty Research Award, and the Cisco Faculty Research Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We would like to thank the anonymous reviewers for their constructive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited References [1] Agarwal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Lakkaraju, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', and Zitnik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Towards a unified framework for fair and stable graph representation learning.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Fifty, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Yu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', and Weinberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Simplifying graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In ICML (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' [33] Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', He, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', and Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Learning an evolutionary embedding via massive knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' International Journal of Computer Vision (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' [34] Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', and Shi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Extract the knowledge of graph neural networks and go beyond it: An effective knowledge distillation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In WWW (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' [35] Yeh, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', and Lien, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The comparisons of data mining techniques for the predictive accuracy of probability of default of credit card clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Expert systems with applications (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' [36] Zemel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Swersky, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', Pitassi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', and Dwork, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Learning fair representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In ICML (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited A Reproducibility In this section, we introduce the reproducibility of the presented experiments as a supplement of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' More specifically, we first introduce the experimental settings in detail, followed by the implementation details of our proposed framework RELIANT, GNNs, KD backbones, and the baseline debiasing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We finally introduce several key packages and their corresponding versions used in our implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The code will be released upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 Experimental Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The implementation of our experiments could mainly be divided into three parts, well-trained teacher GNNs, knowledge distillation backbones, and our RELIANT module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Generally, our experiments are implemented on an Nvidia RTX A6000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We write our experiment code in PyTorch [26] framework, and use Adam [20] optimizer to learn the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We run all experiments three times and record the average and the standard deviation where the random seeds are chosen in {0,10,100,1000,10000}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2 Implementation of RELIANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We imple- ment RELIANT in PyTorch and use Adam as the op- timizer of the learnable proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For results shown in Table 2 and Table 4, we set the proxy learning rate as 10−2 and weight decay as 10−2 for CPF and set the proxy learning rate as 10−2 and weight decay as 5×10−4 for GraphAKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For all the results, we search for the optimal coefficient λ in the set {1,10,100,1000,10000}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3 Implementation of Graph Neural Net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For the training of the teacher GNN models, we use the code in the CPF framework where the details of implementation setting are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4 Implementation of KD Frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' In our experimental setting, the teacher model is fixed during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hence we first train a teacher GNN with the parameters shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' After obtaining a well-trained teacher GNN, we use it as supervision to train the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Details of student training are shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For CPF, we follow the implementation in [34], where we use PLP as the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For the training settings of the student model, we set the maximum epoch as 1,000, early stopping as 500, layer number as 5, feature dropout as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8, edge weight dropout as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2, learning rate in {10−2, 10−3, 10−4}, and weight decay as 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For GraphAKD, we follow the implementation in [16], where we utilize SGC [32] as the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For detailed training settings, we set the maximum Table 3: Experimental settings of teacher GNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hyperparameter GCN GraphSAGE Layer 3 3 Hidden Dimension 64 128 Epoch 1000 1000 Early Stopping 500 500 Learning Rate 10−2 10−2 Weight Decay 10−3 5 × 10−4 Dropout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5 epoch as 600, layer number as 3, learning rate in {10−2, 10−3, 10−4}, weight decay as 5 × 10−4, and dropout as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5 Implementation of the Baseline Debiasing Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For the vanilla KD frameworks, we follow the settings in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4 to directly implement the KD framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' For the baselines with the one-hot proxy for bias, we use the one-hot vector of sensitive attributes as the constant proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' It is added to the input data before training and is fixed during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Hence there is no loss term (as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4 shows) for the constant proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The training settings of the one-hot baseline are the same as Vanilla since the constant proxy vectors do not contain any extra parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6 Packages Required for Implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We list the key packages and corresponding versions in our implementations as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Python == 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 torch == 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 torch-cluster == 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='9 torch-geometric == 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 torch-sparse == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='9 numpy == 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='0 networkx == 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 scikit-learn == 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 pandas==1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='3 scipy==1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 B Complementary Experiments In this section, we perform experiments as a supplement of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Specifically, we first present the perfor- mance of RELIANT over prediction utility and fairness under another widely studied fairness notion – Equal Opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Then, we perform experiments to com- pare the performance of RELIANT on balancing the GNN prediction utility and fairness with state-of-the-art methods that directly debias GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='1 Effectiveness of RELIANT Here we present complementary experiments to answer RQ1, where the Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited Recidivism Credit 40 60 80 100 Accuracy (a) Comparison on the node classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Recidivism Credit 0 10 20 SP Teacher RELIANT EDITS NIFTY (b) Comparison on ∆SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Figure 5: Performance comparison on balancing predic- tion utility and bias mitigation between the proposed framework RELIANT and other state-of-the-art meth- ods that directly debias GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' All numerical results are in percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' fairness notion is instantiated with Equal Opportunity (measured with ∆EO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here we adopt the same settings as those in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We compare the performances between the teacher GNN model and the student models trained with three different framework variants, including the vanilla KD framework, the KD framework with the one-hot proxy of bias, and our proposed RELIANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We present quantitative results on node classification accuracy and ∆EO in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We make the following observations from Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of prediction utility, all stu- dent models trained with the adopted three KD frameworks are able to achieve comparable perfor- mances with the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' This corroborates that all three KD frameworks are capable of achiev- ing effective knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of bias mitigation, the student models trained with the vanilla KD frameworks inherit or exaggerate the bias from the teacher GNN in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Compared with the student models trained with the vanilla KD framework and the one-hot proxy, RELIANT consistently exhibits less bias under the fairness notion of Equal Opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The student model trained with RELIANT even exhibits less bias than the teacher model in certain cases, which further corroborates the effectiveness of RELIANT in training less biased student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' According to the performance of RELIANT in both perspectives above, RELIANT is proved to achieve effective debiasing for the student model but maintains comparable prediction utility compared with the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, we argue that RELIANT achieves a satisfying balance between debiasing and maintaining the prediction utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='2 RELIANT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' GNN-Debiasing Methods In this subsection, we perform experiments and compare the performance of RELIANT with other state-of-the- art GNN debiasing methods on balancing the prediction utility and bias mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here we choose two state-of-the-art GNN de- biasing methods as our baselines, namely EDITS [6] and NIFTY [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' EDITS is a recent GNN debiasing method that learns less biased network data in the pre-processing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' After debiasing, the network data will be fed into the GNN model for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' NIFTY is another recent GNN-based debiasing framework that achieves bias mit- igation in the processing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' During training, node representations are learned to be invariant to the sensi- tive attributes after counterfactual perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here we choose the most widely used GCN as the backbone GNN model for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We choose GraphAKD as the KD backbone of the proposed RELIANT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' It is also worth noting that we also have similar observations with other GNN backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We present the performance comparison results on node classification accuracy and ∆SP in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We make the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of prediction utility, RE- LIANT keeps comparable to the teacher GCN model, while other debiasing methods bear different levels of prediction utility corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Therefore, RELIANT achieves the best performance in main- taining the prediction utility among all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' From the perspective of bias mitigation, RELIANT is able to achieve comparable performance with other baselines when all models bear similar predic- tion utility (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', on Credit dataset);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' when baselines outperform RELIANT on bias mitigation, there is also much more prediction utility sacrifice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=', on Recidivism dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Considering that debiasing the student model with biased supervision is much more difficult than directly debiasing GNNs, we argue that the performances of RELIANT in both cases should be considered satisfying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' According to the performance of RELIANT in both perspectives above, we argue that RELIANT achieves comparable performance with other state- of-the-art GNN debiasing approaches, which further corroborates its satisfying performance on balancing the prediction utility and bias mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Copyright © 2023 by SIAM Unauthorized reproduction of this article is prohibited Table 4: The experimental results based on node classification accuracy and ∆EO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' We use "(T)" and "(S)" suffixes to represent the teacher model and the student model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' Here Vanilla(S) denotes the student model trained with the vanilla KD framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' One-Hot(S) represents the student model trained with the one-hot bias proxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' RELIANT(S) is the student model trained with our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' ↑ denotes the larger, the better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' while ↓ denotes the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' All quantitative results are presented in percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' The best results are in Bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content=' DBLP DBLP-L Credit Recidivism CPF +GCN Accuracy (↑) GCN(T) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='09 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='48 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='21 Vanilla(S) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='18 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='04 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='16 One-Hot(S)) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='35 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='07 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='15 RELIANT(S) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='12 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='57 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='19 ∆EO (↓) GCN(T) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='34 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 Vanilla(S) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='29 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='19 One-Hot(S) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='26 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='28 RELIANT(S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='02 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='12 CPF +SAGE Accuracy (↑) SAGE(T) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} 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+page_content='14 Vanilla(S) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='03 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='08 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='23 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} 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+page_content='23 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='55 RELIANT(S) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='09 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} 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AKD +GCN Accuracy (↑) GCN(T) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='09 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='48 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} 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+page_content='65 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='06 One-Hot(S) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='28 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='13 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='48 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='13 RELIANT(S) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='19 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='37 RELIANT(S) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAzT4oBgHgl3EQfNfuY/content/2301.01150v1.pdf'} +page_content='80 0.' 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0000000000000000000000000000000000000000..66aecc4f728d8b894ecbddbe8aba51b6e88de50a --- /dev/null +++ b/n9E4T4oBgHgl3EQfuw0c/content/tmp_files/2301.05234v1.pdf.txt @@ -0,0 +1,2680 @@ +Title: Study the effect of electrode material, its surface, and dielectric material on plasma and +properties of plasma-activated water +Authors +Vikas Rathore1,2* and Sudhir Kumar Nema1,2 +1. Atmospheric Plasma Division, Institute for Plasma Research (IPR), Gandhinagar, Gujarat +382428, India +2. Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai +400094, India +*Email: vikas.rathore@ipr.res.in + +Abstract +In the present work, the significance of various components (ground electrode material and its +surface (knurling pitch size), power electrode material and type, and dielectric material) of +dielectric barrier discharge plasma device (DBD-PD) on plasma and plasma-activated water +(PAW) properties are studied. The characterization of plasma is performed by studying +voltage-current waveform and plasma discharge power. In addition, the characterization of +PAW is performed by studying the physicochemical properties (PP) and reactive oxygen- +nitrogen species (RONS) concentration. + +The results of plasma characterization and PAW properties reveals that introducing +knurling to ground electrodes showed significant improve the physicochemical properties of +PAW and RONS concentration. Moreover, the use of quartz over glass as dielectric layer +provides a substantial enhancement in PAW properties. Furthermore, the use of wire as a power +electrode compared to mesh and sheet also help in improving the PAW properties. Further, we + +observed that the ground and power electrodes made using copper enriches the RONS +concentration and physicochemical properties of PAW compared to brass and stainless steel. +Keywords: Plasma activated water, plasma device, reactive oxygen-nitrogen species, electrode +and dielecteric material, electrode knurling +1. Introduction +The emerging applications of plasma activated water (PAW) in the field of plasma medicine, +plasma agriculture, food preservation, and senitization industry, etc. provide it a lot of +recognition all over the world[1-6]. These applications of PAW are possible due to the +dissolution of various reactive oxygen-nitrogen species (RONS) in it[7-10]. The oxidizing +species (reactive oxygen species, ROS) such as hydroxyl (̇OH) radical, hydrogen peroxide +(H2O2), superoxide ions (O2ˉ), dissolved O3, and peroxynitrite ions (ONOOˉ), etc. provide +PAW excellent antimicrobial efficacy[5, 6, 11, 12]. The antimicrobial efficacy of PAW +towards bacteria, fungi, viruses, and pest has already been reported in published work of +various researchers[2, 5, 6, 13]. In addition, selective killing of cancer cells and non-cytotoxic +of PAW towards skin cells have also been explored in past literature [14]. + +This antimicrobial activity of PAW is widely used in the surface disinfection of a wide +variety of food products including meat products, sea food, fruits, and vegetables, etc. In +addition, no change in phenotypic characteristics and nutritional value was observed after PAW +treatment with food products[1, 5, 6]. + +Along with reactive oxygen species (ROS), a high concentration of reactive nitrogen +species (RNS) is also present in PAW in the form of nitrate (NO3ˉ) and nitrite (NO2ˉ) ions, +etc[7-10]. Hence, being a rich source of nitrogen species, PAW can also be used as a fertilizer +to enhance crop growth [15]. Past literature also demonstrated the use of PAW to enhance seeds +germination and plant growth in a variety of crops[1, 4, 16, 17]. + + +Hence, considering the above applications of PAW, different types of plasma devices +are used to produce PAW. These devices are mainly composed of different geometries of +electrode, power supplies (high-frequency AC, radiofrequency, and microwave, etc.), type of +plasma discharge (glow discharge, filamentary discharge, spark discharge, and gliding arc +discharge, etc.), etc[7, 10, 11, 14, 17-21]. Also, various PAW process parameters were also +studied in detail to enhance the physicochemical properties of PAW and RONS concentration +in it. These process parameters include plasma-water treatment/exposure time, plasma +discharge power, different type of plasma forming gases (air, N2, Ar, He, N2 + O2, Ar + O2, He ++ O2), gas flow rate, water stirring, and controlling water temperature, etc[4-10]. + +However, no emphasis is given to the components used to prepare plasma devices for +PAW generation as per the best of the author’s knowledge. Since, different materials of +construction of electrode, dielectric, and type of electrode, etc. may significantly influence the +properties of plasma and PAW. This literature gap created the basis of the present investigation. +In the present work, a self-made dielectric barrier discharge plasma device (DBD-PD) is used +to produce air plasma and the generated air plasma is exposed to water to produce plasma +activated water. The DBD-PD has three major components which play a significant role in +plasma production named ground electrode, power electrode, and dielectric cone. The variation +in these components is investigated in the present study. The ground electrode made using +hollow metal pipe and diamond knurling is introduced on the pipe surface. Since, diamond +knurling created metal spikes hence increased the localized electric field which may help in the +generation of excess plasma radicals and species[22-24]. Hence, may improve the PAW +properties. In addition, the variation in knurling pitch size is also studied on plasma and PAW +properties[22]. For comparison of dielectric material glass and quartz are chosen with the same +dimensions and geometry[25, 26]. This is due to the frequent use of glass and quartz as +dielectric materials in DBD discharge[22, 24-27]. The power electrode is made by wrapping + +the wire, mesh, or sheet over the dielectric cone surface[24, 26, 28]. Hence, variation in a +different types of power electrode is also studied on plasma and PAW properties. At last, the +different materials of construction used for making ground and power electrodes and their +effect on PAW physicochemical properties and RONS concentration in it are investigated. The +chosen materials are copper (Cu), brass (Br), and stainless steel (SS), these materials are chosen +due to their frequent use in the preparation of electrodes for various plasma devices [24, 29, +30]. +The produced air plasma using DBD-PD is characterized by studying the voltage- +current characteristics and air plasma species are identified using optical emission +spectroscopy. The effect of variation in DBD-PD components is studied by studying the +voltage-discharge current characteristics, plasma discharge power, physicochemical properties +of PAW, and RONS concentration in it. +2. Materials and Methods +2.1 Experimental setup +The schematic of the experimental setup is shown in figure 1. It shows the electrical and optical +characterization of plasma device, and the production of plasma activated water (PAW) using +plasma device. The plasma device composes a coaxial cylindrical dielectric cone (or tube) (see +figure 1) with an outer diameter 24 mm and thickness 2 mm. The cone has a double-side B24 +male socket. The top side of the cone is fitted in a B24 socket receiver adapter with an air leak +tube which uses an air inlet in a plasma device (see figure 1). The ground electrode of the +plasma device was made using a hollow metal pipe with an outer diameter 16 mm with or +without diamond knurling on its surface. This ground electrode was a tight fit in the teflon cap +as shown in figure 1. The power electrode is either made of flexible mesh, sheet, and wire +which is wrapped around a dielectric cone as shown in figure 1. The airflow rate feed to the + +plasma device was controlled using an air rotameter and set at 15 l min-1. This plasma device +was powered using 0-30 kV, 0-30 kHz higher voltage high-frequency power supply. The +present work uses a constant frequency of 20 kHz for all experiments. + +The generated air plasma in the plasma device was characterized using electrical and +optical emission measurements. A 1000x high voltage probe (Tektronix P6015A) and a 4- +channel 100 MHz, 2 GS s-1 digital storage oscilloscope (Tektronix TDS2014C) was used to +measure the applied voltage across plasma device. To measure the total current (conducion and +discharge) and transported charge during plasma production, a voltage drop was measured +across 31 ohms non-inductive resistor and 100 nF capacitor using a 10x voltage probe +(Tektronix TPP0201) in series with the ground. The air plasma emission spectrum was +measured using optical fiber and a spectrometer. The emission spectrum was measured in the +range of 190 nm to 925 nm using two different spectrometers (Model EPP2000-UV from +StellarNet Inc. (range 190 nm to 610 nm) and UVH-1 from ASEQ instruments (range 290 nm +to 925 nm)). + +A 40 ml of ultrapure milli-Q (Demineralized water, DM water) was used for PAW +production. To activate the water, plasma-water inteaction time varied from 1 to 5 min. The set +distance between the ground electrode tip and the water surface was 30 mm. To enhance the +dissolution of reactive species produced due to plasma-water interaction, the continuous +stirring and water temperature were maintained at 0 °C. The stirring of the water was controlled +using a magnetic stirrer and magnetic teflon bar kept in PAW. The water stirring speed was +kept constant at 300 rpm throughout the experimental duration. The water temperature was +maintained at 0 °C by keeping the ice-water mixture in a storage container as shown in figure +1. +2.2 The role of plasma device components on plasma and PAW properties + +The plasma device used to produce air plasma mainly consists of three important components +which play a significant role in plasma production. The ground electrode, power electrode, and +dielectric material. Hence, the present investigation emphasizes the role of these plasma device +components on plasma and PAW properties. The present work investigates the role of ground +electrode knurling pitch size, type of power electrode, dielectric material, and material of +construction of ground and power electrode. +2.2.1 The ground electrode knurling +The knurling of the ground electrode may increase the localized electric field for different +knurling pitch sizes. That influences the generation of radicals and species in the plasma phase. +Hence, the role of different knurling pitch sizes was studied on plasma and PAW properties in +the present work. The different knurling pitch sizes chosen were 0 mm, 0.5 mm, 1 mm, and 2 +mm, respectively (see figure 1). While studying the role of knurling pitch size other plasma +device components and PAW process parameters were kept constant. +2.2.2 The dielectric material +The dielectric material is one of the most important parameters during dielectric barrier +discharge. To study the role of dielectric material on plasma and PAW properties two different +types of dielectric material were used named glass and quartz in form of a B24 double side +cone (see figure 1). The other PAW process parameters and plasma device components remain +unchanged while comparing the mentioned dielectric material. +2.2.3 The type of power electrode +To study the impact of power electrode type on plasma and PAW properties, three different +types of power electrodes were used (mesh, sheet, and wire) keeping other variables constant. +This mesh, sheet, and wire were wrapped around the dielectric cone to make a power electrode +as shown in figure 1. + +2.2.4 Material of ground and power electrode +Three different types of materials named copper (Cu), brass (Br), and stainless steel (SS) are +used to study the role of ground and power electrode material on the physicochemical +properties and RONS concentration of PAW. A detail of the same is shown in figure 1. In +which, a picture of different ground electrode materials and power electrode materials is shown. +2.3 Measurement of physicochemical properties and RONS concentration of PAW +2.3.1 Physicochemical properties +A pH meter (Hanna Instrument, model HI98120) was used to measured the pH of PAW. An +ORP meter (Hanna Instrument, model ORP-200) was used to determine the oxidizing tendency +of PAW. The dissolved ions in PAW were measured using TDS (total dissolved solid) meter +(HM digital, model AP1) and EC (electrical conductivity) meter (Contech, model CC-01), +respectively. The least count of instruments used in the present work was given as 0.01 (pH), +1 mV (ORP), 0.1 µS cm-1 (EC), and 1 ppm (TDS), respectively. +2.3.2 Reactive oxygen-nitrogen species (RONS) +The preliminary detection of RONS present in PAW was performed using a strip test for NO2ˉ +ions (Macherey-Nagel, QUANTOFIX Nitrite) and H2O2 (Macherey-Nagel, QUANTOFIX +Peroxide 25), and colorimetric test kit for NO3ˉ ions (Macherey-Nagel, VISOCOLOR Nitrate) +and dissolved O3 (Hanna Instrument, HI-38054 Ozone test kit) test kit. + +The quantitative estimation of RONS concentrations in PAW was determined +spectrophotometrically. A standard curve of nitrate (NO3ˉ) ions, nitrite (NO2ˉ) ions, and +hydrogen peroxide (H2O2) was prepared to determine the unknown concentration of these +species in PAW. The molar attenuation coefficient of NO3ˉ ions, NO2ˉ ions, and H2O2 were +given as 0.0602 (mg l-1)-1 (range of NO3ˉ ions is 0.61 to 6.10 mg l-1), 0.0009 (µg l-1)-1 (range of + +NO2ˉ ions is 67 to 536 µg l-1), and 0.4857 (mmol l-1)-1 (range of H2O2 is 9.8 to 98 µmol l-1), +respectively[5, 7]. The concentration of unknown NO3ˉ ions present in PAW was determined +spectrophotometrically at 220 nm using mentioned molar attenuation coefficient. The NO2ˉ +ions react with a reaction mixture of sulfanilamide and N-(1-naphthyl) ethylenediamine +dihydrochloride to give reddish-purple color (azo dye) which showed maximum absorbance at +540 nm in an acidic region. Hence, the unknown concentration of NO2ˉ ions in PAW was +measured at 540 nm using mentioned molar attenuation coefficient. Similarly, H2O2 reacts with +titanium ions (titanium (IV) oxysulfate) in the acidic region to form a yellow color complex +(pertitanic acid, H2TiO4) which shows maximum absorbance at 407 nm[5, 7]. This property of +H2O2 is used to determine the unknown H2O2 concentration in PAW using the mentioned molar +attenuation coefficient. In addition, NO2ˉ ions present in PAW interfere in the determination +of H2O2 concentration in PAW. The NO2ˉ ions reacted with H2O2 and suppress the H2O2 +concentration in PAW beyond the detection limit. Hence, azide ions (N3ˉ) in the form of sodium +azide (NaN3) were added to PAW which reacts with nitrate ions and degrades it so the +interference in H2O2 determination can be prohibited[5, 7]. Appropriate dilution was performed +to determine RONS concentration if the RONS concentration in PAW exceeded the detection +limit. + +An indigo colorimetry method was used to determine the dissolved O3 concentration in +PAW. In which, the rapid decolorization of indigo reagent occurs by dissolved O3 in the acidic +region. The indigo reagent was prepared using potassium indigo trisulfonate, sodium +phosphate, phosphoric acid, and water, respectively. The volumetric method (equation (1))[5, +7] used to determine the dissolved O3 in PAW was given as: +𝑚𝑔 +𝑙 𝑜𝑓 𝑂3 = +100 × ∆𝐴 +𝑓 ×𝑏 × 𝑣 + + + + + + + + +(1) + +Where, ‘ΔA’ is the absorbance difference in PAW and blank at 600 nm, ‘b’ is the optical path +length of the cell (1 cm), ‘v’ is the volume of PAW, and ‘f’ is the sensitivity factor (0.42), +respectively. +2.4 Residue metal analysis in PAW +The plasma generation may erode the inner (ground) electrode material. The energetic particles +collide with the ground electrode and result in erosion/sputtering. The erosion of material may +dissolved in water in the form of metal residue in PAW. The residual metal analysis in PAW +and control were performed using inductive couple plasma-mass spectroscopy (ICP-MS) +(2000B ICP-MS, Perkin Elmer) in collision mode (Helium KED). The chosen ICP-MS method +of testing were Environmental Protection Agency's (EPAs) 1638 and EPA 6020 B. As the +ground electrodes were made of SS (iron and carbon alloy), Br (copper and zinc alloy), and +Cu. The residual metal analyzed in PAW using ICP-MS were iron, copper, and zinc. The PAW +and control sample were reconstituted in hydrochloric acid, and diluted (25X dilution) with +DM water and preserve with nitric acid before ICP-MS analysis. +2.5 Data Analysis +All experiments were repeated atleast three times (n ≥ 3). The results were expressed in plots +and tables as mean ± standard deviation (µ ± σ). The statistically significant difference among +the group µ ± σ were estimated using one-way ANOVA followed by post-hoc test (Fisher's +least significant difference (LSD)). + + +Figure 1. Schematic of electrical and optical emission characterization of plasma device and +production of plasma activated water + + + +Glass +Quar +Oscilloscope +MagneticStirrer15 +0 +5 +Current (mA) +Current (mA) +0 +¥5 +5 +-10 +-15 +.15 +15 +10 +５０ +Current (mA) +5 +Current (mA) +5 +5 +15 +-15 +10 +10 +Current (mA) +Current (mA) +0 +-10 +-10Figure 2. Voltage-current characterization of plasma device. (a, b) With and without ground +electrode knurling, (c, d) variation in dielectric material (glass, quartz), (e, f, g) variation in +power electrode type (mesh, wire, sheet) +3. Results and discussions +3.1 Electrical and optical emission characterization of air plasma +3.1.1 Voltage-current characteristics +The variation in air plasma when produced using different electrode materials, electrode types, +dielectric materials, and with and without ground electrode knurling is studied using the current +waveform as shown in figure 2. The current profile shown in figure 2 is a combination of two +currents, a continuous alternating current (AC) (sine wave) and a discharge current. The +discharge current normally appears in each rising and falling half-cycle. The discharge current +is the indicator of gas breakdown flowing through the coaxial pathway between the ground +electrode and dielectric. The breakdown of gases created various ions and electrons which are +indicated by various high and low multiple current peaks (filaments) in rising and falling half- +cycles over continuous AC. The created ions and electrons due to gas discharge were +responsible for the discharge current shown in figure 2. Hence, the periodic formation of +discharge current over the continuous AC showed the formation of plasma between the ground +electrode and dielectric. The time period of discharge current filaments is in order ~ 100 ns. +Hence the combination of these current filaments represents the characteristics of dielectric +barrier discharge (DBD) filamentary micro-discharge[31]. +Ground electrode knurling +Figure 2 (a, b) showed the voltage-current characteristics of air plasma with and without ground +electrode knurling while keeping the other design parameters and process parameters constant. +The discharge current formed in the rising curve of positive half-cycle for with and without + +knurling ground electrode of plasma device. The peak value of discharge current with and +without ground electrode knurling was 14.2 mA and 11.6 mA (positive half-cycle), +respectively. The increase in discharge current peak in knurling plasma devices showed more +generation of electron-ion pair during gas discharge which showed by a shoot up in the current +peak. Hence, the knurling of the ground electrode supports more formation of discharge gases +products. This was due to improvisation in localized electric fields with sharp knurling edges. +The sharp edges of diamond knurling distorting the uniform electric field [23, 24]. Due to +which localized electric field near sharp edges enhanced. As a result, generate higher pulse +filaments near the sharp edges of diamond knurling. Hence, more reactive species generation +occurs in the plasma phase which could be utilized for various purposes. Similar results were +shown in the work reported by Mei et al.[24] and Takaki et al.[23]. However, they use screw +edges and a large number of pyramids in multipoint geometry of the inner electrode to enhance +localized electric field instead of diamond knurling. The diamond knurling has a slight edge +over the screw-type electrode since it creates a significantly higher sharp edge density +compared to the screw. +Dielectric material +The most commonly used dielectrics during DBD plasma production are glass and quartz. The +comparison of discharge characteristics of glass and quartz dielectric, when used in plasma +device is shown in figure 2 (c, d). In glass dielectric, the discharge current peaks were observed +in the rising half-cycle. However, in the case of quartz dielectric, the discharge current peaks +were observed in both positive and negative half-cycles. Moreover, the observed discharge +current peaks in the positive rising half-cycle were significantly higher compared to the +negative falling half-cycle. Hence, over a time period, two-discharge currents regions (positive +rising half-cycle and negative falling half-cycle) appeared in quartz compared to the one- +discharge current region (positive rising half-cycle) in the glass. This signifies a higher + +concentration of discharge gas products formed in plasma using quartz as a dielectric compared +to glass. This will be due to better distribution of charge over quartz surface compared to glass. +Since, the quartz has a crystalline structure and glass is amorphous[25]. Moreover, the use of +quartz (dielectric strength 470 to 670 MV m-1) over the glass is preferred due to its substantially +higher dielectric strength compared to glass (dielectric strength 20 to 40 MV m -1). The +discharge current observation of the present investigation was also supported by work reported +by Ozkan et al.[25]. In which, higher discharge current peaks were observed in quartz dielectric +compared to glass dielectric. Also, filamentary discharge peaks were observed in both positive +and negative half-cycles for quartz dielectric. However, for glass dielectric, observed +filamentary discharge peaks in one-half cycle were substantially higher than other half cycles. +Power electrode type +The effect of different types of power electrodes on air plasma discharge characteristics is +shown in figure 2 (e-g). The power electrode is made from mesh or winding of wire over a +dielectric cone or thin metal sheet. The use of different types of power electrodes had a +significant impact on discharge current characteristics. The number of high discharge-current +filaments in power electrode made using wire was significantly greater compared to power +electrodes made using mesh or sheet as shown in figure 2 (e-g). The discharge current filaments +in the power electrode made using wire or sheet mainly occur in the negative falling half-cycle +(figure 2 (f, g)). However, inverse behavior was observed in the power electrode made using +mesh in which discharge current filaments appeared in the positive rising half-cycle (figure 2 +(e)). + +As the large number of high peaks current filaments observed in the discharge-current +profile of air plasma produced using wire as power electrode showed more formation of +electron-ion pairs. As a result, the density of reactive plasma species produced using wire as a + +power electrode will be higher compared to power electrodes made using mesh or sheet. The +use of wire, mesh, and sheet (foil) as outer electrodes for DBD discharge wrapped around +dielectric was previously explored by Mei et al.[24], Chang et al.[28], and Nur et al.[26]. In +which, Mei et al.[24] used aluminium foil as outer electrode wrapped around dielectric have +higher filamentary discharge current peaks compared to mesh. The present work also showed +denser current peaks in the sheet as outer electrode compared to mesh. This was due to uniform +covering of dielectric using sheet compared to mesh results in more charge accumulation on +the dielectric surface and more discharge area. Hence, the number of discharge filaments +increases in the sheet compared to mesh shown as dense filamentary discharge in voltage- +current characteristics of the sheet as power electrode (figure 2 (g)). + +In conclusion, the use of a knurled ground electrode, quartz as dielectric material, and +wire as a power electrode results in the generation of a high concentration of charged plasma +species/radicals. Hence, this configuration could be used in future technology where high +plasma species density is required. + + +Figure 3. Variation in plasma discharge power while varying, (a) ground electrode knurling +pitch, (b) ground electrode material, (c) power electrode type, (d) dielectric material, (e) power +1 +2 +3 +4 +0 +2 +4 +6 +8 +10 +12 +14 +b +b +a +Plasma discharge power (W) +Ground electrode +knurling pitch (mm) +0 0.5 1 2 +(a) +a +SS +Br +Cu +0 +2 +4 +6 +8 +10 +c +b +a +(e) +(d) +(c) +(b) +Plasma discharge power (W) +Ground electrode +material +Mesh +Wire +Sheet +0 +2 +4 +6 +8 +b +b +a +Plasma discharge power (W) +Power electrode +type +Glass +Quartz +0 +2 +4 +6 +8 +10 +b +Plasma discharge power (W) +Dielectric material +SS +Br +Cu +0 +2 +4 +6 +8 +10 +a +c +b +a +Plasma discharge power (W) +Power electrode +material + +electrode material. Different lowercase letters showed statistically significant difference +(p<0.05, n ≥3) among the group mean ± standard deviation (µ ± σ) +3.1.2 Plasma discharge power +As discussed above the discharge of gas is shown by the increase in the discharge-current. This +discharge current signifies the movement of newly generated ionized species in the plasma. +Hence, the energy is consumed during the generation of these reactive species/radicals in +plasma. The higher dissipation of energy (or power) results in the formation of a high +concentration of reactive species/radicals in plasma. Figure 3 (a-e) showed the variation in the +plasma discharge power while varying knurling pitch, ground and power electrode material, +dielectric material, and type of power electrode, respectively at constant applied voltage. + +Figure 3 (a) showed the variation in the plasma discharge power with increasing +knurling pitch size. An increase in knurling pitch size from 0 to 1 mm increased the plasma +discharge power by 76.8%. Moreover, a further increase in the knurling pitch size from 1 mm +to 2 mm decreased the plasma discharge power by 9.4%. However, this decrease in power was +not statistically significant (p > 0.05). Hence, the knurling pitch is an important parameter that +influences the generation of more reactive species/radicals in plasma. Mei et al.[24] and Takaki +et al.[23] also showed improvement in energy efficiency and input energy at the same applied +voltage. They used a screw-type (similar to knurling) inner electrode compared to the rode- +type inner electrode and multipoint (pyramid shape) geometry (similar to knurling) compared +to plane plate geometry. + +The effect of ground and power electrode material on plasma discharge power is shown +in figure 3 (b, e). The calculated plasma discharge power for copper (Cu) material as ground +and power electrodes was substantially (p < 0.05) higher than brass (Br) and stainless steel (SS) +ground and power electrodes. For the ground electrode, plasma discharge power for Cu was + +23.7% higher compared to SS and 87.5% higher compared to Br. Similarly, for the power +electrode, plasma discharge power for Cu was 50.9% higher compared to SS and 83.7% higher +compared to Br. Hence, from the above discussion the choice of material of electrode while +producing plasma also plays a significant role in the concentration of generated plasma species +and radicals. Since higher plasma discharge signifies more formation of reactive plasma species +and radicals. The role of electrode material on plasma discharge power also was reported by +Jahanmiri et al.[32]. In which they showed average power consumption by Cu and Br +substantially higher than SS. In addition, no significant difference was observed in power +consumption when Cu and Br were used as material. + +The impact of different types of power electrodes on plasma discharge power is shown +in figure 3 (c). As shown in the results of discharge current characteristics, the multiple high +discharge current peaks observed in power electrode made using wire compared to mesh and +sheet as power electrode (figure 2 (e-f)). Similar results were observed in the plasma discharge +power, in which the power electrode made using wire had 14.5% higher discharge power +compared to mesh and 19.8% higher discharge power compared to the sheet. Mei et al.[24] +used SS mesh and aluminium foil as outer electrode wrapped around quartz dielectric tube. +They showed better energy efficacy in aluminium foil compared to SS mesh. However, in the +present work, we did not observe any significant difference in the plasma discharge power +when using SS sheet and SS mesh as power electrodes wrapped around the dielectric. + +The effect of dielectric material on the plasma discharge power is shown in figure 3 (d). +In which, the calculated plasma discharge power of air plasma when produced using quartz as +dielectric significantly (p < 0.05) higher compared to glass as dielectric. The higher discharge +power in quartz compared to glass was due to the power dissipation in each rising and falling +half-cycle in quartz compared to single rising half-cycle power dissipation in the glass over a +period. Similar results were observed in the work reported by Ozkan et al.[25], in which higher + +discharge power was observed in quartz dielectric compared to glass dielectric at constant +parameters. +3.1.3 Emission spectra of air plasma +The emission spectrum of air plasma is shown in figure 4. The wavelength range shown in +figure 4 lies between 185 nm to 950 nm. The recording emission spectra of air plasma mainly +contain strong emission band peaks of the N2 second positive system (C 3Πu → B 3Πg) lies in +the range of 290 nm and 440 nm (shown in figure 4 box). In addition, weak intensity N2+ (B +2Σu+ → X 2Σg+) first negative system band peaks were also observed in air plasma. The observed +band peaks details are shown in Table 1. +Mechanism of formation of N2 second positive system and N2+ first negative system in air +plasma +The ground state N2 (X 1Σg+)υ present in the air was excited by direct impact excitation to upper- +level N2 (C 3Πu)υ' (equation (2)) in an applied electric field between space change developed +over the dielectric surface and ground electrode. The population of upper-level N2 is the result +of high energy electron collision with the N2 ground state[33, 34]. +N2 (X 1Σg+)υ ++ +e +→ +N2 (C 3Πu)υ' ++ +e + + + +(2) +The radiative decay of upper-level N2 (C 3Πu) to lower-level N2 (B 3Πg) (equation (3)) +results in the formation of strong emission band peaks of the N2 second positive system as +shown in Table 1. +N2 (C 3Πu)υ' +→ +N2 (B 3Πg)υ" + hυ + + + + + + +(3) + +The formation of excited-state N2+ (B 2Σu+)υ' from N2 (X 1Σg+)υ ground state and its +population may followed the following paths. +One-step process + +N2 (X 1Σg+)υ ++ +e +→ +N2+ (B 2Σu+)υ' + +2e + + + +(4) +Two-step process +N2 (X 1Σg+)υ ++ +e +→ +N2+ (X 2Σu+)υ' + +2e + + + +(5) +N2+ (X 2Σu+)υ + +e +→ +N2+ (B 2Σu+)υ' + +e + + + +(6) + +In a one-step process, electron impact ionization of N2 (X 1Σg+)υ ground state molecule +occurs to N2+ (B 2Σu+)υ' excited state (equation (4)). However, in a two-step process, electron +impact ionization of N2 (X 1Σg+)υ ground state molecule occurs to N2+ (X 2Σu+)υ' ground state +(equation (5)). Then this generated N2+ (X 2Σu+)υ ground state populated to N2+ (B 2Σu+)υ' excited +state with electron impact excitation (equation (6)). + +The radiative decay of excited state N2+ (B 2Σu+)υ' to ground state N2+ (X 2Σu+)υ (equation +(7)) results in formation of weak intensity N2+ (B 2Σu+ → X 2Σg+) first negative system band +peaks[33]. +N2+ (B 2Σu+)υ' → +N2+ (X 2Σu+)υ" + +hυ + + + + + +(7) + + +Figure 4. Optical emission spectra of air plasma +Table 1 The observed band peaks lines in air emission spectra[34] +Species +Transitions +Spectral +lines +(nm) +υ' → υ" +Transition Probabilities +(s-1) +N2 +C 3Πu → B 3Πg +296.1 +3 → 1 +6.61 × 106 +313.3 +2 → 1 +8.84 × 106 +315.4 +1 → 0 +1.02 × 107 +337.0 +0 → 0 +1.10 × 107 +353.6 +1 → 2 +4.61 × 106 +357.7 +0 → 1 +7.33 × 106 +371.0 +2 → 4 +3.37 × 106 +375.6 +1 → 3 +4.10 × 106 + +N2 (C "I-B"Ilg)380.4 +0 → 2 +2.94 × 106 +394.0 +2 → 5 +2.63 × 106 +399.6 +1 → 4 +2.49 × 106 +405.5 +0 → 3 +9.23 × 105 +426.7 +1 → 5 +7.69 × 105 +434.2 +0 → 4 +2.47 × 105 +N2+ +B +2Σu+ → X +2Σg+ +391.7 +0 → 0 +1.10 × 107 +419.9 +2 → 3 +3.13 × 106 + +3.2 Formation of RONS in water and change in physicochemical properties of water +The mechanism of formation of various reactivity oxygen-nitrogen species (RONS) in PAW +is shown in Table 2. The reactions are divided into three phases – plasma phase (equations (8- +20)), plasma-liquid interphase, and liquid phase (equations (21-37))[10-12, 21, 35-41]. In the +plasma phase, the dissociation of N2, N2+, O2, and H2O, etc. occurs by electrode impact +dissociation into corresponding atoms (N, O, H, etc.) and molecules (OH, etc.) (equations (8- +10, 12). The dissociation of molecules also occurs by high-energy excited molecules. As shown +in equation (11), dissociation of H2O molecule with high energy N2 molecule. Along with the +dissociation reaction, the dissociative replacement also occurs by high-energy atoms. The +formation of NO molecule occurs by dissociative replacement of N2, O2, and OH by O and N +atoms (equations (13-15)). Similarly, the formation of HO2 and NO2 occurs by dissociative +replacement of OH and NO by gases O3 (equations (19, 20)). At last, recombination reactions +occur in the plasma phase which results in the formation of gases O3, NO2, and H2O2, etc +(equations (16-18)). + + +The region between the plasma phase and liquid phase is known as the plasma-liquid +interphase. In which, the relatively long-lived species exist before dissolved into the liquid +phase. These species were given as excited N2, H2O, O3, NO, OH, NO2, and HO2 molecules, +etc. as shown in Table 2 of plasma-liquid interphase. + +The formation of stable reactive RONS is shown in the liquid phase of Table 2. The +stable species which are identified and whose concentrations are measured in the present work +are shown in bold (marked in red) in Table 2. The reactions (equations (21-37)) which result +in the formation/degradation of stable species such as NO2ˉ ions, NO3ˉ ions, dissolved O3, and +H2O2 are shown in the liquid phase of Table 2. The reactions that result in the formation of +NO2ˉ ions (equations (21,30,31,)) in PAW were the reaction between NO (aq.) and OH (aq.), +and NO (aq.) and NO2 (aq.) with H2O (aq.). Similarly, the reaction that results in the formation +of NO3ˉ ions (equations (22,31,36,37)) in PAW were given as the reaction between dissolved +O3 and NO2ˉ ions, and NO2 and H2O (aq.), NO2ˉ and OH (aq.), and dissociation of peroxynitric +acid (aq. ONOOH). The dissolution of gases O3 in water results in the formation of dissolved +O3 (aq.). This was due to the particle solubility of O3 in water at room temperature. The +formation of H2O2 (equations (23,26,34)) in PAW occurs due to the reaction between OH +molecules, HO2 molecules, and HO2 molecule and O2ˉ ion in water. + +Along with the formation of RONS in PAW, the degradation of RONS also occurs in +PAW to form more stable species in PAW. The dissolved O3 and NO2ˉ ions react to form more +stable NO3ˉ ions in PAW (equation (22)). Similarly, aqueous OH reacts with NO2ˉ ions reacts +to form NO3ˉ ions in PAW in the acidic region (equation 36). Also, the NO2ˉ ions react with +H2O2 to form peroxynitric acid, which is degraded to form NO3ˉ ions in PAW (equation +(35,37)). + + +As discussed in Table 2, the formation of various reactive oxygen-nitrogen species in +PAW. These species in PAW give PAW an immense potential to be used in various +applications such as microbial (bacteria, fungi, virus, and pest, etc.) inactivation, food +preservation, selective killing of cancer cells, and seeds germination and plant growth, etc.[1, +4-6, 14-17] The affinity of PAW for microbial inactivation and selective killing of cancer cells +is due to the presence of various strong oxidizing species such as H2O2, dissolved O3, hydroxyl +radical (OH), peroxynitrile (ONOOˉ), and superoxide ions (O2ˉ), etc. as shown in Table 2[2, 5, +6, 11, 12, 14]. Along with strong oxidizing species, PAW is also a rich source of nitrogen +species (NO3ˉ, NO2ˉ, etc.) which signifies the usefulness of PAW in the agriculture field[1, 4, +15-17]. +Table 2 Mechanism of formation of reactive oxygen-nitrogen species in PAW[10-12, 21, 35- +41] +Reaction +phase +Reaction +Rate constant or reaction +rate +Equation +number +Plasma +phase +𝑁2 + 𝑒− → 2𝑁 + 𝑒− +6.3 × 10-6 Te-1.6 e-9.8/Te +cm3 s-1 +8 +𝑒− + 𝑁2 ++ → 𝑁 + 𝑁∗ +2.0 × 10-7 (Te/0.03)-0.39 +cm3 s-1 +9 +𝐻2𝑂 + 𝑒− → 𝑂𝐻 + 𝐻 + 𝑒− +2.6 × 10-12 cm3 s-1 +10 +𝐻2𝑂 + 𝑁2(𝐴) → 𝑂𝐻 + 𝐻 + 𝑁2 +4.2 × 10-11 cm3 s-1 +11 +𝑂2 + 𝑒− → 𝑂 + 𝑂 + 𝑒− +3.2 × 10-11 cm3 s-1 +12 +𝑁2 + 𝑂 → 𝑁𝑂 + 𝑁 +2.7 × 10-11 cm3 s-1 +13 +𝑁 + 𝑂2 → 𝑁𝑂 + 𝑂 +8.5 × 10-17 cm3 s-1 +14 +𝑁 + 𝑂𝐻 → 𝐻 + 𝑁𝑂 +4.7 × 10-11 cm3 s-1 +15 + +𝑂 + 𝑁𝑂 + 𝑀 → 𝑁𝑂2 + 𝑀 +9.0 × 10-32 cm6 s-1 +16 +𝑂 + 𝑂2 → 𝑂3 +1.7 × 10-12 cm3 s-1 +17 +𝑂𝐻 + 𝑂𝐻 → 𝐻2𝑂2 +2.6 × 10-11 cm3 s-1 +18 +𝑂𝐻 + 𝑂3 → 𝐻𝑂2 + 𝑂2 +1.9 × 10-12 cm3 s-1 +19 +𝑁𝑂 + 𝑂3 → 𝑁𝑂2 + 𝑂2 +5.1 × 10-12 cm3 s-1 +20 +Plasma- +liquid +interphase +N2, e-, N, N*, H2O, OH, H, O2, NO, +O3, NO2, H, HO2, H2O2, etc. + + +Liquid +phase +𝑂𝐻 + 𝑁𝑂 → 𝑵𝑶𝟐 +− + 𝐻+ +1.0 × 1010 M-1 s-1 +21 +𝑶𝟑 + 𝑵𝑶𝟐 +− → 𝑂2 + 𝑵𝑶𝟑 +− +2.5 × 105 M-1 s-1 +22 +𝑂𝐻 + 𝑂𝐻 → 𝑯𝟐𝑶𝟐 +5.0 × 109 M-1 s-1 +23 +𝑂𝐻 + 𝑶𝟑 → 𝑂2 + 𝐻𝑂2 +1.0 × 108 M-1 s-1 +24 +𝑂𝐻 + 𝐻𝑂2 → 𝑂2 + 𝐻2𝑂 +7.5 × 109 M-1 s-1 +25 +𝐻𝑂2 + 𝐻𝑂2 → 𝑂2 + 𝑯𝟐𝑶𝟐 +1.0 × 106 M-1 s-1 +26 +𝐻𝑂2 + 𝑁𝑂 → 𝑂𝑁𝑂𝑂− + 𝐻+ +3.2 × 109 M-1 s-1 +27 +𝑂𝐻 + 𝑁𝑂2 → 𝑂𝑁𝑂𝑂− + 𝐻+ +1.2 × 1010 M-1 s-1 +28 +𝑁𝑂 + 𝑁𝑂 + 𝑂2 → 2𝑁𝑂2 +2.3 × 106 M-2 s-1 +29 +𝑁𝑂2 + 𝑁𝑂 + 𝐻2𝑂 +→ 𝟐𝑵𝑶𝟐 +− + 2𝐻+ +2.0 × 108 M-1 s-1 +30 +2𝑁𝑂2 + 𝐻2𝑂 → 𝑵𝑶𝟑 +− + 𝑵𝑶𝟐 +− ++ 2𝐻+ +0.5 × 108 M-1 s-1 +31 +𝑯𝟐𝑶𝟐 + 𝑂𝐻 → 𝐻2𝑂 + 𝑂2 +− + 𝐻+ +2.7 × 107 M-1 s-1 +32 +𝑂2 +− + 𝑁𝑂 → 𝑂𝑁𝑂𝑂− +5.0 × 109 M-1 s-1 +33 + +𝐻2𝑂 + 𝐻𝑂2 + 𝑂2 +− +→ 𝑂2 + 𝑯𝟐𝑶𝟐 ++ 𝐻𝑂− +9.7 × 107 M-1 s-1 +34 +𝑵𝑶𝟐 +− + 𝑯𝟐𝑶𝟐 + 𝐻+ +→ 𝑂𝑁𝑂𝑂𝐻 + 𝐻2𝑂 +1.1 × 103 M-1 s-1 +35 +𝑵𝑶𝟐 +− + 𝑂𝐻 + 𝐻+ → 𝑵𝑶𝟑 +− + 2𝐻+ 5.3 × 109 M-1 s-1 +36 +𝑂𝑁𝑂𝑂𝐻 → 𝑵𝑶𝟑 +− + 𝐻+ +0.9 s-1 +37 + +3.3 The effect of knurling on the physicochemical properties of PAW and RONS concentration +The variation in the physicochemical properties of PAW and reactive oxygen-nitrogen species +(RONS) concentration while varying knurling pitch size is shown in figure 5. Figure 5 showed +that introducing knurling to the ground electrode significantly (p < 0.05) improves the +physicochemical properties of PAW and RONS concentration in it. This signifies the increase +in species/radicals produced in the plasma phase when the ground electrode has knurling +compared to the non-knurled electrode. This behavior was also observed in the electric +characterization of plasma. In which, higher discharge current and power dissipation were +observed in plasma with ground electrode knurling compared to the non-knurled electrode. + +The plasma-water interaction decreased the pH of PAW due to the formation of various +acidic species in water. The most leading acidic species are NO2ˉ ions and NO3ˉ ions in the +form of nitrous and nitric acid. Increasing plasma-water treatment time results in the formation +of more acidic species in PAW as a result, we observed a continuous decrease in the pH of +PAW as shown in figure 5 (a). Moreover, the pH of PAW prepared using plasma device with +knurling and without knurling have significant (p < 0.05) difference among them. With +knurling (2 mm knurling pitch), the pH of PAW decreased by 53.0% after 5 min of plasma- + +water treatment compared to control. However, without knurling pH of PAW decreased by +25.8% after 5 min of plasma-water treatment compared to control only. In addition, increasing +knurling pitch from 0.5 mm to 2 mm showed a decrease in pH of PAW, however, this decrease +in pH of PAW was not statistically significant (p < 0.05). Hence, introducing knurling to the +ground electrode has a substantial impact on the pH of PAW. Since, as discussed in figure 3 +(a), introducing knurling increased the plasma discharge and resulted in the formation of more +acidic plasma species that dissolved in water and decreased the pH of PAW. + +The effect of knurling on oxidation-reduction potential (ORP) of PAW is shown in +figure 5 (b). The ORP gives the oxidizing tendency of PAW which could be used as an indicator +of the antimicrobial activity of PAW. Since, it gives the net combination of all oxidizing species +(dissolved O3, H2O2, ̇OH, free electrons, etc.) present in PAW. Increasing plasma-water +treatment time increased the ORP of PAW which showed the increase in dissolution of +oxidizing species in PAW. The knurling of the ground electrode also helps in increasing the +oxidizing tendency of PAW as shown in figure 5 (b). The ORP of PAW prepared using a 1 mm +knurling ground electrode (ORP – 600 mV) was 18.8% higher compared to without a knurling +ground electrode (ORP – 505 mV). Moreover, the ORP of PAW prepared using a 1 mm +knurling electrode was substantially (p < 0.05) higher compared to 0.5 mm and 2 mm knurling +electrodes. This was due to higher power dissipation in the 1 mm knurling electrode (figure 3 +(a)) resulting in the formation of more oxidizing species in PAW. + +The net combination of all inorganic ions (H+, NO3ˉ ions, NO2ˉ ions, OONOˉ, etc.) +present in PAW were measured using total dissolved solids (TDS) and electrical conductivity +(EC). Figure 5 (c, d) showed the variation in TDS and EC of PAW while varying knurling pitch +and plasma-water treatment time. Increasing plasma-water exposure time significantly (p < +0.05) increases the TDS and EC of PAW showing the continuous formation of inorganic ions +in PAW. In addition, introducing knurling to the ground electrode substantially (p < 0.05) + +increased the TDS and EC of PAW. The TDS and EC of PAW with ground electrode knurling +(0.5 mm knurling pitch) were 616.7% and 567.7% higher than without ground electrode +knurling. Initially, the TDS and EC of PAW were prepared using a 0.5 mm knurling electrode +plasma device that showed a higher value compared to other knurling pitches. As the plasma- +water treatment time increases, this difference in TDS and EC of PAW keeps on decreasing +and becomes statistically insignificant (p > 0.05) at 5 min. + +The variation in NO3ˉ ions and NO2ˉ ions concentration (reactive nitrogen species) in +PAW with varying ground electrode knurling pitch and plasma-water treatment time is shown +in figure 5 (e, f). Increasing plasma-water treatment continuously increased the NO3ˉ ions and +NO2ˉ ions concentration in PAW for all knurling pitch ranges. Moreover, the knurling of the +ground electrode significantly increased the NO3ˉ ions and NO2ˉ ions concentration in PAW +compared to a non-knurled ground electrode. The maximum concentration of NO3ˉ ions and +NO2ˉ ions in PAW (plasma-water treatment time of 5 min) with and without knurling ground +electrode were given as 24.98 mg l-1 and 5.21 mg l-1 NO3ˉ ions and 2.80 mg l-1 and 1.11 mg l-1 +NO2ˉ ions, respectively. Moreover, the larger knurling pitch size (2 mm) did not support the +higher formation of NO3ˉ ion concentration compared to the smaller knurling pitch size (0.5 +mm and 1 mm). Hence, considering the appropriate knurling pitch size is also important to get +higher production of NO3ˉ ions concentration in PAW. For NO2ˉ ions, 1 mm knurling pitch +gives the highest concentration of NO2ˉ ions in PAW compared to 0.5 mm and 2 mm knurling +pitch. + +The variation in reactive oxygen species (ROS (H2O2 and dissolved O3)) with plasma- +water treatment time and ground electrode knurling pitch size is shown in figure 5 (g, h). As +the plasma-water treatment time increases, the continuous increase in the H2O2 and dissolved +O3 concentration were observed in PAW when the ground electrode without knurling was used. +However, the H2O2 and dissolved O3 concentration in PAW when prepared using a ground + +electrode with knurling showed an initial increase and then decreased with time with an +exception of H2O2 at 5 min of plasma-water treatment time. This was due to the reactivity +environment favoring the reaction within ROS and ROS reaction with NO2ˉ ions to give more +stable NO3ˉ ions (equations (22,24,32,35-37)). Thus, the concentration of NO3ˉ ions in PAW +is substantially higher compared to other RONS in PAW. Hence, the ROS concentration +decreased at a higher plasma-water treatment time. + +Figure 5. The variation in physicochemical properties of PAW and RONS concentration with +varying ground electrode knurling pitch (0 mm, 0.5 mm, 1 mm, and 2 mm). Different lowercase +letters showed statistically significant difference (p<0.05, n ≥3) among the properties of PAW +mean ± standard deviation (µ ± σ) +3.4 The effect of ground electrode material on the physicochemical properties of PAW and +RONS concentration +-1 +1 +3 +5 +4 +6 +8 +b'' +a'' +a' +a''' +a'' +a' +a'' +a' +a''' +a'' +a' +a'' +a' +d''' +c''' +b''' +a''' +d'' +c'' +b'' +a'' +d' +c' +b' +a' +b +b +b +Plasma treatment time +pH +Plasma treatment time + 0 mm + 0.5 mm + 1 mm + 2 mm +Knurling Pitch +(a) +a +-1 +1 +3 +5 +200 +400 +600 +a''' +c''' +b''' +a''' +a''' +c'' +c' +b'' +b'' +b' +a' +a +a +a +a +(b) +ORP (mV) + 0 mm + 0.5 mm + 1 mm + 2 mm +-1 +1 +3 +5 +0 +30 +60 +90 +d''' +c''' +b''' +d'' +c'' +b'' +d' +c' +b' +d +c +b +a +(c) +TDS (ppm) +Plasma treatment time + 0 mm + 0.5 mm + 1 mm + 2 mm +-1 +1 +3 +5 +0 +50 +100 +150 +d''' +c''' +b''' +d'' +c'' +b'' +d' +c' +b' +c +b +a +a +Plasma treatment time +(d) +EC (µS cm +-1) + 0 mm + 0.5 mm + 1 mm + 2 mm +-1 +1 +3 +5 +0 +10 +20 +30 +a +d''' +c''' +b''' +d'' +c'' +b'' +d' +c' +b' +d +c +b +a +Plasma treatment time +Plasma treatment time +(e) +NO3 +- ions (mg L +-1) + 0 mm + 0.5 mm + 1 mm + 2 mm +-1 +1 +3 +5 +0 +1 +2 +3 +a +c''' +b''' +a''' +a''' +d'' +c'' +c' +b' +a' +c +b +b +a +(f) +NO2 +- ions (mg L +-1) +Plasma treatment time + 0 mm + 0.5 mm + 1 mm + 2 mm +-1 +1 +3 +5 +0 +3 +6 +9 +d''' +c''' +a''' +a''' +d'' +c'' +b'' +a'' +c' +b' +a'b' +a' +d +c +b +(g) +H2O2 (mg L +-1) + 0 mm + 0.5 mm + 1 mm + 2 mm +-1 +1 +3 +5 +0 +3 +6 +9 +b''' +b''' +a''' +c'' +b'' +b'' +a'' +c' +b' +b' +a' +c +c +b +(h) +Dissolved O3 (mg L +-1) +Plasma treatment time + 0 mm + 0.5 mm + 1 mm + 2 mm +0 +0 +0 +0 +0 +0 +0 +0 + +The role of ground electrode material of plasma device on the physicochemical properties and +RONS concentration of PAW is shown in figure 6. The observed values of physicochemical +properties of PAW when copper (Cu) was used as a ground electrode material was higher +compared to stainless steel (SS) and brass (Br). This can be implied from the plasma discharge +power in which Cu had the higher plasma discharge power compared to Br and SS (figure 3 +(b)). Higher discharge power signifies more generation of plasma radicals/species which come +in contact with water and improved the physicochemical properties of PAW and RONS +concentration in it. The lowest pH of PAW (figure 6 (a)) PAW (5 min of plasma-water +treatment time) when prepared using Cu, Br, and SS as ground electrode material were given +as 3.1, 3.56, and 3.5, respectively. This was also reflected as higher TDS, EC, NO2ˉ + NO3ˉ +ions concentration (figure 6 (c-f)) in PAW prepared using Cu as ground electrode compared to +Br and SS. The TDS and EC of PAW when prepared using Cu as ground electrode (5 min of +plasma-water treatment time) were 12.8% and 12.5% higher than SS, and 29.3% and 24.1% +higher than Br, respectively. Similarly, for a plasma-water treatment time of 5 min, the +observed NO3ˉ ions concentration in PAW when prepared using Cu as ground electrode was +14.3% and 27.1% higher than SS and Br. + +The oxidizing potential (ORP) of PAW prepared using Cu and SS as the ground +electrode was slightly higher than Br. The ORP of PAW prepared using Cu, SS, and Br as +ground electrodes was given as 548 mV, 545 mV, and 535 mV, respectively. The slightly +higher ORP and lower pH of PAW prepared using Cu and SS as ground electrodes compared +to Br increases PAW reactivity. The reactive environment of PAW favors reactions within ROS +and generated ROS with NO2ˉ ions. As a result, the concentration of NO2ˉ ions and ROS +decreased in the high reactive environment of PAW. The results of NO2ˉ ions concentration in +PAW shown in figure 6 (f) confirm the same. Initial for a plasma-water treatment time of 1 +min, the observed NO2ˉ ions concentration prepared using Br as ground electrode lower than + +Cu and SS. However, as the plasma-water treatment time increased (3 min and 5 min), the +observed NO2ˉ ions concentration in PAW prepared using Br as the ground electrode was +higher than Cu and SS. Since, increasing plasma-water treatment time, increased the reactivity +of PAW as a result NO2ˉ ions present in PAW react with dissolved O3 and H2O2. Therefore, +decreased the concentration of NO2ˉ ions in PAW prepared using Cu and SS as ground +electrodes material compared to Br. Similar results were observed in the ROS concentration in +PAW as shown in figure 6 (g, h) with an exception of H2O2 in PAW (5 min treatment time) +prepared using Cu as a ground electrode. In which a high reactive PAW, the H2O2 and dissolved +O3 concentration present in PAW either decrease or remain constant. As discussed, the PAW +prepared using Br as ground electrode material had comparatively low reactivity, hence, the +increase in the concentration of H2O2 and dissolved O3 were observed with time. Moreover, +the concentration of H2O2 and dissolved O3 in PAW decreases or remains constant as the +plasma-water treatment time increases (or, the reactivity of PAW increases) for SS and Cu as +ground electrode material. One exception was also observed in H2O2 concentration in PAW +(figure 6 (g)), in which a higher concentration of H2O2 was observed in high reactivity PAW +when prepared using Cu as ground electrode material. The possible reason for the same is an +excess concentration of H2O2 in PAW which is left even after reaction with dissolved O3 and +NO2ˉ ions present in PAW. + + + +Figure 6. The variation in physicochemical properties of PAW and RONS concentration with +varying ground electrode material (SS, Br, and Cu). Different lowercase letters showed +statistically significant difference (p<0.05, n ≥3) among the properties of PAW mean ± +standard deviation (µ ± σ) +3.5 The effect of power electrode material on the physicochemical properties of PAW and +RONS concentration +The variation in the power electrode material on the physicochemical properties of PAW and +RONS concentration is shown in figure 7. Similar to ground electrode material, the use of Cu +for power electrode material significantly enhances the physicochemical properties of PAW +and RONS concentration in it. The results also supported the plasma discharge power in which +Cu had substantially higher power compared to Br and SS (figure 3 (e)). The lowest pH of +PAW (5 min of plasma treatment time) when prepared using Cu, Br, and SS as power electrode +-1 +1 +3 +5 +4 +6 +8 +a'' +a'' +a'' +a'' +a'' +a'' +a' +a +a'' +a +a +c'' +b'' +b'' +a'' +d' +c' +b' +a' +d +c +b +pH +Plasma treatment time (min) + SS + Br + Cu +Ground Electrode Material +(a) +a +-1 +1 +3 +5 +200 +400 +600 +c'' +b'' +a''b'' +c' +b' +a'b' +a' +c +b +(b) +ORP (mV) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +40 +80 +d'' +c'' +b'' +d' +c' +b' +d +c +b +Plasma treatment time (min) +(c) +TDS (ppm) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +100 +200 +a' +d'' +c'' +b'' +d' +c' +b' +a' +d +c +b +a +(d) +EC (µS cm +-1) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +10 +20 +30 +d'' +c'' +b'' +d' +c' +b' +a' +d +c +b +a +(e) +NO3 +- ions (mg L +-1) + SS + Br + Cu +-1 +1 +3 +5 +0 +2 +4 +d'' +c'' +b'' +d' +c' +b' +d +c +b +a +(f) +NO2 +- ions (mg L +-1) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +3 +6 +9 +d'' +c'' +b'' +c' +b' +a' +a' +c +b +a +a +(g) +H2O2 (mg L +-1) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +3 +6 +d'' +c'' +b'' +d' +c' +b' +a' +c +b +b +a +(h) +Dissolved O3 (mg L +-1) +Plasma treatment time (min) + SS + Br + Cu +0 +0 +0 +0 +0 +0 +0 +0 + +material (figure 7 (a)) were given as 2.9, 3, and 3.3, respectively. Hence, the PAW produced +using Cu as power electrode material generates PAW with higher acidity compared to other +materials. + +At a higher plasma-water treatment time, the oxidizing potential (ORP) of PAW +prepared using Br and Cu as power electrode material was significantly (p < 0.05) higher +compared to SS (figure 7 (b)). In addition, the ORP of PAW prepared using Cu (ORP – 590 +mV) as a power electrode was slightly higher compared to Br (ORP – 580 mV). Hence, the +generation of oxidizing species in PAW during plasma-water exposure was substantially higher +when Cu or Br was used as power electrode material. + +The significance of Cu material as a power electrode over SS and Br is easily +understood by comparing the results of TDS, EC, NO3ˉ ions, and NO2ˉ ions concentration in +PAW. For a plasma-water treatment time of 5 min, a substantial growth (p < 0.05) in the values +of TDS, EC, NO3ˉ ions, and NO2ˉ ions in PAW when prepared using Cu as power electrode +material compared to SS and Br as shown in figure 7 (c-f). The increase in TDS and EC of +PAW using Cu as power electrode were 100.0% and 107.9% higher compared to Br, and 92.3% +and 96.4% higher compared to SS. Similarly, the increase in NO3ˉ and NO2ˉ ions concentration +in PAW prepared using Cu as power electrode material were 109.0% and 18.8% higher +compared to Br, and 92.2% and 40.7% higher compared to SS. Hence, the use of Cu as an outer +electrode material helps in the generation of more inorganic ions in PAW which was +communicated above as enhancement in TDS, EC, NO3ˉ ions, and NO2ˉ ions concentration in +PAW. + +The variation in H2O2 and dissolved O3 when using different power electrode materials +is shown in figure 7 (g, h). The H2O2 and dissolved O3 concentration present in PAW when +using different materials for power electrode showed a rise and fall (or rise and remain + +constant) in its concentration with increasing plasma-water treatment time. Since, initially (t = +0 min), there was no H2O2 and dissolved O3 present in PAW. Hence, as soon as plasma-water +interaction starts occurring, the formation of H2O2 and dissolved O3, as a result, their +concentration starts increasing. The fall in concentration of H2O2 and dissolved O3 showed (SS +power electrode) reduction of these species due to reaction occurring among themselves and +with NO2ˉ ions to form more stable NO3ˉ ions. Moreover, the constant concentration of H2O2 +and dissolved O3 (Br and Cu power electrode) with increasing time showed the established +equilibrium in which excess concentration above the equilibrium point converted more stable +products like NO3ˉ ions. + +Figure 7. The variation in physicochemical properties of PAW and RONS concentration with +varying power electrode material (SS, Br, and Cu). Different lowercase letters showed +-1 +1 +3 +5 +4 +6 +8 +a'' +a'' +a'' +a'' +a'' +a'' +a'' +a' +a' +b' +a' +a +c'' +b'' +b'' +a'' +d' +c' +b' +a' +c +b +b +pH +Plasma treatment time (min) + SS + Br + Cu +(a) +Power Electrode Material +a +-1 +1 +3 +5 +200 +400 +600 +c'' +b'' +c' +d +c +b +(b) +ORP (mV) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +100 +200 +300 +d'' +c'' +b'' +d' +c' +b' +a' +d +c +b +a +(c) +TDS (ppm) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +200 +400 +600 +a' +a' +d'' +c'' +b'' +d' +c' +b' +a' +d +c +b +a +(d) +EC (µS cm +-1) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +30 +60 +90 +a'b' +a' +a +d'' +c'' +b'' +d' +c' +b' +a' +d +c +b +a +(e) +NO3 +- ions (mg L +-1) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +2 +4 +c +b +a +a +c' +b' +a' +d'' +c'' +b'' +(f) +NO2 +- ions (mg L +-1) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +3 +6 +9 +c' +c'' +b'' +b'' +a'' +b' +c +c +b +(g) +H2O2 (mg L +-1) +Plasma treatment time (min) + SS + Br + Cu +-1 +1 +3 +5 +0 +3 +6 +b' +a' +a' +b'' +c'' +b'' +d +c +b +a +(h) +Dissolved O3 (mg L +-1) +Plasma treatment time (min) + SS + Br + Cu +0 +0 +0 +0 +0 +0 +0 +0 + +statistically significant difference (p<0.05, n ≥3) among the properties of PAW mean ± +standard deviation (µ ± σ) +3.6 The effect of power electrode type on the physicochemical properties of PAW and RONS +concentration +In this section, we discussed the role of different types of electrodes on the physicochemical +properties of PAW and RONS concentration. For this study, three different types of electrodes +were used such as mesh, wire, and sheet. The observed result is shown in figure 8. The results +of physicochemical properties of PAW (figure 8 (a-d)) showed the use of wire as a power +electrode type significantly enhanced the physicochemical properties of PAW compared to +other electrode types. This was due to the use of wire as a power electrode having higher plasma +discharge power compared to mesh and sheet (figure 3 (c)). The pH of PAW, when prepared +using wire as a power electrode (5 min of plasma treatment time) showed 11.4% lower +compared to mesh, and 7.7% lower compared to the sheet. Similarly, the oxidizing potential +(ORP) of PAW when prepared using wire as a power electrode was 7.2% higher than mesh and +5.6% higher compared to the sheet. Moreover, at the same operating parameters, the TDS and +EC of PAW prepared using wire as power electrode showed 19.4% and 17.6% higher compared +to mesh, and 32.3% and 30.0% higher compared to the sheet. + +Similar to physicochemical properties, the NO3ˉ ions concentration present in PAW +prepare (5 min of plasma treatment time) using wire as power electrode type showed +significantly higher value compared to mesh and sheet (figure 8 (e)). The observed NO3ˉ ions +concentration in PAW prepared using wire, mesh, and sheet as power electrodes were given as +27.1 mg l-1, 24.5 mg l-1, and 19.8 mg l-1, respectively. + +In contrast to physicochemical properties and NO3ˉ ions concentration, the NO2ˉ ions, +H2O2, and dissolved O3 present in PAW showed higher value for the sheet as power compared + +to wire and mesh (figure 8 (f-h)). This signifies the PAW produced using the sheet as a power +electrode did not favors the reaction within dissolved ROS (H2O2 and dissolved O3) and with +NO2ˉ ions. Hence, due to limiting reactions (equations (22, 35-37)) between these species, the +observed concentration of reactants (H2O2, dissolved O3, and NO2ˉ ions) were high and the +product concentration (NO3ˉ ions) was low. + +Figure 8. The variation in physicochemical properties of PAW and RONS concentration with +varying power electrode types (mesh, wire, and sheet). Different lowercase letters showed +statistically significant difference (p<0.05, n ≥3) among the properties of PAW mean ± +standard deviation (µ ± σ) +3.7 The effect of dielectric material on the physicochemical properties of PAW and RONS +concentration +-1 +1 +3 +5 +4 +6 +8 +a' +a' +a +a' +a' +a +d'' +c'' +b'' +a'' +d' +c' +b' +a' +d +c +b +pH +Plasma treatment time (min) + Mesh + Wire + Sheet +Power Electrode Type +(a) +a +-1 +1 +3 +5 +200 +400 +600 +c'' +b'' +a'' +a'' +c' +b' +a' +c +b +a +(b) +ORP (mV) +Plasma treatment time (min) + Mesh + Wire + Sheet +-1 +1 +3 +5 +0 +30 +60 +90 +d'' +c'' +b'' +a'' +d' +c' +b' +d +c +b +(c) +TDS (ppm) +Plasma treatment time (min) + Mesh + Wire + Sheet +-1 +1 +3 +5 +0 +50 +100 +150 +a'' +a'' +d'' +c'' +b'' +a'' +d' +c' +b' +d +c +b +a +(d) +EC (µS cm +-1) +Plasma treatment time (min) + Mesh + Wire + Sheet +-1 +1 +3 +5 +0 +10 +20 +30 +a'' +d'' +c'' +b'' +a'' +d' +c' +b' +d +c +b +a +(e) +NO3 +- (mg L +-1) +Plasma treatment time (min) + Mesh + Wire + Sheet +-1 +1 +3 +5 +0 +2 +4 +d'' +c'' +b'' +c' +b' +a' +a' +c +b +a +a +(f) +NO2 +- (mg L +-1) +Plasma treatment time (min) + Mesh + Wire + Sheet +-1 +1 +3 +5 +0 +3 +6 +9 +c'' +b'' +b'' +c' +b' +a' +a' +c +b +ab +a +(g) +H2O2 (mg L +-1) +Plasma treatment time (min) + Mesh + Wire + Sheet +-1 +1 +3 +5 +0 +3 +6 +c'' +b'' +b'' +c' +b' +a' +a' +c +b +b +a +(h) +Dissolved O3 (mg L +-1) +Plasma treatment time (min) + Mesh + Wire + Sheet + +Figure 9 showed the variation in physicochemical properties of PAW and RONS concentration +when using glass and quartz dielectric materials in plasma device. The results showed +statistically significant (p < 0.05) enhancement in the physicochemical properties of PAW and +RONS concentration when quartz was used as a dielectric material compared to glass. As +discussed in the electrical characterization, higher discharge current peaks and plasma +discharge power in quartz as dielectric compared to glass (figures 2 (c, d) and 3 (d))[25]. This +signifies higher plasma species/radicals density in quartz plasma device compared to glass. As +a result, a high concentration of reactive species dissolved in PAW was prepared using a quartz +plasma device as shown by enhanced physicochemical properties and RONS concentration +compared to a glass plasma device. The ORP, TDS, and EC of PAW when prepared using +quartz as dielectric showed a higher value and lower value of pH of PAW compared to glass +(figure 9 (a-d)). This showed the improvement in the physicochemical properties of PAW when +prepared using quartz. The pH of PAW (plasma treatment time of 5 min) when prepared using +glass and quartz as dielectric were given as 3.5 and 3. At similar conditions, the ORP, TDS, +and EC of PAW when prepared using glass and quartz were given as 545 mV and 580 mV +(ORP), 86 ppm and 150 ppm (TDS), and 160 µS cm-1 and 290 µS cm-1 (EC), respectively. + +The variation in dissolved RONS in PAW when prepared using glass and quartz as the +dielectric material of the plasma device is shown in figure 9 (e-h). The observed NO3ˉ and +NO2ˉ ions concentration in PAW prepared using quartz as dielectric significantly (p < 0.05) +higher compared to glass (figure 9 (e-f)). At a plasma-water treatment time of 5 min, the NO3ˉ +and NO2ˉ ions concentration in PAW were 71.8% and 45.4% higher compared to glass. Similar +to NO3ˉ and NO2ˉ ions, the observed H2O2 and dissolved O3 concentrations in PAW were +higher for quartz compared to glass. For glass, with increasing plasma-water treatment time, +the H2O2 concentration and dissolved O3 in PAW showed a rise and fall. This was as the +reactivity of PAW increases these species react with each other and other NO2- ions present in + +PAW. As a result, their concentration decreased at higher plasma-water treatment time. For +quartz, the H2O2 and dissolved O3 present in PAW follow a trend of rise-fall-rise with time +(figure 9 (g, h)). This was due to initially no H2O2 and dissolved O3 present in PAW. Hence, +these species’ concentration increases with increasing time. As the concentration of these +reached sufficient reactivity, they react with each other, and NO2ˉ ions concentration was +present in PAW. As a result, a decrease in their concentration was observed. Further increase +in the plasma-water treatment time showed a generation of a higher concentration of ROS in +PAW. Hence, even after reacting with each other and NO2ˉ ions present in PAW a slight +increase in these species (H2O2 and dissolved O3) concentration was observed in PAW. +-1 +1 +3 +5 +4 +6 +8 +c' +b' +a' +a' +a' +a' +a' +a' +a' +c' +b' +a' +a +d' +c' +b' +a' +d +c +b +pH +Plasma treatment time (min) + Glass + Quartz +Dielectric Material +(a) +a +-1 +1 +3 +5 +200 +400 +600 +d' +d +c +b +(b) +ORP (mV) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +50 +100 +150 +d +c +b +a +d' +c' +b' +(c) +TDS (ppm) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +100 +200 +300 +d +c +b +a +d' +c' +b' +(d) +EC (µS cm +-1) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +15 +30 +45 +d +c +b +a +d' +c' +b' +(e) +NO3 +- ions (mg L +-1) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +1 +2 +3 +a +d +c +b +a +d' +c' +b' +(f) +NO2 +- ions (mg L +-1) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +3 +6 +c +b +a +a +c' +b' +(h) +(g) +H2O2 (mg L +-1) +Plasma treatment time (min) + Glass + Quartz +0 +0 +0 +-1 +1 +3 +5 +0 +2 +4 +6 +c +c' +b +b +Dissolved O3 (mg L +-1) +Plasma treatment time (min) + Glass + Quartz +0 +0 +0 +0 +0 + + + +Figure 9. The variation in physicochemical properties of PAW and RONS concentration with +varying dielectric material (glass and quartz). Different lowercase letters showed statistically +significant difference (p<0.05, n ≥3) among the properties of PAW mean ± standard deviation +(µ ± σ) +3.8 Residual metal analysis in PAW +As PAW has various applications in the field of agriculture, food preservation, and cancer cells +inactivation, etc. The presence of heavy metal in PAW coming from electron erosion used in +plasma device may interfere in the applications of PAW. Hence, the residual metal analysis of +PAW become important. Table 1 shows the residual metal in PAW when prepared using +different ground electrode material. The different material of construction of ground electrode +are stainless steel (SS, allow of iron (Fe) and carbon), brass (Br, copper and zinc (Zn)), and +copper (Cu). Hence, the Table 1 showed the concentration of Cu, Zn, and Fe in PAW and +-1 +1 +3 +5 +4 +6 +8 +c' +b' +a' +a' +a' +a' +a' +a' +a' +c' +b' +a' +a +d' +c' +b' +a' +d +c +b +pH +Plasma treatment time (min) + Glass + Quartz +Dielectric Material +(a) +a +-1 +1 +3 +5 +200 +400 +600 +d' +d +c +b +(b) +ORP (mV) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +50 +100 +150 +d +c +b +a +d' +c' +b' +(c) +TDS (ppm) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +100 +200 +300 +d +c +b +a +d' +c' +b' +(d) +EC (µS cm +-1) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +15 +30 +45 +d +c +b +a +d' +c' +b' +(e) +NO3 +- ions (mg L +-1) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +1 +2 +3 +a +d +c +b +a +d' +c' +b' +(f) +NO2 +- ions (mg L +-1) +Plasma treatment time (min) + Glass + Quartz +-1 +1 +3 +5 +0 +3 +6 +c +b +a +a +c' +b' +(h) +(g) +H2O2 (mg L +-1) +Plasma treatment time (min) + Glass + Quartz +0 +0 +0 +-1 +1 +3 +5 +0 +2 +4 +6 +c +c' +b +b +Dissolved O3 (mg L +-1) +Plasma treatment time (min) + Glass + Quartz +0 +0 +0 +0 +0 + +control. The observed concentration of Cu and Zn present in PAW and control were +insignificant (p > 0.05) for different ground electrode material of plasma device. Moreover, the +concentration of Cu and Zn were less than 6 µg l-1 and 2 µg l-1. Also, the Fe concentration in +PAW and control was beyond the detection limit (B.D.L). In conclusion, the observed heavy +metal concentration in PAW similar to ultrapure milli-Q water (control). Hence, no erosion of +inner (ground) electrode material occurs during plasma-water treatment which settle in PAW +in form of residue metal. +Table 1. Residual metal analysis in PAW when prepared using different ground electrode +material. Different lowercase letters showed statistically significant difference (p<0.05, n ≥3) +among the group mean ± standard deviation (µ ± σ) (*B.D.L – Below Detection Limit) +Material +of +ground +electrode +Plasma-water +treatment time +Residue metal present in water + + +Copper (µg l-1) +Zinc (µg l- +1) +Iron (µg l-1) +Control +0 min +3.9 ± 0.2a +1.6 ± 0.14 a B.D.L* +Stainless Steel +1 min +4.6 ± 1.1 a +1.3 ± 0.3 a +B.D.L* +3 min +3.9 ± 0.8 a +1.7 ± 0.5 a +B.D.L* +5 min +4.7 ± 0.7 a +1.8 ± 0.4 a +B.D.L* +Brass +1 min +4.5 ± 0.5 a +1.7 ± 0.2 a +B.D.L* +3 min +4.9 ± 1.2 a +1.2 ± 0.3 a +B.D.L* +5 min +5.0 ± 1.0 a +1.7 ± 0.2 a +B.D.L* +Copper +1 min +5.1 ± 0.9 a +1.6 ± 0.1 a +B.D.L* +3 min +4.8 ± 0.5 a +1.6 ± 0.3 a +B.D.L* +5 min +5.8 ± 0.7 a +1.7 ± 0.5 a +B.D.L* + + +4. Conclusion +The present work shows the effect of various components of dielectric barrier discharge plasma +device (DBD-PD) on plasma and physicochemical properties of PAW and RONS +concentration present in activated water. The discharge current characteristics showed +substantial improvement in filamentary discharge when introduced knurling to ground +electrode, using quartz as dielectric layer compared to glass, and employed wired power +electrode compared to mesh and sheet. These results are further supported by the results of +plasma discharge power. +Similarly, the plasma activated water produced using a diamond knurled electrode, +quartz as dielectric, and wire as power electrode showed substantial improvement in the +physicochemical properties of PAW and RONS concentration. We have also observed that the +use of copper material for manufacturing ground and power electrodes significantly improves +the PAW properties compared to brass and stainless steel materials. +Acknowledgments +This work was supported by the Department of Atomic Energy (Government of India) graduate +fellowship scheme (DGFS). The authors sincerely thank Mr Chirayu Patil, O. R. Kaila, and +Mr. Nimish for providing constant support and useful suggestions during this work. +Data availability statement +The data that support the findings of this study are available upon reasonable request from the +authors. +Conflict of interests +The authors declare that there are no conflicts of interests. + +Authors’ contributions +Both authors contributed to the study conception and design. Material preparation, data +collection, and analysis were performed by Vikas Rathore. The first draft of the manuscript +was written by Vikas Rathore, and both authors commented on previous versions of the +manuscript. 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L, Lamichhane P, Gaur N, Robson A J, Trivedi D, Thet +N T, Jenkins A T A, Choi E H J P S S and Technology 2021 Enhancement of hydrogen +peroxide production from an atmospheric pressure argon plasma jet and implications to +the antibacterial activity of plasma activated water 30 035009 +[40] +Moravej M, Yang X, Barankin M, Penelon J, Babayan S, Hicks R J P S S and +Technology 2006 Properties of an atmospheric pressure radio-frequency argon and +nitrogen plasma 15 204 +[41] +Goldstein S, Lind J and Merényi G J C r 2005 Chemistry of peroxynitrites as compared +to peroxynitrates 105 2457-70 + + diff --git a/n9E4T4oBgHgl3EQfuw0c/content/tmp_files/load_file.txt b/n9E4T4oBgHgl3EQfuw0c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a96520db5363098275faeafd49b518e48fe5389b --- /dev/null +++ b/n9E4T4oBgHgl3EQfuw0c/content/tmp_files/load_file.txt @@ -0,0 +1,1908 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf,len=1907 +page_content='Title: Study the effect of electrode material, its surface, and dielectric material on plasma and properties of plasma-activated water Authors Vikas Rathore1,2* and Sudhir Kumar Nema1,2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Atmospheric Plasma Division, Institute for Plasma Research (IPR), Gandhinagar, Gujarat 382428, India 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India Email: vikas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='rathore@ipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='in Abstract In the present work, the significance of various components (ground electrode material and its surface (knurling pitch size), power electrode material and type, and dielectric material) of dielectric barrier discharge plasma device (DBD-PD) on plasma and plasma-activated water (PAW) properties are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The characterization of plasma is performed by studying voltage-current waveform and plasma discharge power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, the characterization of PAW is performed by studying the physicochemical properties (PP) and reactive oxygen- nitrogen species (RONS) concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The results of plasma characterization and PAW properties reveals that introducing knurling to ground electrodes showed significant improve the physicochemical properties of PAW and RONS concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the use of quartz over glass as dielectric layer provides a substantial enhancement in PAW properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Furthermore, the use of wire as a power electrode compared to mesh and sheet also help in improving the PAW properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Further, we observed that the ground and power electrodes made using copper enriches the RONS concentration and physicochemical properties of PAW compared to brass and stainless steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Keywords: Plasma activated water, plasma device, reactive oxygen-nitrogen species, electrode and dielecteric material, electrode knurling 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Introduction The emerging applications of plasma activated water (PAW) in the field of plasma medicine, plasma agriculture, food preservation, and senitization industry, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' provide it a lot of recognition all over the world[1-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' These applications of PAW are possible due to the dissolution of various reactive oxygen-nitrogen species (RONS) in it[7-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The oxidizing species (reactive oxygen species, ROS) such as hydroxyl (̇OH) radical, hydrogen peroxide (H2O2), superoxide ions (O2ˉ), dissolved O3, and peroxynitrite ions (ONOOˉ), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' provide PAW excellent antimicrobial efficacy[5, 6, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The antimicrobial efficacy of PAW towards bacteria, fungi, viruses, and pest has already been reported in published work of various researchers[2, 5, 6, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, selective killing of cancer cells and non-cytotoxic of PAW towards skin cells have also been explored in past literature [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This antimicrobial activity of PAW is widely used in the surface disinfection of a wide variety of food products including meat products, sea food, fruits, and vegetables, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, no change in phenotypic characteristics and nutritional value was observed after PAW treatment with food products[1, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Along with reactive oxygen species (ROS), a high concentration of reactive nitrogen species (RNS) is also present in PAW in the form of nitrate (NO3ˉ) and nitrite (NO2ˉ) ions, etc[7-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, being a rich source of nitrogen species, PAW can also be used as a fertilizer to enhance crop growth [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Past literature also demonstrated the use of PAW to enhance seeds germination and plant growth in a variety of crops[1, 4, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, considering the above applications of PAW, different types of plasma devices are used to produce PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' These devices are mainly composed of different geometries of electrode, power supplies (high-frequency AC, radiofrequency, and microwave, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' ), type of plasma discharge (glow discharge, filamentary discharge, spark discharge, and gliding arc discharge, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' ), etc[7, 10, 11, 14, 17-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Also, various PAW process parameters were also studied in detail to enhance the physicochemical properties of PAW and RONS concentration in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' These process parameters include plasma-water treatment/exposure time, plasma discharge power, different type of plasma forming gases (air, N2, Ar, He, N2 + O2, Ar + O2, He + O2), gas flow rate, water stirring, and controlling water temperature, etc[4-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, no emphasis is given to the components used to prepare plasma devices for PAW generation as per the best of the author’s knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Since, different materials of construction of electrode, dielectric, and type of electrode, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' may significantly influence the properties of plasma and PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This literature gap created the basis of the present investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In the present work, a self-made dielectric barrier discharge plasma device (DBD-PD) is used to produce air plasma and the generated air plasma is exposed to water to produce plasma activated water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The DBD-PD has three major components which play a significant role in plasma production named ground electrode, power electrode, and dielectric cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in these components is investigated in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The ground electrode made using hollow metal pipe and diamond knurling is introduced on the pipe surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Since, diamond knurling created metal spikes hence increased the localized electric field which may help in the generation of excess plasma radicals and species[22-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, may improve the PAW properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, the variation in knurling pitch size is also studied on plasma and PAW properties[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' For comparison of dielectric material glass and quartz are chosen with the same dimensions and geometry[25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This is due to the frequent use of glass and quartz as dielectric materials in DBD discharge[22, 24-27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The power electrode is made by wrapping the wire, mesh, or sheet over the dielectric cone surface[24, 26, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, variation in a different types of power electrode is also studied on plasma and PAW properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' At last, the different materials of construction used for making ground and power electrodes and their effect on PAW physicochemical properties and RONS concentration in it are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The chosen materials are copper (Cu), brass (Br), and stainless steel (SS), these materials are chosen due to their frequent use in the preparation of electrodes for various plasma devices [24, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The produced air plasma using DBD-PD is characterized by studying the voltage- current characteristics and air plasma species are identified using optical emission spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The effect of variation in DBD-PD components is studied by studying the voltage-discharge current characteristics, plasma discharge power, physicochemical properties of PAW, and RONS concentration in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Materials and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 Experimental setup The schematic of the experimental setup is shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' It shows the electrical and optical characterization of plasma device, and the production of plasma activated water (PAW) using plasma device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The plasma device composes a coaxial cylindrical dielectric cone (or tube) (see figure 1) with an outer diameter 24 mm and thickness 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The cone has a double-side B24 male socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The top side of the cone is fitted in a B24 socket receiver adapter with an air leak tube which uses an air inlet in a plasma device (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The ground electrode of the plasma device was made using a hollow metal pipe with an outer diameter 16 mm with or without diamond knurling on its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This ground electrode was a tight fit in the teflon cap as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The power electrode is either made of flexible mesh, sheet, and wire which is wrapped around a dielectric cone as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The airflow rate feed to the plasma device was controlled using an air rotameter and set at 15 l min-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This plasma device was powered using 0-30 kV, 0-30 kHz higher voltage high-frequency power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The present work uses a constant frequency of 20 kHz for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The generated air plasma in the plasma device was characterized using electrical and optical emission measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' A 1000x high voltage probe (Tektronix P6015A) and a 4- channel 100 MHz, 2 GS s-1 digital storage oscilloscope (Tektronix TDS2014C) was used to measure the applied voltage across plasma device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' To measure the total current (conducion and discharge) and transported charge during plasma production, a voltage drop was measured across 31 ohms non-inductive resistor and 100 nF capacitor using a 10x voltage probe (Tektronix TPP0201) in series with the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The air plasma emission spectrum was measured using optical fiber and a spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The emission spectrum was measured in the range of 190 nm to 925 nm using two different spectrometers (Model EPP2000-UV from StellarNet Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' (range 190 nm to 610 nm) and UVH-1 from ASEQ instruments (range 290 nm to 925 nm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' A 40 ml of ultrapure milli-Q (Demineralized water, DM water) was used for PAW production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' To activate the water, plasma-water inteaction time varied from 1 to 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The set distance between the ground electrode tip and the water surface was 30 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' To enhance the dissolution of reactive species produced due to plasma-water interaction, the continuous stirring and water temperature were maintained at 0 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The stirring of the water was controlled using a magnetic stirrer and magnetic teflon bar kept in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The water stirring speed was kept constant at 300 rpm throughout the experimental duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The water temperature was maintained at 0 °C by keeping the ice-water mixture in a storage container as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 The role of plasma device components on plasma and PAW properties The plasma device used to produce air plasma mainly consists of three important components which play a significant role in plasma production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The ground electrode, power electrode, and dielectric material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the present investigation emphasizes the role of these plasma device components on plasma and PAW properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The present work investigates the role of ground electrode knurling pitch size, type of power electrode, dielectric material, and material of construction of ground and power electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 The ground electrode knurling The knurling of the ground electrode may increase the localized electric field for different knurling pitch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' That influences the generation of radicals and species in the plasma phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the role of different knurling pitch sizes was studied on plasma and PAW properties in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The different knurling pitch sizes chosen were 0 mm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm, 1 mm, and 2 mm, respectively (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' While studying the role of knurling pitch size other plasma device components and PAW process parameters were kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 The dielectric material The dielectric material is one of the most important parameters during dielectric barrier discharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' To study the role of dielectric material on plasma and PAW properties two different types of dielectric material were used named glass and quartz in form of a B24 double side cone (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The other PAW process parameters and plasma device components remain unchanged while comparing the mentioned dielectric material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 The type of power electrode To study the impact of power electrode type on plasma and PAW properties, three different types of power electrodes were used (mesh, sheet, and wire) keeping other variables constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This mesh, sheet, and wire were wrapped around the dielectric cone to make a power electrode as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4 Material of ground and power electrode Three different types of materials named copper (Cu), brass (Br), and stainless steel (SS) are used to study the role of ground and power electrode material on the physicochemical properties and RONS concentration of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' A detail of the same is shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which, a picture of different ground electrode materials and power electrode materials is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 Measurement of physicochemical properties and RONS concentration of PAW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 Physicochemical properties A pH meter (Hanna Instrument, model HI98120) was used to measured the pH of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' An ORP meter (Hanna Instrument, model ORP-200) was used to determine the oxidizing tendency of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The dissolved ions in PAW were measured using TDS (total dissolved solid) meter (HM digital, model AP1) and EC (electrical conductivity) meter (Contech, model CC-01), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The least count of instruments used in the present work was given as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='01 (pH), 1 mV (ORP), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 µS cm-1 (EC), and 1 ppm (TDS), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 Reactive oxygen-nitrogen species (RONS) The preliminary detection of RONS present in PAW was performed using a strip test for NO2ˉ ions (Macherey-Nagel, QUANTOFIX Nitrite) and H2O2 (Macherey-Nagel, QUANTOFIX Peroxide 25), and colorimetric test kit for NO3ˉ ions (Macherey-Nagel, VISOCOLOR Nitrate) and dissolved O3 (Hanna Instrument, HI-38054 Ozone test kit) test kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The quantitative estimation of RONS concentrations in PAW was determined spectrophotometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' A standard curve of nitrate (NO3ˉ) ions, nitrite (NO2ˉ) ions, and hydrogen peroxide (H2O2) was prepared to determine the unknown concentration of these species in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The molar attenuation coefficient of NO3ˉ ions, NO2ˉ ions, and H2O2 were given as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0602 (mg l-1)-1 (range of NO3ˉ ions is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='61 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='10 mg l-1), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0009 (µg l-1)-1 (range of NO2ˉ ions is 67 to 536 µg l-1), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4857 (mmol l-1)-1 (range of H2O2 is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8 to 98 µmol l-1), respectively[5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The concentration of unknown NO3ˉ ions present in PAW was determined spectrophotometrically at 220 nm using mentioned molar attenuation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The NO2ˉ ions react with a reaction mixture of sulfanilamide and N-(1-naphthyl) ethylenediamine dihydrochloride to give reddish-purple color (azo dye) which showed maximum absorbance at 540 nm in an acidic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the unknown concentration of NO2ˉ ions in PAW was measured at 540 nm using mentioned molar attenuation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, H2O2 reacts with titanium ions (titanium (IV) oxysulfate) in the acidic region to form a yellow color complex (pertitanic acid, H2TiO4) which shows maximum absorbance at 407 nm[5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This property of H2O2 is used to determine the unknown H2O2 concentration in PAW using the mentioned molar attenuation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, NO2ˉ ions present in PAW interfere in the determination of H2O2 concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The NO2ˉ ions reacted with H2O2 and suppress the H2O2 concentration in PAW beyond the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, azide ions (N3ˉ) in the form of sodium azide (NaN3) were added to PAW which reacts with nitrate ions and degrades it so the interference in H2O2 determination can be prohibited[5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Appropriate dilution was performed to determine RONS concentration if the RONS concentration in PAW exceeded the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' An indigo colorimetry method was used to determine the dissolved O3 concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which, the rapid decolorization of indigo reagent occurs by dissolved O3 in the acidic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The indigo reagent was prepared using potassium indigo trisulfonate, sodium phosphate, phosphoric acid, and water, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The volumetric method (equation (1))[5, 7] used to determine the dissolved O3 in PAW was given as: 𝑚𝑔 𝑙 𝑜𝑓 𝑂3 = 100 × ∆𝐴 𝑓 ×𝑏 × 𝑣 (1) Where, ‘ΔA’ is the absorbance difference in PAW and blank at 600 nm, ‘b’ is the optical path length of the cell (1 cm), ‘v’ is the volume of PAW, and ‘f’ is the sensitivity factor (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='42), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4 Residue metal analysis in PAW The plasma generation may erode the inner (ground) electrode material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The energetic particles collide with the ground electrode and result in erosion/sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The erosion of material may dissolved in water in the form of metal residue in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The residual metal analysis in PAW and control were performed using inductive couple plasma-mass spectroscopy (ICP-MS) (2000B ICP-MS, Perkin Elmer) in collision mode (Helium KED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=" The chosen ICP-MS method of testing were Environmental Protection Agency's (EPAs) 1638 and EPA 6020 B." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As the ground electrodes were made of SS (iron and carbon alloy), Br (copper and zinc alloy), and Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The residual metal analyzed in PAW using ICP-MS were iron, copper, and zinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The PAW and control sample were reconstituted in hydrochloric acid, and diluted (25X dilution) with DM water and preserve with nitric acid before ICP-MS analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 Data Analysis All experiments were repeated atleast three times (n ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The results were expressed in plots and tables as mean ± standard deviation (µ ± σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=" The statistically significant difference among the group µ ± σ were estimated using one-way ANOVA followed by post-hoc test (Fisher's least significant difference (LSD))." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Schematic of electrical and optical emission characterization of plasma device and production of plasma activated water Glass Quar Oscilloscope MagneticStirrer15 0 5 Current (mA) Current (mA) 0 ¥5 5 10 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='15 15 10 ５０ Current (mA) 5 Current (mA) 5 5 15 15 10 10 Current (mA) Current (mA) 0 10 10Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Voltage-current characterization of plasma device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' (a, b) With and without ground electrode knurling, (c, d) variation in dielectric material (glass, quartz), (e, f, g) variation in power electrode type (mesh, wire, sheet) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Results and discussions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 Electrical and optical emission characterization of air plasma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 Voltage-current characteristics The variation in air plasma when produced using different electrode materials, electrode types, dielectric materials, and with and without ground electrode knurling is studied using the current waveform as shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The current profile shown in figure 2 is a combination of two currents, a continuous alternating current (AC) (sine wave) and a discharge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The discharge current normally appears in each rising and falling half-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The discharge current is the indicator of gas breakdown flowing through the coaxial pathway between the ground electrode and dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The breakdown of gases created various ions and electrons which are indicated by various high and low multiple current peaks (filaments) in rising and falling half- cycles over continuous AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The created ions and electrons due to gas discharge were responsible for the discharge current shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the periodic formation of discharge current over the continuous AC showed the formation of plasma between the ground electrode and dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The time period of discharge current filaments is in order ~ 100 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence the combination of these current filaments represents the characteristics of dielectric barrier discharge (DBD) filamentary micro-discharge[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Ground electrode knurling Figure 2 (a, b) showed the voltage-current characteristics of air plasma with and without ground electrode knurling while keeping the other design parameters and process parameters constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The discharge current formed in the rising curve of positive half-cycle for with and without knurling ground electrode of plasma device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The peak value of discharge current with and without ground electrode knurling was 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 mA and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 mA (positive half-cycle), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The increase in discharge current peak in knurling plasma devices showed more generation of electron-ion pair during gas discharge which showed by a shoot up in the current peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the knurling of the ground electrode supports more formation of discharge gases products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was due to improvisation in localized electric fields with sharp knurling edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The sharp edges of diamond knurling distorting the uniform electric field [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Due to which localized electric field near sharp edges enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As a result, generate higher pulse filaments near the sharp edges of diamond knurling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, more reactive species generation occurs in the plasma phase which could be utilized for various purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similar results were shown in the work reported by Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [24] and Takaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, they use screw edges and a large number of pyramids in multipoint geometry of the inner electrode to enhance localized electric field instead of diamond knurling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The diamond knurling has a slight edge over the screw-type electrode since it creates a significantly higher sharp edge density compared to the screw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Dielectric material The most commonly used dielectrics during DBD plasma production are glass and quartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The comparison of discharge characteristics of glass and quartz dielectric, when used in plasma device is shown in figure 2 (c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In glass dielectric, the discharge current peaks were observed in the rising half-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, in the case of quartz dielectric, the discharge current peaks were observed in both positive and negative half-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the observed discharge current peaks in the positive rising half-cycle were significantly higher compared to the negative falling half-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, over a time period, two-discharge currents regions (positive rising half-cycle and negative falling half-cycle) appeared in quartz compared to the one- discharge current region (positive rising half-cycle) in the glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This signifies a higher concentration of discharge gas products formed in plasma using quartz as a dielectric compared to glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This will be due to better distribution of charge over quartz surface compared to glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Since, the quartz has a crystalline structure and glass is amorphous[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the use of quartz (dielectric strength 470 to 670 MV m-1) over the glass is preferred due to its substantially higher dielectric strength compared to glass (dielectric strength 20 to 40 MV m -1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The discharge current observation of the present investigation was also supported by work reported by Ozkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which, higher discharge current peaks were observed in quartz dielectric compared to glass dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Also, filamentary discharge peaks were observed in both positive and negative half-cycles for quartz dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, for glass dielectric, observed filamentary discharge peaks in one-half cycle were substantially higher than other half cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Power electrode type The effect of different types of power electrodes on air plasma discharge characteristics is shown in figure 2 (e-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The power electrode is made from mesh or winding of wire over a dielectric cone or thin metal sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The use of different types of power electrodes had a significant impact on discharge current characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The number of high discharge-current filaments in power electrode made using wire was significantly greater compared to power electrodes made using mesh or sheet as shown in figure 2 (e-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The discharge current filaments in the power electrode made using wire or sheet mainly occur in the negative falling half-cycle (figure 2 (f, g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, inverse behavior was observed in the power electrode made using mesh in which discharge current filaments appeared in the positive rising half-cycle (figure 2 (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As the large number of high peaks current filaments observed in the discharge-current profile of air plasma produced using wire as power electrode showed more formation of electron-ion pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As a result, the density of reactive plasma species produced using wire as a power electrode will be higher compared to power electrodes made using mesh or sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The use of wire, mesh, and sheet (foil) as outer electrodes for DBD discharge wrapped around dielectric was previously explored by Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [24], Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [28], and Nur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which, Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [24] used aluminium foil as outer electrode wrapped around dielectric have higher filamentary discharge current peaks compared to mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The present work also showed denser current peaks in the sheet as outer electrode compared to mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was due to uniform covering of dielectric using sheet compared to mesh results in more charge accumulation on the dielectric surface and more discharge area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the number of discharge filaments increases in the sheet compared to mesh shown as dense filamentary discharge in voltage- current characteristics of the sheet as power electrode (figure 2 (g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In conclusion, the use of a knurled ground electrode, quartz as dielectric material, and wire as a power electrode results in the generation of a high concentration of charged plasma species/radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, this configuration could be used in future technology where high plasma species density is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Variation in plasma discharge power while varying, (a) ground electrode knurling pitch, (b) ground electrode material, (c) power electrode type, (d) dielectric material, (e) power 1 2 3 4 0 2 4 6 8 10 12 14 b b a Plasma discharge power (W) Ground electrode knurling pitch (mm) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 1 2 (a) a SS Br Cu 0 2 4 6 8 10 c b a (e) (d) (c) (b) Plasma discharge power (W) Ground electrode material Mesh Wire Sheet 0 2 4 6 8 b b a Plasma discharge power (W) Power electrode type Glass Quartz 0 2 4 6 8 10 b Plasma discharge power (W) Dielectric material SS Br Cu 0 2 4 6 8 10 a c b a Plasma discharge power (W) Power electrode material electrode material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Different lowercase letters showed statistically significant difference (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05, n ≥3) among the group mean ± standard deviation (µ ± σ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 Plasma discharge power As discussed above the discharge of gas is shown by the increase in the discharge-current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This discharge current signifies the movement of newly generated ionized species in the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the energy is consumed during the generation of these reactive species/radicals in plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The higher dissipation of energy (or power) results in the formation of a high concentration of reactive species/radicals in plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 3 (a-e) showed the variation in the plasma discharge power while varying knurling pitch, ground and power electrode material, dielectric material, and type of power electrode, respectively at constant applied voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 3 (a) showed the variation in the plasma discharge power with increasing knurling pitch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' An increase in knurling pitch size from 0 to 1 mm increased the plasma discharge power by 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, a further increase in the knurling pitch size from 1 mm to 2 mm decreased the plasma discharge power by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, this decrease in power was not statistically significant (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the knurling pitch is an important parameter that influences the generation of more reactive species/radicals in plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [24] and Takaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [23] also showed improvement in energy efficiency and input energy at the same applied voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' They used a screw-type (similar to knurling) inner electrode compared to the rode- type inner electrode and multipoint (pyramid shape) geometry (similar to knurling) compared to plane plate geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The effect of ground and power electrode material on plasma discharge power is shown in figure 3 (b, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The calculated plasma discharge power for copper (Cu) material as ground and power electrodes was substantially (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) higher than brass (Br) and stainless steel (SS) ground and power electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' For the ground electrode, plasma discharge power for Cu was 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7% higher compared to SS and 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5% higher compared to Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, for the power electrode, plasma discharge power for Cu was 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9% higher compared to SS and 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7% higher compared to Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, from the above discussion the choice of material of electrode while producing plasma also plays a significant role in the concentration of generated plasma species and radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Since higher plasma discharge signifies more formation of reactive plasma species and radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The role of electrode material on plasma discharge power also was reported by Jahanmiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which they showed average power consumption by Cu and Br substantially higher than SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, no significant difference was observed in power consumption when Cu and Br were used as material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The impact of different types of power electrodes on plasma discharge power is shown in figure 3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As shown in the results of discharge current characteristics, the multiple high discharge current peaks observed in power electrode made using wire compared to mesh and sheet as power electrode (figure 2 (e-f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similar results were observed in the plasma discharge power, in which the power electrode made using wire had 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5% higher discharge power compared to mesh and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8% higher discharge power compared to the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [24] used SS mesh and aluminium foil as outer electrode wrapped around quartz dielectric tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' They showed better energy efficacy in aluminium foil compared to SS mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, in the present work, we did not observe any significant difference in the plasma discharge power when using SS sheet and SS mesh as power electrodes wrapped around the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The effect of dielectric material on the plasma discharge power is shown in figure 3 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which, the calculated plasma discharge power of air plasma when produced using quartz as dielectric significantly (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) higher compared to glass as dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The higher discharge power in quartz compared to glass was due to the power dissipation in each rising and falling half-cycle in quartz compared to single rising half-cycle power dissipation in the glass over a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similar results were observed in the work reported by Ozkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [25], in which higher discharge power was observed in quartz dielectric compared to glass dielectric at constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 Emission spectra of air plasma The emission spectrum of air plasma is shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The wavelength range shown in figure 4 lies between 185 nm to 950 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The recording emission spectra of air plasma mainly contain strong emission band peaks of the N2 second positive system (C 3Πu → B 3Πg) lies in the range of 290 nm and 440 nm (shown in figure 4 box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, weak intensity N2+ (B 2Σu+ → X 2Σg+) first negative system band peaks were also observed in air plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The observed band peaks details are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=" Mechanism of formation of N2 second positive system and N2+ first negative system in air plasma The ground state N2 (X 1Σg+)υ present in the air was excited by direct impact excitation to upper- level N2 (C 3Πu)υ' (equation (2)) in an applied electric field between space change developed over the dielectric surface and ground electrode." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The population of upper-level N2 is the result of high energy electron collision with the N2 ground state[33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=" N2 (X 1Σg+)υ + e → N2 (C 3Πu)υ' + e (2) The radiative decay of upper-level N2 (C 3Πu) to lower-level N2 (B 3Πg) (equation (3)) results in the formation of strong emission band peaks of the N2 second positive system as shown in Table 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' N2 (C 3Πu)υ\' → N2 (B 3Πg)υ" + hυ (3) The formation of excited-state N2+ (B 2Σu+)υ\' from N2 (X 1Σg+)υ ground state and its population may followed the following paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=" One-step process N2 (X 1Σg+)υ + e → N2+ (B 2Σu+)υ' + 2e (4) Two-step process N2 (X 1Σg+)υ + e → N2+ (X 2Σu+)υ' + 2e (5) N2+ (X 2Σu+)υ + e → N2+ (B 2Σu+)υ' + e (6) In a one-step process, electron impact ionization of N2 (X 1Σg+)υ ground state molecule occurs to N2+ (B 2Σu+)υ' excited state (equation (4))." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=" However, in a two-step process, electron impact ionization of N2 (X 1Σg+)υ ground state molecule occurs to N2+ (X 2Σu+)υ' ground state (equation (5))." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=" Then this generated N2+ (X 2Σu+)υ ground state populated to N2+ (B 2Σu+)υ' excited state with electron impact excitation (equation (6))." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=" The radiative decay of excited state N2+ (B 2Σu+)υ' to ground state N2+ (X 2Σu+)υ (equation (7)) results in formation of weak intensity N2+ (B 2Σu+ → X 2Σg+) first negative system band peaks[33]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' N2+ (B 2Σu+)υ\' → N2+ (X 2Σu+)υ" + hυ (7) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Optical emission spectra of air plasma Table 1 The observed band peaks lines in air emission spectra[34] Species Transitions Spectral lines (nm) υ\' → υ" Transition Probabilities (s-1) N2 C 3Πu → B 3Πg 296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 3 → 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='61 × 106 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 2 → 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='84 × 106 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4 1 → 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='02 × 107 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 0 → 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='10 × 107 353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 1 → 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='61 × 106 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 0 → 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='33 × 106 371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 2 → 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='37 × 106 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 1 → 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='10 × 106 N2 (C "I-B"Ilg)380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4 0 → 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='94 × 106 394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 2 → 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='63 × 106 399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 1 → 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='49 × 106 405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 0 → 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='23 × 105 426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 1 → 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='69 × 105 434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 0 → 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='47 × 105 N2+ B 2Σu+ → X 2Σg+ 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 0 → 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='10 × 107 419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9 2 → 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='13 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 Formation of RONS in water and change in physicochemical properties of water The mechanism of formation of various reactivity oxygen-nitrogen species (RONS) in PAW is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The reactions are divided into three phases – plasma phase (equations (8- 20)), plasma-liquid interphase, and liquid phase (equations (21-37))[10-12, 21, 35-41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In the plasma phase, the dissociation of N2, N2+, O2, and H2O, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' occurs by electrode impact dissociation into corresponding atoms (N, O, H, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') and molecules (OH, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') (equations (8- 10, 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The dissociation of molecules also occurs by high-energy excited molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As shown in equation (11), dissociation of H2O molecule with high energy N2 molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Along with the dissociation reaction, the dissociative replacement also occurs by high-energy atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The formation of NO molecule occurs by dissociative replacement of N2, O2, and OH by O and N atoms (equations (13-15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, the formation of HO2 and NO2 occurs by dissociative replacement of OH and NO by gases O3 (equations (19, 20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' At last, recombination reactions occur in the plasma phase which results in the formation of gases O3, NO2, and H2O2, etc (equations (16-18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The region between the plasma phase and liquid phase is known as the plasma-liquid interphase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which, the relatively long-lived species exist before dissolved into the liquid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' These species were given as excited N2, H2O, O3, NO, OH, NO2, and HO2 molecules, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' as shown in Table 2 of plasma-liquid interphase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The formation of stable reactive RONS is shown in the liquid phase of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The stable species which are identified and whose concentrations are measured in the present work are shown in bold (marked in red) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The reactions (equations (21-37)) which result in the formation/degradation of stable species such as NO2ˉ ions, NO3ˉ ions, dissolved O3, and H2O2 are shown in the liquid phase of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The reactions that result in the formation of NO2ˉ ions (equations (21,30,31,)) in PAW were the reaction between NO (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') and OH (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' ), and NO (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') and NO2 (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') with H2O (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, the reaction that results in the formation of NO3ˉ ions (equations (22,31,36,37)) in PAW were given as the reaction between dissolved O3 and NO2ˉ ions, and NO2 and H2O (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' ), NO2ˉ and OH (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' ), and dissociation of peroxynitric acid (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' ONOOH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The dissolution of gases O3 in water results in the formation of dissolved O3 (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was due to the particle solubility of O3 in water at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The formation of H2O2 (equations (23,26,34)) in PAW occurs due to the reaction between OH molecules, HO2 molecules, and HO2 molecule and O2ˉ ion in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Along with the formation of RONS in PAW, the degradation of RONS also occurs in PAW to form more stable species in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The dissolved O3 and NO2ˉ ions react to form more stable NO3ˉ ions in PAW (equation (22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, aqueous OH reacts with NO2ˉ ions reacts to form NO3ˉ ions in PAW in the acidic region (equation 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Also, the NO2ˉ ions react with H2O2 to form peroxynitric acid, which is degraded to form NO3ˉ ions in PAW (equation (35,37)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As discussed in Table 2, the formation of various reactive oxygen-nitrogen species in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' These species in PAW give PAW an immense potential to be used in various applications such as microbial (bacteria, fungi, virus, and pest, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') inactivation, food preservation, selective killing of cancer cells, and seeds germination and plant growth, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' [1, 4-6, 14-17] The affinity of PAW for microbial inactivation and selective killing of cancer cells is due to the presence of various strong oxidizing species such as H2O2, dissolved O3, hydroxyl radical (OH), peroxynitrile (ONOOˉ), and superoxide ions (O2ˉ), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' as shown in Table 2[2, 5, 6, 11, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Along with strong oxidizing species, PAW is also a rich source of nitrogen species (NO3ˉ, NO2ˉ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') which signifies the usefulness of PAW in the agriculture field[1, 4, 15-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Table 2 Mechanism of formation of reactive oxygen-nitrogen species in PAW[10-12, 21, 35- 41] Reaction phase Reaction Rate constant or reaction rate Equation number Plasma phase 𝑁2 + 𝑒− → 2𝑁 + 𝑒− 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 × 10-6 Te-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 e-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8/Te cm3 s-1 8 𝑒− + 𝑁2 + → 𝑁 + 𝑁∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 × 10-7 (Te/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='03)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='39 cm3 s-1 9 𝐻2𝑂 + 𝑒− → 𝑂𝐻 + 𝐻 + 𝑒− 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 × 10-12 cm3 s-1 10 𝐻2𝑂 + 𝑁2(𝐴) → 𝑂𝐻 + 𝐻 + 𝑁2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 × 10-11 cm3 s-1 11 𝑂2 + 𝑒− → 𝑂 + 𝑂 + 𝑒− 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 × 10-11 cm3 s-1 12 𝑁2 + 𝑂 → 𝑁𝑂 + 𝑁 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 × 10-11 cm3 s-1 13 𝑁 + 𝑂2 → 𝑁𝑂 + 𝑂 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 × 10-17 cm3 s-1 14 𝑁 + 𝑂𝐻 → 𝐻 + 𝑁𝑂 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 × 10-11 cm3 s-1 15 𝑂 + 𝑁𝑂 + 𝑀 → 𝑁𝑂2 + 𝑀 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 × 10-32 cm6 s-1 16 𝑂 + 𝑂2 → 𝑂3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 × 10-12 cm3 s-1 17 𝑂𝐻 + 𝑂𝐻 → 𝐻2𝑂2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 × 10-11 cm3 s-1 18 𝑂𝐻 + 𝑂3 → 𝐻𝑂2 + 𝑂2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9 × 10-12 cm3 s-1 19 𝑁𝑂 + 𝑂3 → 𝑁𝑂2 + 𝑂2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 × 10-12 cm3 s-1 20 Plasma- liquid interphase N2, e-, N, N*, H2O, OH, H, O2, NO, O3, NO2, H, HO2, H2O2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Liquid phase 𝑂𝐻 + 𝑁𝑂 → 𝑵𝑶𝟐 − + 𝐻+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 × 1010 M-1 s-1 21 𝑶𝟑 + 𝑵𝑶𝟐 − → 𝑂2 + 𝑵𝑶𝟑 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 × 105 M-1 s-1 22 𝑂𝐻 + 𝑂𝐻 → 𝑯𝟐𝑶𝟐 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 × 109 M-1 s-1 23 𝑂𝐻 + 𝑶𝟑 → 𝑂2 + 𝐻𝑂2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 × 108 M-1 s-1 24 𝑂𝐻 + 𝐻𝑂2 → 𝑂2 + 𝐻2𝑂 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 × 109 M-1 s-1 25 𝐻𝑂2 + 𝐻𝑂2 → 𝑂2 + 𝑯𝟐𝑶𝟐 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 × 106 M-1 s-1 26 𝐻𝑂2 + 𝑁𝑂 → 𝑂𝑁𝑂𝑂− + 𝐻+ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 × 109 M-1 s-1 27 𝑂𝐻 + 𝑁𝑂2 → 𝑂𝑁𝑂𝑂− + 𝐻+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 × 1010 M-1 s-1 28 𝑁𝑂 + 𝑁𝑂 + 𝑂2 → 2𝑁𝑂2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 × 106 M-2 s-1 29 𝑁𝑂2 + 𝑁𝑂 + 𝐻2𝑂 → 𝟐𝑵𝑶𝟐 − + 2𝐻+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 × 108 M-1 s-1 30 2𝑁𝑂2 + 𝐻2𝑂 → 𝑵𝑶𝟑 − + 𝑵𝑶𝟐 − + 2𝐻+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 × 108 M-1 s-1 31 𝑯𝟐𝑶𝟐 + 𝑂𝐻 → 𝐻2𝑂 + 𝑂2 − + 𝐻+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 × 107 M-1 s-1 32 𝑂2 − + 𝑁𝑂 → 𝑂𝑁𝑂𝑂− 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 × 109 M-1 s-1 33 𝐻2𝑂 + 𝐻𝑂2 + 𝑂2 − → 𝑂2 + 𝑯𝟐𝑶𝟐 + 𝐻𝑂− 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 × 107 M-1 s-1 34 𝑵𝑶𝟐 − + 𝑯𝟐𝑶𝟐 + 𝐻+ → 𝑂𝑁𝑂𝑂𝐻 + 𝐻2𝑂 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 × 103 M-1 s-1 35 𝑵𝑶𝟐 − + 𝑂𝐻 + 𝐻+ → 𝑵𝑶𝟑 − + 2𝐻+ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 × 109 M-1 s-1 36 𝑂𝑁𝑂𝑂𝐻 → 𝑵𝑶𝟑 − + 𝐻+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9 s-1 37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 The effect of knurling on the physicochemical properties of PAW and RONS concentration The variation in the physicochemical properties of PAW and reactive oxygen-nitrogen species (RONS) concentration while varying knurling pitch size is shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 5 showed that introducing knurling to the ground electrode significantly (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) improves the physicochemical properties of PAW and RONS concentration in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This signifies the increase in species/radicals produced in the plasma phase when the ground electrode has knurling compared to the non-knurled electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This behavior was also observed in the electric characterization of plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which, higher discharge current and power dissipation were observed in plasma with ground electrode knurling compared to the non-knurled electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The plasma-water interaction decreased the pH of PAW due to the formation of various acidic species in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The most leading acidic species are NO2ˉ ions and NO3ˉ ions in the form of nitrous and nitric acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Increasing plasma-water treatment time results in the formation of more acidic species in PAW as a result, we observed a continuous decrease in the pH of PAW as shown in figure 5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the pH of PAW prepared using plasma device with knurling and without knurling have significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) difference among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' With knurling (2 mm knurling pitch), the pH of PAW decreased by 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0% after 5 min of plasma- water treatment compared to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, without knurling pH of PAW decreased by 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8% after 5 min of plasma-water treatment compared to control only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, increasing knurling pitch from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm to 2 mm showed a decrease in pH of PAW, however, this decrease in pH of PAW was not statistically significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, introducing knurling to the ground electrode has a substantial impact on the pH of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Since, as discussed in figure 3 (a), introducing knurling increased the plasma discharge and resulted in the formation of more acidic plasma species that dissolved in water and decreased the pH of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The effect of knurling on oxidation-reduction potential (ORP) of PAW is shown in figure 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The ORP gives the oxidizing tendency of PAW which could be used as an indicator of the antimicrobial activity of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Since, it gives the net combination of all oxidizing species (dissolved O3, H2O2, ̇OH, free electrons, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') present in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Increasing plasma-water treatment time increased the ORP of PAW which showed the increase in dissolution of oxidizing species in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The knurling of the ground electrode also helps in increasing the oxidizing tendency of PAW as shown in figure 5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The ORP of PAW prepared using a 1 mm knurling ground electrode (ORP – 600 mV) was 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8% higher compared to without a knurling ground electrode (ORP – 505 mV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the ORP of PAW prepared using a 1 mm knurling electrode was substantially (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) higher compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm and 2 mm knurling electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was due to higher power dissipation in the 1 mm knurling electrode (figure 3 (a)) resulting in the formation of more oxidizing species in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The net combination of all inorganic ions (H+, NO3ˉ ions, NO2ˉ ions, OONOˉ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=') present in PAW were measured using total dissolved solids (TDS) and electrical conductivity (EC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 5 (c, d) showed the variation in TDS and EC of PAW while varying knurling pitch and plasma-water treatment time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Increasing plasma-water exposure time significantly (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) increases the TDS and EC of PAW showing the continuous formation of inorganic ions in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, introducing knurling to the ground electrode substantially (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) increased the TDS and EC of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The TDS and EC of PAW with ground electrode knurling (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm knurling pitch) were 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7% and 567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7% higher than without ground electrode knurling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Initially, the TDS and EC of PAW were prepared using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm knurling electrode plasma device that showed a higher value compared to other knurling pitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As the plasma- water treatment time increases, this difference in TDS and EC of PAW keeps on decreasing and becomes statistically insignificant (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) at 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in NO3ˉ ions and NO2ˉ ions concentration (reactive nitrogen species) in PAW with varying ground electrode knurling pitch and plasma-water treatment time is shown in figure 5 (e, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Increasing plasma-water treatment continuously increased the NO3ˉ ions and NO2ˉ ions concentration in PAW for all knurling pitch ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the knurling of the ground electrode significantly increased the NO3ˉ ions and NO2ˉ ions concentration in PAW compared to a non-knurled ground electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The maximum concentration of NO3ˉ ions and NO2ˉ ions in PAW (plasma-water treatment time of 5 min) with and without knurling ground electrode were given as 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='98 mg l-1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='21 mg l-1 NO3ˉ ions and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='80 mg l-1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='11 mg l-1 NO2ˉ ions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the larger knurling pitch size (2 mm) did not support the higher formation of NO3ˉ ion concentration compared to the smaller knurling pitch size (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm and 1 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, considering the appropriate knurling pitch size is also important to get higher production of NO3ˉ ions concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' For NO2ˉ ions, 1 mm knurling pitch gives the highest concentration of NO2ˉ ions in PAW compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm and 2 mm knurling pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in reactive oxygen species (ROS (H2O2 and dissolved O3)) with plasma- water treatment time and ground electrode knurling pitch size is shown in figure 5 (g, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As the plasma-water treatment time increases, the continuous increase in the H2O2 and dissolved O3 concentration were observed in PAW when the ground electrode without knurling was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, the H2O2 and dissolved O3 concentration in PAW when prepared using a ground electrode with knurling showed an initial increase and then decreased with time with an exception of H2O2 at 5 min of plasma-water treatment time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was due to the reactivity environment favoring the reaction within ROS and ROS reaction with NO2ˉ ions to give more stable NO3ˉ ions (equations (22,24,32,35-37)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Thus, the concentration of NO3ˉ ions in PAW is substantially higher compared to other RONS in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the ROS concentration decreased at a higher plasma-water treatment time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in physicochemical properties of PAW and RONS concentration with varying ground electrode knurling pitch (0 mm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm, 1 mm, and 2 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Different lowercase letters showed statistically significant difference (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05, n ≥3) among the properties of PAW mean ± standard deviation (µ ± σ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="4 The effect of ground electrode material on the physicochemical properties of PAW and RONS concentration 1 1 3 5 4 6 8 b'' a'' a' a''' a'' a' a'' a' a''' a'' a' a'' a' d''' c''' b''' a''' d'' c'' b'' a'' d' c' b' a' b b b Plasma treatment time pH Plasma treatment time 0 mm 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="5 mm 1 mm 2 mm Knurling Pitch (a) a 1 1 3 5 200 400 600 a''' c''' b''' a''' a''' c'' c' b'' b'' b' a' a a a a (b) ORP (mV) 0 mm 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="5 mm 1 mm 2 mm 1 1 3 5 0 30 60 90 d''' c''' b''' d'' c'' b'' d' c' b' d c b a (c) TDS (ppm) Plasma treatment time 0 mm 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="5 mm 1 mm 2 mm 1 1 3 5 0 50 100 150 d''' c''' b''' d'' c'' b'' d' c' b' c b a a Plasma treatment time (d) EC (µS cm 1) 0 mm 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="5 mm 1 mm 2 mm 1 1 3 5 0 10 20 30 a d''' c''' b''' d'' c'' b'' d' c' b' d c b a Plasma treatment time Plasma treatment time (e) NO3 ions (mg L 1) 0 mm 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="5 mm 1 mm 2 mm 1 1 3 5 0 1 2 3 a c''' b''' a''' a''' d'' c'' c' b' a' c b b a (f) NO2 ions (mg L 1) Plasma treatment time 0 mm 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="5 mm 1 mm 2 mm 1 1 3 5 0 3 6 9 d''' c''' a''' a''' d'' c'' b'' a'' c' b' a'b' a' d c b (g) H2O2 (mg L 1) 0 mm 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="5 mm 1 mm 2 mm 1 1 3 5 0 3 6 9 b''' b''' a''' c'' b'' b'' a'' c' b' b' a' c c b (h) Dissolved O3 (mg L 1) Plasma treatment time 0 mm 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mm 1 mm 2 mm 0 0 0 0 0 0 0 0 The role of ground electrode material of plasma device on the physicochemical properties and RONS concentration of PAW is shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The observed values of physicochemical properties of PAW when copper (Cu) was used as a ground electrode material was higher compared to stainless steel (SS) and brass (Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This can be implied from the plasma discharge power in which Cu had the higher plasma discharge power compared to Br and SS (figure 3 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Higher discharge power signifies more generation of plasma radicals/species which come in contact with water and improved the physicochemical properties of PAW and RONS concentration in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The lowest pH of PAW (figure 6 (a)) PAW (5 min of plasma-water treatment time) when prepared using Cu, Br, and SS as ground electrode material were given as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='56, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was also reflected as higher TDS, EC, NO2ˉ + NO3ˉ ions concentration (figure 6 (c-f)) in PAW prepared using Cu as ground electrode compared to Br and SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The TDS and EC of PAW when prepared using Cu as ground electrode (5 min of plasma-water treatment time) were 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8% and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5% higher than SS, and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3% and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1% higher than Br, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, for a plasma-water treatment time of 5 min, the observed NO3ˉ ions concentration in PAW when prepared using Cu as ground electrode was 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3% and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1% higher than SS and Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The oxidizing potential (ORP) of PAW prepared using Cu and SS as the ground electrode was slightly higher than Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The ORP of PAW prepared using Cu, SS, and Br as ground electrodes was given as 548 mV, 545 mV, and 535 mV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The slightly higher ORP and lower pH of PAW prepared using Cu and SS as ground electrodes compared to Br increases PAW reactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The reactive environment of PAW favors reactions within ROS and generated ROS with NO2ˉ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As a result, the concentration of NO2ˉ ions and ROS decreased in the high reactive environment of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The results of NO2ˉ ions concentration in PAW shown in figure 6 (f) confirm the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Initial for a plasma-water treatment time of 1 min, the observed NO2ˉ ions concentration prepared using Br as ground electrode lower than Cu and SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' However, as the plasma-water treatment time increased (3 min and 5 min), the observed NO2ˉ ions concentration in PAW prepared using Br as the ground electrode was higher than Cu and SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Since, increasing plasma-water treatment time, increased the reactivity of PAW as a result NO2ˉ ions present in PAW react with dissolved O3 and H2O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Therefore, decreased the concentration of NO2ˉ ions in PAW prepared using Cu and SS as ground electrodes material compared to Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similar results were observed in the ROS concentration in PAW as shown in figure 6 (g, h) with an exception of H2O2 in PAW (5 min treatment time) prepared using Cu as a ground electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In which a high reactive PAW, the H2O2 and dissolved O3 concentration present in PAW either decrease or remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As discussed, the PAW prepared using Br as ground electrode material had comparatively low reactivity, hence, the increase in the concentration of H2O2 and dissolved O3 were observed with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the concentration of H2O2 and dissolved O3 in PAW decreases or remains constant as the plasma-water treatment time increases (or, the reactivity of PAW increases) for SS and Cu as ground electrode material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' One exception was also observed in H2O2 concentration in PAW (figure 6 (g)), in which a higher concentration of H2O2 was observed in high reactivity PAW when prepared using Cu as ground electrode material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The possible reason for the same is an excess concentration of H2O2 in PAW which is left even after reaction with dissolved O3 and NO2ˉ ions present in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in physicochemical properties of PAW and RONS concentration with varying ground electrode material (SS, Br, and Cu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Different lowercase letters showed statistically significant difference (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05, n ≥3) among the properties of PAW mean ± standard deviation (µ ± σ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 The effect of power electrode material on the physicochemical properties of PAW and RONS concentration The variation in the power electrode material on the physicochemical properties of PAW and RONS concentration is shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similar to ground electrode material, the use of Cu for power electrode material significantly enhances the physicochemical properties of PAW and RONS concentration in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The results also supported the plasma discharge power in which Cu had substantially higher power compared to Br and SS (figure 3 (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The lowest pH of PAW (5 min of plasma treatment time) when prepared using Cu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Br,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' and SS as power electrode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='material (figure 7 (a)) were given as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9, 3, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the PAW produced using Cu as power electrode material generates PAW with higher acidity compared to other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' At a higher plasma-water treatment time, the oxidizing potential (ORP) of PAW prepared using Br and Cu as power electrode material was significantly (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) higher compared to SS (figure 7 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In addition, the ORP of PAW prepared using Cu (ORP – 590 mV) as a power electrode was slightly higher compared to Br (ORP – 580 mV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the generation of oxidizing species in PAW during plasma-water exposure was substantially higher when Cu or Br was used as power electrode material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The significance of Cu material as a power electrode over SS and Br is easily understood by comparing the results of TDS, EC, NO3ˉ ions, and NO2ˉ ions concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' For a plasma-water treatment time of 5 min, a substantial growth (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) in the values of TDS, EC, NO3ˉ ions, and NO2ˉ ions in PAW when prepared using Cu as power electrode material compared to SS and Br as shown in figure 7 (c-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The increase in TDS and EC of PAW using Cu as power electrode were 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0% and 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9% higher compared to Br, and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3% and 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4% higher compared to SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, the increase in NO3ˉ and NO2ˉ ions concentration in PAW prepared using Cu as power electrode material were 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0% and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8% higher compared to Br, and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2% and 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7% higher compared to SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the use of Cu as an outer electrode material helps in the generation of more inorganic ions in PAW which was communicated above as enhancement in TDS, EC, NO3ˉ ions, and NO2ˉ ions concentration in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in H2O2 and dissolved O3 when using different power electrode materials is shown in figure 7 (g, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The H2O2 and dissolved O3 concentration present in PAW when using different materials for power electrode showed a rise and fall (or rise and remain constant) in its concentration with increasing plasma-water treatment time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Since, initially (t = 0 min), there was no H2O2 and dissolved O3 present in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, as soon as plasma-water interaction starts occurring, the formation of H2O2 and dissolved O3, as a result, their concentration starts increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The fall in concentration of H2O2 and dissolved O3 showed (SS power electrode) reduction of these species due to reaction occurring among themselves and with NO2ˉ ions to form more stable NO3ˉ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the constant concentration of H2O2 and dissolved O3 (Br and Cu power electrode) with increasing time showed the established equilibrium in which excess concentration above the equilibrium point converted more stable products like NO3ˉ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in physicochemical properties of PAW and RONS concentration with varying power electrode material (SS, Br, and Cu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Different lowercase letters showed ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Br ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Cu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="b'' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='(h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Dissolved O3 (mg L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Plasma treatment time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='SS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Br ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Cu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='statistically significant difference (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05, n ≥3) among the properties of PAW mean ± standard deviation (µ ± σ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 The effect of power electrode type on the physicochemical properties of PAW and RONS concentration In this section, we discussed the role of different types of electrodes on the physicochemical properties of PAW and RONS concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' For this study, three different types of electrodes were used such as mesh, wire, and sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The observed result is shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The results of physicochemical properties of PAW (figure 8 (a-d)) showed the use of wire as a power electrode type significantly enhanced the physicochemical properties of PAW compared to other electrode types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was due to the use of wire as a power electrode having higher plasma discharge power compared to mesh and sheet (figure 3 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The pH of PAW, when prepared using wire as a power electrode (5 min of plasma treatment time) showed 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4% lower compared to mesh, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7% lower compared to the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, the oxidizing potential (ORP) of PAW when prepared using wire as a power electrode was 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2% higher than mesh and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6% higher compared to the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, at the same operating parameters, the TDS and EC of PAW prepared using wire as power electrode showed 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4% and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6% higher compared to mesh, and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3% and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0% higher compared to the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similar to physicochemical properties, the NO3ˉ ions concentration present in PAW prepare (5 min of plasma treatment time) using wire as power electrode type showed significantly higher value compared to mesh and sheet (figure 8 (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The observed NO3ˉ ions concentration in PAW prepared using wire, mesh, and sheet as power electrodes were given as 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 mg l-1, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 mg l-1, and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8 mg l-1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In contrast to physicochemical properties and NO3ˉ ions concentration, the NO2ˉ ions, H2O2, and dissolved O3 present in PAW showed higher value for the sheet as power compared to wire and mesh (figure 8 (f-h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This signifies the PAW produced using the sheet as a power electrode did not favors the reaction within dissolved ROS (H2O2 and dissolved O3) and with NO2ˉ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, due to limiting reactions (equations (22, 35-37)) between these species, the observed concentration of reactants (H2O2, dissolved O3, and NO2ˉ ions) were high and the product concentration (NO3ˉ ions) was low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in physicochemical properties of PAW and RONS concentration with varying power electrode types (mesh, wire, and sheet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Different lowercase letters showed statistically significant difference (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05, n ≥3) among the properties of PAW mean ± standard deviation (µ ± σ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 The effect of dielectric material on the physicochemical properties of PAW and RONS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='concentration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="a' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='(f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='(mg L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Plasma treatment time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Mesh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Wire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Sheet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='(g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='H2O2 (mg L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="c' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="b' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="a' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="a' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='(h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Dissolved O3 (mg L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Plasma treatment time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Mesh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Wire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Sheet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Figure 9 showed the variation in physicochemical properties of PAW and RONS concentration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='when using glass and quartz dielectric materials in plasma device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The results showed statistically significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) enhancement in the physicochemical properties of PAW and RONS concentration when quartz was used as a dielectric material compared to glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As discussed in the electrical characterization, higher discharge current peaks and plasma discharge power in quartz as dielectric compared to glass (figures 2 (c, d) and 3 (d))[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This signifies higher plasma species/radicals density in quartz plasma device compared to glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As a result, a high concentration of reactive species dissolved in PAW was prepared using a quartz plasma device as shown by enhanced physicochemical properties and RONS concentration compared to a glass plasma device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The ORP, TDS, and EC of PAW when prepared using quartz as dielectric showed a higher value and lower value of pH of PAW compared to glass (figure 9 (a-d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This showed the improvement in the physicochemical properties of PAW when prepared using quartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The pH of PAW (plasma treatment time of 5 min) when prepared using glass and quartz as dielectric were given as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' At similar conditions, the ORP, TDS, and EC of PAW when prepared using glass and quartz were given as 545 mV and 580 mV (ORP), 86 ppm and 150 ppm (TDS), and 160 µS cm-1 and 290 µS cm-1 (EC), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in dissolved RONS in PAW when prepared using glass and quartz as the dielectric material of the plasma device is shown in figure 9 (e-h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The observed NO3ˉ and NO2ˉ ions concentration in PAW prepared using quartz as dielectric significantly (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) higher compared to glass (figure 9 (e-f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' At a plasma-water treatment time of 5 min, the NO3ˉ and NO2ˉ ions concentration in PAW were 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8% and 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4% higher compared to glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similar to NO3ˉ and NO2ˉ ions, the observed H2O2 and dissolved O3 concentrations in PAW were higher for quartz compared to glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' For glass, with increasing plasma-water treatment time, the H2O2 concentration and dissolved O3 in PAW showed a rise and fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was as the reactivity of PAW increases these species react with each other and other NO2- ions present in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As a result, their concentration decreased at higher plasma-water treatment time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' For quartz, the H2O2 and dissolved O3 present in PAW follow a trend of rise-fall-rise with time (figure 9 (g, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' This was due to initially no H2O2 and dissolved O3 present in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, these species’ concentration increases with increasing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As the concentration of these reached sufficient reactivity, they react with each other, and NO2ˉ ions concentration was present in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' As a result, a decrease in their concentration was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Further increase in the plasma-water treatment time showed a generation of a higher concentration of ROS in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, even after reacting with each other and NO2ˉ ions present in PAW a slight increase in these species (H2O2 and dissolved O3) concentration was observed in PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The variation in physicochemical properties of PAW and RONS concentration with varying dielectric material (glass and quartz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Different lowercase letters showed statistically significant difference (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05, n ≥3) among the properties of PAW mean ± standard deviation (µ ± σ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8 Residual metal analysis in PAW As PAW has various applications in the field of agriculture, food preservation, and cancer cells inactivation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The presence of heavy metal in PAW coming from electron erosion used in plasma device may interfere in the applications of PAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, the residual metal analysis of PAW become important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Table 1 shows the residual metal in PAW when prepared using different ground electrode material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The different material of construction of ground electrode are stainless steel (SS, allow of iron (Fe) and carbon), brass (Br, copper and zinc (Zn)), and copper (Cu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' the Table 1 showed the concentration of Cu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' and Fe in PAW and ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Glass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Quartz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content="c' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Dissolved O3 (mg L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Plasma treatment time (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Glass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='Quartz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The observed concentration of Cu and Zn present in PAW and control were insignificant (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05) for different ground electrode material of plasma device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Moreover, the concentration of Cu and Zn were less than 6 µg l-1 and 2 µg l-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Also, the Fe concentration in PAW and control was beyond the detection limit (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' In conclusion, the observed heavy metal concentration in PAW similar to ultrapure milli-Q water (control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Hence, no erosion of inner (ground) electrode material occurs during plasma-water treatment which settle in PAW in form of residue metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Residual metal analysis in PAW when prepared using different ground electrode material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Different lowercase letters showed statistically significant difference (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='05, n ≥3) among the group mean ± standard deviation (µ ± σ) (*B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L – Below Detection Limit) Material of ground electrode Plasma-water treatment time Residue metal present in water Copper (µg l-1) Zinc (µg l- 1) Iron (µg l-1) Control 0 min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='14 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* Stainless Steel 1 min 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* 3 min 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* 5 min 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='4 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* Brass 1 min 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* 3 min 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* 5 min 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='0 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='2 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* Copper 1 min 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='9 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='1 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* 3 min 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='3 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* 5 min 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='5 a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='L* 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Conclusion The present work shows the effect of various components of dielectric barrier discharge plasma device (DBD-PD) on plasma and physicochemical properties of PAW and RONS concentration present in activated water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The discharge current characteristics showed substantial improvement in filamentary discharge when introduced knurling to ground electrode, using quartz as dielectric layer compared to glass, and employed wired power electrode compared to mesh and sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' These results are further supported by the results of plasma discharge power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Similarly, the plasma activated water produced using a diamond knurled electrode, quartz as dielectric, and wire as power electrode showed substantial improvement in the physicochemical properties of PAW and RONS concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' We have also observed that the use of copper material for manufacturing ground and power electrodes significantly improves the PAW properties compared to brass and stainless steel materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Acknowledgments This work was supported by the Department of Atomic Energy (Government of India) graduate fellowship scheme (DGFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The authors sincerely thank Mr Chirayu Patil, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Kaila, and Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Nimish for providing constant support and useful suggestions during this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Data availability statement The data that support the findings of this study are available upon reasonable request from the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Conflict of interests The authors declare that there are no conflicts of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Authors’ contributions Both authors contributed to the study conception and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Material preparation, data collection, and analysis were performed by Vikas Rathore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' The first draft of the manuscript was written by Vikas Rathore, and both authors commented on previous versions of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Both authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' ORCID iDs Vikas Rathore https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content='org/0000-0001-6480-5009 References [1] Thirumdas R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E4T4oBgHgl3EQfuw0c/content/2301.05234v1.pdf'} +page_content=' Kothakota A,' metadata={'source': 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b/rNAzT4oBgHgl3EQfrf0Z/content/tmp_files/2301.01643v1.pdf.txt @@ -0,0 +1,685 @@ +arXiv:2301.01643v1 [math.QA] 4 Jan 2023 +Idempotent set-theoretical solutions of the pentagon equation⋆ +Marzia MAZZOTTAa +aDipartimento di Matematica e Fisica “Ennio De Giorgi” +Universit`a del Salento +Via Provinciale Lecce-Arnesano +73100 Lecce (Italy) +Abstract +A set-theoretical solution of the pentagon equation on a non-empty set X is a function +s : X × X → X × X satisfying the relation s23 s13 s12 = s12 s23, with s12 = s × idX, +s23 = idX × s and s13 = (idX × τ)s12(idX × τ), where τ : X ×X → X ×X is the flip map +given by τ(x, y) = (y, x), for all x, y ∈ X. Writing a solution as s(x, y) = (xy, θx(y)), +where θx : X → X is a map, for every x ∈ X, one has that X is a semigroup. +In this paper, we study idempotent solutions, i.e., s2 = s, by showing that the idempo- +tents of X have a key role in such an investigation. In particular, we describe all such +solutions on monoids having central idempotents. Moreover, we focus on idempotent so- +lutions defined on monoids for which the map θ1 is a monoid homomorphism, by showing +that they have to be derived considering the kernel congruence of the map θ1. +Keywords: +pentagon equation, set-theoretical solution, semigroup +2022 MSC: 16T25, 81R50, 20M99 +Introduction +If V is a vector space over a field F, a linear map S : V ⊗ V → V ⊗ V is said to be a +solution of the pentagon equation on V if it satisfies the relation +S12S13S23 = S23S12, +(1) +where S12 = S ⊗ idV , S23 = idV ⊗ S, S13 = (idV ⊗ Σ) S12 (idV ⊗ Σ), with Σ the flip +operator on V ⊗ V , i.e., Σ(u ⊗ v) = v ⊗ u, for all u, v ∈ V . The pentagon equation +classically originates from the field of Mathematical Physics, but it has several appli- +cations and appears in different areas of mathematics, also with different terminologies +(see, for instance, [23, 1, 22, 2, 17, 11, 14, 12]). To know more about contexts in which +⋆This work was partially supported by the Dipartimento di Matematica e Fisica “Ennio De Giorgi” - +Universit`a del Salento. The author is member of GNSAGA (INdAM) and of the non-profit association +ADV-AGTA. +Email address: marzia.mazzotta@unisalento.it (Marzia MAZZOTTA) +January 5, 2023 + +the pentagon equation appears, we refer to the introduction of the paper by Dimakis and +M¨uller-Hoissen [8] along with the references therein. +In 1998, Kashaev and Sergeev [13] explicitly first highlighted an existing link between +solutions on a vector space viewed as the space F X of all functions from a finite set X +to F and maps from X × X into itself satisfying a certain relation. Into the specific, to +any map s : X × X → X × X one can associate a linear operator S : F X×X → F X×X +defined as S(f)(x, y) = f(s(x, y)), for all x, y ∈ X. If the map s satisfies the “reversed” +pentagon relation +s23s13s12 = s12s23, +(2) +where s12 = s×idX, s23 = idX ×s, s13 = (idX ×τ) s12 (idX ×τ), with τ(x, y) = (y, x), for +all x, y ∈ X, then the linear map S is a solution of the pentagon equation on F X. We call +the map s as above set-theoretical solution of the pentagon equation, or briefly solution, +on X. In the pioneering paper [13], one can also find the first systematic way to obtain +solutions on closed under multiplications subsets of arbitrary groups. However, such +set-theoretical maps had already appeared before with the terminology transformations +pentagonales in two non purely algebraic papers: those by Zakrzewski [23] and Baaj and +Skandalis [2]. In these papers, we can find the first instances of bijective solutions. +Attention only to set-theoretical solutions has been recently given in [3]. Following the +notation therein, writing a solution s : X × X → X × X as s(x, y) = (xy, θx(y)), where +θx is a map from X into itself, for every x ∈ X, one has that X is a semigroup and +θx(y)θxy(z) = θx(yz), +(P1) +θθx(y)θx y = θy, +(P2) +for all x, y, z ∈ X. +In [3, Theorem 15], a complete description of all solutions de- +fined on a group is given. +In this case, it holds that θx(y) = θ1(x)−1θ1(xy), for all +x, y ∈ X, where 1 is the identity of the group X, and it is sufficient to study the set +ker θ1 = {x ∈ X | θ1(x) = 1} that it a normal subgroup of X, even if, in general, θ1 is +not a homomorphism. However, describing all solutions on arbitrary semigroups seems +to be very difficult since there are many even in the case of small-order semigroups. For +instance, it is sufficient to look [16, Appendix B], where, as suggested by Rump, all the +non-isomorphic solutions on semigroups of order 3 have been computed. +A first step could be studying specific classes of solutions. In this regard, a characteriza- +tion of all involutive solutions, i.e., s2 = idX×X, has been provided by Colazzo, Jespers, +and Kubat in their recent paper [6]. Moreover, in [4], Catino, Mazzotta, and Stefanelli +studied the pentagon and the quantum Yang-Baxter equations (see [9]) for some similar- +ities. Indeed, there exist several maps defined on particular semigroups that satisfy both +equations. In this last paper, one can also find some methods to construct solutions of +the pentagon equation such as, for instance, on the matched product of two semigroups, +that is a semigroup including the classical Zappa product [24]. +Recently, idempotent left non-degenerate solutions of the Yang-Baxter equation have +been completely described (see [15, 21, 7] and the classifications therein). +A similar +study could also be done for the pentagon equation. +2 + +In this paper, we deal with idempotent set-theoretical solutions of the pentagon equa- +tion, i.e., s2 = s. One can easily check that a solution s(x, y) = (xy, θx(y)) on a semigroup +X is idempotent if and only if +xyθx(y) = xy, +(I1) +θxyθx(y) = θx(y), +(I2) +for all x, y ∈ X. We show that the idempotents of X have a crucial role in finding +them and, for that, we exhibit some useful properties of the maps θx involving the +idempotents. In particular, we focus on solutions defined on monoids, by showing that, +unlike solutions defined on groups, it is not possible to find a way to write the maps θx +by means of the map θ1, where 1 is the identity of the monoid. We provide a description +theorem for idempotent solutions on monoids M having central idempotents, namely, +E(M) ⊆ Z(M), by illustrating that it is enough to construct specific idempotent maps +θe, for every e ∈ E(M). Furthermore, in this situation, the map θ1 is an idempotent +monoid homomorphism, and, for this reason, we deepen idempotent solutions on monoids +satisfying this additional property. +In this case, all the maps θx have to be derived +considering the kernel congruence of the function θ1, namely, the set ker θ1 = {(x, y) ∈ +M × M | θ1(x) = θ1(y)} (see [20] for more details). Indeed, (θx(y), y) ∈ ker θ1, for all +x, y ∈ M, and θ1(M) is a system of representatives of M/ ker θ1. +Finally, we collect some properties of idempotent solutions on arbitrary semigroups, that +could be useful in a future study of these maps in the more general case. +1. Basics on solutions +In this section, we give some basics on the solutions. Moreover, we introduce some classes +of solutions and provide several examples. +From now on, following the notation used in [3, Proposition 8], given a semigroup +X, we will briefly call solution on X any map s : X × X → X × X given by s(x, y) = +(xy, θx(y)), where θx is a map from X into itself, satisfying (P1) and (P2). +Example 1. (cf. [17]) Let X be a set and f, g : X → X idempotent maps such that +fg = gf. Set x · y = f(x), for all x, y ∈ X, one has that (X, ·) is a semigroup and the +map s(x, y) = (x · y, g (y)) is a solution on X. +Note that the previous example belongs to the class of P-QYBE solutions, namely +the maps that are solutions both to the pentagon and the Yang-Baxter equations [4]. +Definition 1. Let (X, ·) and (Y, ∗) be two semigroups and s(x, y) = (x · y, θx(y)) and +r(a, b) = (a ∗ b, ηa(b)) two solutions on X and Y , respectively. +Then, s and r are +isomorphic if there exists an isomorphism f : X → Y such that +fθx(y) = ηf(x)f(y), +(3) +for all x, y ∈ X, or, equivalently, (f × f)s = r(f × f). +3 + +A complete description in the case of a group is given in [3, Theorem 15]. For the +sake of completeness, we recall such a result below. +Theorem 2. Let G be a group. Consider a normal subgroup K of G and a system of +representatives R of G/K such that 1 ∈ R. If µ : G → R is a map such that µ(x) ∈ Kx, +for every x ∈ G, then the map s(x, y) = +� +xy, µ (x)−1 µ (xy) +� +, for all x, y ∈ G, is a +solution on G. +Conversely, if s is a solution on G, then the set K = {x ∈ G | θ1(x) = 1} is a normal +subgroup of G for which Im θ1 is a system of representatives of G/K that contains 1, +θ1(x) ∈ Kx, for every x ∈ G, and s(x, y) = +� +xy, θ1 (x)−1 θ1 (xy) +� +, for all x, y ∈ G. +By Theorem 2 and making explicit the condition (3), it is easy to check that two solutions +s(x, y) = (xy, θx(y)) and r(x, y) = (xy, ηx (y)) on the same group G are isomorphic via +f ∈ Aut(G) if and only if fθ1 = η1f, i.e., f sends the system of representatives θ1(G) +into the other one η1f(G). +However, describing all the solutions, up to isomorphisms, on arbitrary semigroups +turns out to be very hard. Indeed, even in the case of semigroups of small order, there +are a lot of solutions, as one can see in [16, Appendix B]. A first step could be studying +specific classes of solutions. In this regard, one can find in [6, Theorem 5.5] a complete +description of all involutive solutions. +Definition 3. Let X be a semigroup and s(x, y) = (xy, θx(y)) a solution on X. We say +that the map s is +- non-degenerate if θx is bijective, for every x ∈ X; +- involutive if s2 = idX×X; +- idempotent if s2 = s. +Examples 2. +1. [3, Examples 2-2.] Let X be a semigroup and γ : X → X a map. Then, the map +s(x, y) = (xy, γ (y)) , for all x, y ∈ X, is a solution if and only if γ ∈ End(X) and +γ2 = γ. One can easily check that such a solution s is non-degenerate if and only +if γ = idX. +2. As a particular case of 1., if X is a semigroup and e ∈ E(X), where E(X) denotes +the set of the idempotents of X, the map s(x, y) = (xy, e) , for all x, y ∈ X, is an +idempotent solution. +3. Let X be a semigroup belonging to the variety S := [abc = bc] (see [18, p. 370]). +Then, the map s (x, y) = (xy, xy) , for all x, y ∈ X, is an idempotent solution. +4. Every Clifford semigroup X gives rise to the idempotent solution s given by s(x, y) = +� +xy, y−1y +� +, for all x, y ∈ X. Recall that a Clifford semigroup X is a semigroup in +which every x ∈ X admits a unique x−1 ∈ X such that xx−1x = x, x−1xx−1 = x−1, +and xx−1 = x−1x (see [19, Exercise II.2.14]). +4 + +5. [16, Appendix B] Let X = {0, a, b} and S the null semigroup on X, i.e., xy = 0, +for all x, y ∈ X. Consider the maps θ0 = idS and θa = θb such that θa(0) = 0, +θa(a) = b, and θa(b) = a. Then, the map s(x, y) = (0, θx(y)) is an idempotent and +non-degenerate solution on S. +Other classes of solutions that can be studied are the commutative and the cocom- +mutative ones (see [3, Definition 6]). +These kinds of solutions are in analogy to the +commutative and the cocommutative multiplicative unitary operators, i.e., solutions of +(1) defined on Hilbert spaces (see [1, Definition 2.1]). +Definition 4. A solution s : X × X → X × X is said to be +- commutative if s12s13 = s13s12; +- cocommutative if s13s23 = s23s13. +If X is a semigroup and s(x, y) = (xy, θx(y)) a solution on X, it is a routine computation +to check that the map s is commutative if and only if +xzy = xyz +(C1) +θx = θxy +(C2) +for all x, y, z ∈ X. Instead, s is cocommutative if and only if +xθy(z) = xz +(CC1) +θxθy = θyθx +(CC2) +for all x, y, z ∈ X. +There exist solutions that are both commutative and cocommutative, such as the maps +in Example 1. Moreover, according to [6, Corollary 3.4], if s is an involutive solution, +then s is both commutative and cocommutative. +Clearly, if M is a monoid, it follows by (CC1) that the unique cocommutative solution +on M is given by s(x, y) = (xy, y), for all x, y ∈ M. In the next result, we describe all +the commutative solutions on a monoid. +Proposition 5. Let M be a monoid. Then, a solution s(x, y) = (xy, θx(y)) on M is +commutative if and only if M is a commutative monoid and θx = γ, with γ ∈ End(M), +γ2 = γ, for every x ∈ M. +Proof. Initially, if the monoid M is commutative, then the map s(x, y) = (xy, γ (y)), +with γ ∈ End(M), γ2 = γ, is a commutative solution on M (see Examples 2-1.). +Conversely, let us assume that s(x, y) = (xy, θx(y)) is commutative. Then, by substitut- +ing x = 1 in (C1), the monoid M is commutative and, by (C2), θ1 = θy, for every y ∈ M. +Thus, by (P1) and (P2) the claim follows. +5 + +2. Properties of the maps θx involving the idempotents +In this section, we provide some properties of the maps θx which involve the idempotents +of arbitrary semigroups and that will be used in the next. +Firstly, according to [10, p. 69], among the idempotents in any semigroup X, there +is a natural partial order relation by the rule that +∀ e, f ∈ E(X) +e ≤ f ⇐⇒ ef = fe = e. +Thus, we can collect the following easy properties for the maps θe on X. +Lemma 6. Let X be a semigroup, e, f ∈ E(X) such that e ≤ f, and s(x, y) = (xy, θx(y)) +a solution on X. Then, the following hold: +1. θe(f) ∈ E(X), +2. θe(e) ≤ θe(f), +3. θf = θθe(f)θe. +Proof. At first, we have that θe(f) = θe(f)θef(f) = θe(f)2. Besides, +θe(e)θe(f) = θe(ef) = θe(e) = θe(f)θef(e) = θe(f)θe(e), +thus θe(e) ≤ θe(f). Moreover, θf = θθe(f)θef = θθe(f)θe, that is our claim. +Now, following [5, p. 22], given a semigroup X and e ∈ E(X), then e is a left identity +(or right identity) if ex = x (or xe = x), for every x ∈ X, and the sets +eX = {x ∈ X | ex = x} +Xe = {x ∈ X | xe = x} +are respectively the principal right and left ideals of X generated by e. Moreover, we set +eXe = eX ∩ Xe. We can check the following properties. +Lemma 7. Let X be a semigroup, e ∈ E(X), and s(x, y) = (xy, θx(y)) a solution on X. +1. If x ∈ Xe, then +a. θx(e) ∈ E(X), +b. θe = θθx(e)θx; +2. if x ∈ eX, then +c. θe(x) ∈ θe(e)X, +d. θx = θθe(x)θx. +Proof. Initially, assume that x ∈ Xe. Then, by (P1), θx(e) = θx(e)θxe(e) = θx(e)2 and, +by (P2), θe = θθx(e)θxe = θθx(e)θx. +Now, assume that x ∈ eX. Thus, using (P1), we have that θe(x) = θe(ex) = θe(e)θe(x). +Hence, since by Lemma 6-1. θe(e) ∈ E(X), we get θe(x) ∈ θe(e)X. Moreover, by (P2), +θx = θθe(x)θex = θθe(x)θx. +6 + +As a direct consequence of the previous lemma, we have the following properties for +arbitrary solutions defined on a monoid. +Lemma 8. Let M be a monoid with identity 1 and s(x, y) = (xy, θx(y)) a solution on +M. Then, the following hold: +1. θx(1) ∈ E(M), +2. θ1 = θθx(1)θx, +3. θ1(x) ∈ θ1(1)M, +4. θx = θθ1(x)θx, +for every x ∈ M. +In particular, it follows by Lemma 8-4. that the only non-degenerate solution on a +monoid M is that for which θx = idM, for every x ∈ M. +We conclude this section focusing on solutions on semigroups having central idem- +potents, i.e., it holds xe = ex, for all e ∈ E(X) and x ∈ X. Obviously, in this case, +Xe = eX. The result we provide is consistent with Lemma 11 and the equation (4) of +[3]. Let us first recall, that if e ∈ E(X), the set +He = {x ∈ Xe | ∃ y ∈ Xe +xy = yx = e} +is a group with identity e. In particular, if e and f are distinct idempotents, then He +and Hf are disjoint. If x ∈ He, let us denote by x− the inverse of x in He. +Proposition 9. Let X be a semigroup having central idempotents, e ∈ E(X), x ∈ He, +and s(x, y) = (xy, θx(y)) a solution on X. Then, the following hold: +1. θe(e) ≤ θx(e); +2. θe(x) ∈ Hθe(e) and in particular θe(x)− = θx (x−); +3. if f ∈ E(X) is such that f ≤ e and y ∈ Hf, then θx(y) = θe (x)− θe(xy). +Proof. At first, we have that, by Lemma 6-1., θe(e) ∈ E(X), and, by Lemma 7-a., it +holds that θx(e) ∈ E(X). Thus, +θe(e)θx(e) = θe +� +xx−� +θx(e) = θe(x)θex +� +x−� +θx(e) = θe(x)θex(e)θex +� +x−� += θe(xe)θex +� +x−� += θe(x)θex +� +x−� += θe +� +xx−� += θe(e), +and so 1. follows. Besides, by Lemma 7-c., θe(x) ∈ Xθe(e) and also θx (x−) ∈ Xθe(e), +since θx (x−) θe(e) = θx (x−e) = θx (x−). Hence, we get +θe(x)θx +� +x−� += θe(x)θex +� +x−� += θe +� +xx−� += θe(e) +and +θx +� +x−� +θe(x) = θx +� +x−� +θe(x)θe(e) +θx (x−) ∈ Xθe(e) += θx +� +x−x +� +θe(e) += θx(e)θe(e) += θe(e) +by 1. +7 + +Finally, if f ∈ E(X) is such that f ≤ e and y ∈ Hf, then, by 2., we obtain +θx(y) = θx (efy) = θx +� +xx−fy +� += θx +� +x−� +θe(xy) = θe (x)− θe(xy), +which completes the proof. +3. Properties of the maps θx of idempotent solutions +In this section, we collect some properties of the maps θx of idempotent solutions on +arbitrary semigroups. +Initially, it is a routine computation to check that a solution s(x, y) = (xy, θx(y)) on +a semigroup X is idempotent if and only if +xyθx(y) = xy, +(I1) +θxyθx(y) = θx(y), +(I2) +for all x, y ∈ X. In particular, by (I1), if X is a group the unique idempotent solution +on X is the map s(x, y) = (xy, 1); such a solution s belongs to the class of solutions +discussed in Examples 2-1.. On the other hand, considering this class of solutions on +monoids, one can easily check the following result. +Proposition 10. Let M be a monoid and γ : M → M a map. Then, s(x, y) = (xy, γ(y)) +is an idempotent solution on M if and only if γ ∈ End(M), γ2 = γ, and xγ(x) = x, for +every x ∈ M. +By the previous proposition, in particular, the solution s(x, y) = (xy, y) on M is idem- +potent if and only if M is an idempotent monoid. +However, taking monoids even of small orders, one can note that among the solutions +there are several of the idempotent type that do not belong to the class of solutions in +Examples 2-1.. The following is an easy example. +Example 3. [16, Appendix B] Let X = {1, a, b} and M be the commutative monoid on +X with identity 1 and multiplication given by a2 = a, ab = a, b2 = 1. Then, there are +three idempotent solutions: +1. s(x, y) = (xy, 1); +2. r(x, y) = (xy, γ(y)), with γ : M → M defined by γ(1) = γ(b) = 1 and γ(a) = a; +3. t(x, y) = (xy, θx(y)), with θx : M → M the map given by θx(1) = 1, θx(a) = a, for +every x ∈ X, and θ1(b) = θb(b) = 1 and θa(b) = b. +Based on the above arguments, in the next, we will focus on idempotent solutions. +We first prove the following properties which hold, in general, for idempotent solutions +on arbitrary semigroups. +8 + +Proposition 11. Let X be a semigroup and s(x, y) = (xy, θx(y)) an idempotent solution +on X. Then, the following hold: +1. θθx(y) = θy, +2. θy = θyθxy, +3. θx(yz) = θx(yz)θy(z), +for all x, y, z ∈ X. +Proof. Let x, y, z ∈ X. Then, we have +θθx(y) = θθxyθx(y)θxyθx(y) +by (P2) += θθx(y)θxy +by (I2)-(I1) += θy +by (P2), +hence 1. is satisfied. Thus, by (P2), θy = θθx(y)θxy = θyθxy. Finally, +θx(yz) = θx(y)θxy(z) +by (P1) += θx(y)θxy(z)θθx(y)θxy(z) +by (I1) += θx(yz)θy(z) +by (P1)-(P2) +and the claim follows. +Proposition 12. Let X be a semigroup, e ∈ E(X), and s(x, y) = (xy, θx(y)) an idem- +potent solution on X. +1. If x ∈ Xe, then +a. x ∈ Xθx(e), +b. ∀y ∈ X +θy(x) ∈ Xθx(e), +c. θe = θeθx. +2. If x ∈ eS, then +d. θe(x) ∈ E(X), +e. x ∈ X θe(x), +f. ∀y ∈ X +θy(x) ∈ Xθe(x), +g. θx is an idempotent map. +Proof. Initially, assume that x ∈ Xe. Then, by Lemma 7-a., θx(e) ∈ E(X). Moreover, +by (I1), we get x = xe = xeθx(e) = xθx(e). Besides, if y ∈ X, by Proposition 11-3., we +have that +θy(x) = θy(xe) = θy(xe)θx(e) = θy(x)θx(e), +and so b. follows. Finally, by Proposition 11-2. , θe = θeθxe = θeθx, i.e., c. holds. +Now, suppose that x ∈ eX. At first, by (I1), we obtain that x = ex = exθe(x) = xθe(x) +and so +θe(x) = θe (xθe (x)) = θe(x)θexθe(x) = θe(x)2, +9 + +where in the last equality we apply (I2). Thus, d. and e. follow. Moreover, +θx = θθe(x) +by Proposition 11-1. += θθexθe(x)θexθe(x) +by (P2) += θθe(x)θx +by (I2)-(I1) += θxθx +by Proposition 11-1. +hence θx is an idempotent map. Finally, if y ∈ X, by Proposition 11-3. , we have that +θy(x) = θy(ex) = θy(ex)θe(x) = θy(x)θe(x). +Therefore, the claim follows. +As a consequence of the previous proposition, we obtain the following result. +Corollary 13. Let X be a semigroup, e ∈ E(X), x ∈ eXe, and s(x, y) = (xy, θx(y)) an +idempotent solution on X. Then, the following hold: +1. θe(x) ∈ E(X), +2. x ∈ Xθe(x) ∩ Xθx(e), +3. θe = θeθx, +4. ∀y ∈ X +θy(x) ∈ Xθe(x) ∩ Xθx(e), +5. θx is idempotent. +Remark 1. Note that if s is an idempotent solution on a monoid M with identity 1, +in general, θ1(E(M)) ̸= E(M). Indeed, if we consider the set X = {1, a, b} and the +commutative monoid M on X with identity 1, E(M) = {1, a}, and such that ab = a, we +have that the only idempotent solution is s(x, y) = (xy, 1). +4. Idempotent solutions on monoids having central idempotents +In this section, we focus only on idempotent solutions defined on monoids. In particular, +we will give a description theorem for idempotent solutions defined on monoids having +central idempotents. In addition, we will show that specific idempotent solutions are +strictly linked to the kernel congruence of an idempotent monoid homomorphism. +Initially, given an idempotent solution s on M, all the statements of Corollary 13 +hold for the identity 1. In particular, θ1(x) ∈ E(M) and x ∈ Mθ1(x), for every x ∈ M. +As a consequence, we have the following: +Proposition 14. Let M be a cancellative monoid. Then, s(x, y) = (xy, 1) is the unique +idempotent solution on M. +10 + +Proof. Since the identity is the unique idempotent of M, then θ1(x) = 1, for every +x ∈ M. Moreover, by Proposition 11-1., θx = θθ1(x), for every x ∈ M, and so the claim +follows. +Thus, from now on we will consider not cancellative monoids. We have the following +properties. +Proposition 15. If M is a monoid and s(x, y) = (xy, θx(y)) an idempotent solution on +M, then +1. θ1(1) = 1, +2. θx = θθ1(x), +3. θ1(x) ≤ θx(1), +4. θx is idempotent, +for every x ∈ M. +Proof. The first statement directly follows by (I1). The second one follows by Propo- +sition 11-1. and the fourth one by g. in Proposition 12. +Moreover, if x ∈ M, then +θ1(x) = θ1(x1) = θ1(x)θx(1). On the other hand, by (I2), it holds θxθ1(x) = θ1(x), and +so we obtain +θx(1)θ1(x) = θx(1)θxθ1(x) = θx(1θ1(x)) = θxθ1(x) = θ1(x), +i.e., θ1(x) ≤ θx(1). +Given a monoid M, recall that a right unit is an element r of M for which there exists +r′ ∈ M such that rr′ = 1. Analogously, l ∈ M is a left unit of M if there exists a left +inverse l′ ∈ M such that l′l = 1. Next, we prove some properties that hold for any right +unit of a monoid M and that can be shown also for any left unit l ∈ M (exchanging the +roles of r and l′ and of r′ and l, respectively). +Proposition 16. Let M be a monoid and s(x, y) = (xy, θx(y)) an idempotent solution +on M. If r ∈ M is a right unit of M, then the following hold: +1. θr (r′) = 1, where r′ ∈ M is such that rr′ = 1, +2. θ1(r) = 1, +3. θ1 = θr. +Proof. The equality θr (r′) = 1 follows by setting x = r and y = r′ in (I1). +As a +consequence, we have that θ1(r) = θ1(r) · 1 = θ1(r)θr (r′) = θ1 (rr′) = θ1(1) = 1. +Moreover, by Lemma 8-4., it follows that θr = θθ1(r) = θ1. +In general, it is not true that θx (M ×) ⊆ M ×, where M X is the group of units of a +monoid M, as we show in the next example. +11 + +Example 4. [16, Appendix B] Let us consider the set X = {1, a, b} and the idempotent +commutative monoid M on X with identity 1 and such that ab = b. Clearly, M X = {1}. +Then, there exists an idempotent solution on M for which θb(1) = a. Such a solution is +defined considering the maps θ1 = θa : M → M given by θ1(1) = θa(1) = 1 and θ1(b) = b +and the map θb : M → M given by θb(1) = θb(a) = a and θb(b) = b. +In the last part of this section, we will focus on idempotent solutions on monoids M +for which θ1 is also a homomorphism from M to E(M). This assumption is not restrictive, +as we show in the next result. Indeed, it is a necessary condition for idempotent solutions +defined on monoids in which the idempotents are central. In the following, let us denote +by Z(M) the center of M. +Proposition 17. Let M be a monoid such that E(M) ⊆ Z(M) and s(x, y) = (xy, θx(y)) +an idempotent solution on M. Then, the map θ1 is an idempotent monoid homomorphism +from M to E(M). +Proof. Initially, by Proposition 15-4., the map θ1 is idempotent. Besides, recalling that +θ1(x) ∈ E(M), for any x ∈ M, we have that +θ1(xy) = θ1(x)θx(y)θ1(x) +by Corollary 13-1. += θ1θ1(x)θθ1(x)(y)θ1(x) +by Proposition 11-1. += θ1 (θ1(x)y) θ1(x) +by (P1) += θ1 (yθ1(x)) θ1(x) += θ1(y)θyθ1(x)θ1(x) +by (P1) += θ1(y)θ1(x)θyθ1(x) += θ1(y)θ1(x)θθ1(y)θ1(x) +by Proposition 11-1. += θ1(y)θ1(x) +by (I1) +for all x, y ∈ M. Finally, by Proposition 15-1., it holds that θ1(1) = 1, hence the claim +follows. +Note that the converse of Proposition 17 is not true. Indeed, the map s(x, y) = (xy, 1) +is a solution in any monoid. +Lemma 18. Let M be a monoid such that E(M) ⊆ Z(M) and s(x, y) = (xy, θx(y)) a +solution on M for which θ1 is an idempotent monoid homomorphism from M to E(M) +such that x = xθ1(x), for every x ∈ M. Then, s is idempotent if and only if (I2) is +satisfied. +Proof. It is enough to notice that (I1) holds since, by (P1), +xy = xθ1(x)yθ1(y) = xyθ1(xy) = xyθ1(x)θx(y) = xθ1(x)yθx(y) = xyθx(y), +for all x, y ∈ M. +12 + +The next result is a description of all idempotent solutions on a monoid M having +central idempotents. +Theorem 19. Let M be a monoid for which E(M) ⊆ Z(M) and µ an idempotent monoid +homomorphism from M to E(M) such that, for every x ∈ M, µ(x) = ex, with ex ∈ E(M) +a right identity for x. Moreover, let {θe : M → M | e ∈ Im µ} be a family of maps such +that θ1 = µ, +θe(xy) = θe(x)θf(y), +(4) +for all x, y ∈ M and e ∈ Im µ, with f = µ(ex), and +θe = θeθef, +(5) +for all e, f ∈ Im µ, and +θefθe(x) = θe(x), +(6) +for all x ∈ M, e ∈ Im µ, with f = µ(x). Then, set θx = θµ(x), for every x ∈ M, one +has that the map s : M × M → M × M given by s(x, y) = (xy, θx(y)) is an idempotent +solution on M. Conversely, every idempotent solution on M can be so constructed. +Proof. Let x, y, z ∈ M. Then, using (4) we obtain (P1), since +θx(y)θxy(z) = θµ(x)(y)θµ(µ(x)y)(z) = θµ(x)(yz) = θx(yz). +Moreover, +θθx(y)θxy(z) = θµθx(y)θµ(x)µ(y)(z) += θθ1θx(y)θθ1(x)θ1(y)(z) += θθ1(y)θθ1(x)θ1(y)(z) +by (5) since θ1 = θ1θx += θy(z) +by (5) +and so (P2) is satisfied. Thus, by Lemma 18, (I1) holds. Finally, applying (6), we get +θxyθx(y) = θµ(x)µ(y)θµ(x)(y) = θµ(x)(y) = θx(y). +Therefore, s(x, y) = (xy, θx(y)) is an idempotent solution on M. +Vice versa, if we assume that s(x, y) = (xy, θx(y)) is an idempotent solution on M, then +by Proposition 17, µ = θ1 is an idempotent monoid homomorphism, and, by Proposi- +tion 12-2.(e), we have that x ∈ Mθ1(x), for every x ∈ M. In addition, by Proposition 15- +2., θx = θθ1(x), for every x ∈ M. Hence, by(P1), we obtain (4). Now, let e, f ∈ Im θ1, +thus there exist x, y ∈ M such that e = θ1(x) and f = θ1(y). Besides, by (P2), +θe = θθ1(x) = θx = θθy(x)θyx = θθ1(x)θθ1(y)θ1(x) = θeθfe, +and so (6) holds. Finally, by (I2), if e ∈ Im θ1, x ∈ M, and f = θ1(x), we obtain +θefθe(x) = θθ1(xy)θθ1(x)(y) = θxyθx(y) = θx(y) = θθ1(x)(y) = θe(y), +for every y ∈ M, hence (6) holds. +13 + +Remark 2. Unlike solutions defined on groups, in the case of idempotent solutions on +monoids, it is not possible to find a way to write the maps θx by means of the map θ1 +as in Theorem 2. Indeed, if we look at the idempotent solutions on the monoid M in +Example 4, one can see that there are three different solutions having the same map +θ1 : M → M given by θ1(1) = θa(1) = 1 and θ1(b) = b. +The next result narrows down that choice of the maps θe in Theorem 19. Indeed, +given an idempotent solution on a monoid M such that θ1 is a monoid homomorphism +from M to E(M), one has that the kernel of θ1, i.e., the set +ker θ1 = {(x, y) ∈ M × M | θ1(x) = θ1(y)} +is a congruence relation on M. Thus, one can naturally consider the quotient monoid +M/ ker θ1 (see [20] for more details). Additionally, we have the following properties. +Theorem 20. Let M be a monoid and s(x, y) = (xy, θx(y)) an idempotent solution on +M such that θ1 is a monoid homomorphism from M to E(M). Then, +1. θ1(M) is a system of representatives of M/ ker θ1 that contains the identity; +2. (θx(y), y) ∈ ker θ1, for all x, y ∈ M. +Proof. The first part is a consequence of the idempotence of the map θ1 by 4. in Propo- +sition 15. In fact, Proposition 15-1., we have that 1 = θ1(1) ∈ θ1(M). Moreover, since +by Proposition 15-4. θ1 is an idempotent map, we easily obtain that (θ1(x), x) ∈ ker θ1, +for every x ∈ M. Besides, if x, y ∈ M are distinct and such that (θ1(y), x) ∈ ker θ1, then +we get θ1(y) = θ1(θ1(y)) = θ1(x). +The second part follows by Corollary 13-3., since θ1θx(y) = θ1(y), for all x, y ∈ M, and +so (θx(y), y) ∈ ker θ1. Therefore, we get the claim. +Corollary 21. Let M be a monoid for which E(M) ⊆ Z(M) and s(x, y) = (xy, θx(y)) +an idempotent solution on M. Then, θ1(M) is a system of representatives of M/ ker θ1 +that contains the identity and (θx(y), y) ∈ ker θ1. +Proof. It is a consequence of Theorem 20, since, by Proposition 17, the map θ1 is a +monoid homomorphism from M to E(M). +Remark 3. Let M be a monoid for which E(M) ⊆ Z(M) and s(x, y) = (xy, θx(y)) and +r(x, y) = (xy, ηx(y)) two idempotent solutions on M. Then, by (3), if such solutions are +isomorphic, there exists an isomorphism f of M such that fθ1 = η1f, i.e., f sends the +system of representatives θ1(M) of M/ ker θ1 into the other one η1f(M). +14 + +References +[1] S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualit´e pour les produits crois´es de C*-alg`ebres, +Ann. Sci. ´Ec. Norm. Sup. 26 (4) (1993) 425–488. +URL http://eudml.org/doc/82346 +[2] S. Baaj, G. Skandalis, Transformations pentagonales, C. R. Acad. Sci. Paris S´er. I Math. 327 (7) +(1998) 623–628. +URL https://doi.org/10.1016/S0764-4442(99)80090-1 +[3] F. Catino, M. Mazzotta, M. M. 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Monzo, Pre-compatible almost endomorphisms and semigroups whose cube is a band, +Semigroup Forum 67 (3) (2003) 355–372. +URL https://doi.org/10.1007/s00233-001-0004-y +15 + +[19] M. Petrich, Inverse semigroups, Pure and Applied Mathematics (New York), John Wiley & Sons, +Inc., New York, 1984, a Wiley-Interscience Publication. +[20] J. Rhodes, B. Tilson, The kernel of monoid morphisms, J. Pure Appl. Algebra 62 (3) (1989) 227– +268. +URL https://doi.org/10.1016/0022-4049(89)90137-0 +[21] D. Stanovsk´y, P. Vojtˇechovsk´y, Idempotent solutions of the Yang-Baxter equation and twisted +group division, Fund. Math. 255 (1) (2021) 51–68. +URL https://doi.org/10.4064/fm872-2-2021 +[22] S. L. Woronowicz, From multiplicative unitaries to quantum groups, Int. J. Math. 7 (01) (1996) +129–149. +URL https://doi.org/10.1142/S0129167X96000086 +[23] S. Zakrzewski, Poisson Lie groups and pentagonal transformations, Lett. Math. Phys. 24 (1) (1992) +13–19. +URL https://doi.org/10.1007/BF00429998 +[24] G. Zappa, Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra loro, in: +Atti Secondo Congresso Un. Mat. Ital., Bologna, 1940, Edizioni Cremonense, Rome, 1942, pp. +119–125. +16 + diff --git a/rNAzT4oBgHgl3EQfrf0Z/content/tmp_files/load_file.txt b/rNAzT4oBgHgl3EQfrf0Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb6ace9a8a56c4aec608f58f29147f5ec529521d --- /dev/null +++ b/rNAzT4oBgHgl3EQfrf0Z/content/tmp_files/load_file.txt @@ -0,0 +1,608 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf,len=607 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='01643v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='QA] 4 Jan 2023 Idempotent set-theoretical solutions of the pentagon equation⋆ Marzia MAZZOTTAa aDipartimento di Matematica e Fisica “Ennio De Giorgi” Universit`a del Salento Via Provinciale Lecce-Arnesano 73100 Lecce (Italy) Abstract A set-theoretical solution of the pentagon equation on a non-empty set X is a function s : X × X → X × X satisfying the relation s23 s13 s12 = s12 s23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' with s12 = s × idX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' s23 = idX × s and s13 = (idX × τ)s12(idX × τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' where τ : X ×X → X ×X is the flip map given by τ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' y) = (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' for all x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Writing a solution as s(x, y) = (xy, θx(y)), where θx : X → X is a map, for every x ∈ X, one has that X is a semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In this paper, we study idempotent solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', s2 = s, by showing that the idempo- tents of X have a key role in such an investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In particular, we describe all such solutions on monoids having central idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, we focus on idempotent so- lutions defined on monoids for which the map θ1 is a monoid homomorphism, by showing that they have to be derived considering the kernel congruence of the map θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Keywords: pentagon equation, set-theoretical solution, semigroup 2022 MSC: 16T25, 81R50, 20M99 Introduction If V is a vector space over a field F, a linear map S : V ⊗ V → V ⊗ V is said to be a solution of the pentagon equation on V if it satisfies the relation S12S13S23 = S23S12, (1) where S12 = S ⊗ idV , S23 = idV ⊗ S, S13 = (idV ⊗ Σ) S12 (idV ⊗ Σ), with Σ the flip operator on V ⊗ V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', Σ(u ⊗ v) = v ⊗ u, for all u, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The pentagon equation classically originates from the field of Mathematical Physics, but it has several appli- cations and appears in different areas of mathematics, also with different terminologies (see, for instance, [23, 1, 22, 2, 17, 11, 14, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' To know more about contexts in which ⋆This work was partially supported by the Dipartimento di Matematica e Fisica “Ennio De Giorgi” - Universit`a del Salento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The author is member of GNSAGA (INdAM) and of the non-profit association ADV-AGTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Email address: marzia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='mazzotta@unisalento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='it (Marzia MAZZOTTA) January 5, 2023 the pentagon equation appears, we refer to the introduction of the paper by Dimakis and M¨uller-Hoissen [8] along with the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In 1998, Kashaev and Sergeev [13] explicitly first highlighted an existing link between solutions on a vector space viewed as the space F X of all functions from a finite set X to F and maps from X × X into itself satisfying a certain relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Into the specific, to any map s : X × X → X × X one can associate a linear operator S : F X×X → F X×X defined as S(f)(x, y) = f(s(x, y)), for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If the map s satisfies the “reversed” pentagon relation s23s13s12 = s12s23, (2) where s12 = s×idX, s23 = idX ×s, s13 = (idX ×τ) s12 (idX ×τ), with τ(x, y) = (y, x), for all x, y ∈ X, then the linear map S is a solution of the pentagon equation on F X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' We call the map s as above set-theoretical solution of the pentagon equation, or briefly solution, on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In the pioneering paper [13], one can also find the first systematic way to obtain solutions on closed under multiplications subsets of arbitrary groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' However, such set-theoretical maps had already appeared before with the terminology transformations pentagonales in two non purely algebraic papers: those by Zakrzewski [23] and Baaj and Skandalis [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In these papers, we can find the first instances of bijective solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Attention only to set-theoretical solutions has been recently given in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Following the notation therein, writing a solution s : X × X → X × X as s(x, y) = (xy, θx(y)), where θx is a map from X into itself, for every x ∈ X, one has that X is a semigroup and θx(y)θxy(z) = θx(yz), (P1) θθx(y)θx y = θy, (P2) for all x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In [3, Theorem 15], a complete description of all solutions de- fined on a group is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In this case, it holds that θx(y) = θ1(x)−1θ1(xy), for all x, y ∈ X, where 1 is the identity of the group X, and it is sufficient to study the set ker θ1 = {x ∈ X | θ1(x) = 1} that it a normal subgroup of X, even if, in general, θ1 is not a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' However, describing all solutions on arbitrary semigroups seems to be very difficult since there are many even in the case of small-order semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' For instance, it is sufficient to look [16, Appendix B], where, as suggested by Rump, all the non-isomorphic solutions on semigroups of order 3 have been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' A first step could be studying specific classes of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In this regard, a characteriza- tion of all involutive solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', s2 = idX×X, has been provided by Colazzo, Jespers, and Kubat in their recent paper [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, in [4], Catino, Mazzotta, and Stefanelli studied the pentagon and the quantum Yang-Baxter equations (see [9]) for some similar- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Indeed, there exist several maps defined on particular semigroups that satisfy both equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In this last paper, one can also find some methods to construct solutions of the pentagon equation such as, for instance, on the matched product of two semigroups, that is a semigroup including the classical Zappa product [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Recently, idempotent left non-degenerate solutions of the Yang-Baxter equation have been completely described (see [15, 21, 7] and the classifications therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' A similar study could also be done for the pentagon equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 2 In this paper, we deal with idempotent set-theoretical solutions of the pentagon equa- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', s2 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' One can easily check that a solution s(x, y) = (xy, θx(y)) on a semigroup X is idempotent if and only if xyθx(y) = xy, (I1) θxyθx(y) = θx(y), (I2) for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' We show that the idempotents of X have a crucial role in finding them and, for that, we exhibit some useful properties of the maps θx involving the idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In particular, we focus on solutions defined on monoids, by showing that, unlike solutions defined on groups, it is not possible to find a way to write the maps θx by means of the map θ1, where 1 is the identity of the monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' We provide a description theorem for idempotent solutions on monoids M having central idempotents, namely, E(M) ⊆ Z(M), by illustrating that it is enough to construct specific idempotent maps θe, for every e ∈ E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Furthermore, in this situation, the map θ1 is an idempotent monoid homomorphism, and, for this reason, we deepen idempotent solutions on monoids satisfying this additional property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In this case, all the maps θx have to be derived considering the kernel congruence of the function θ1, namely, the set ker θ1 = {(x, y) ∈ M × M | θ1(x) = θ1(y)} (see [20] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Indeed, (θx(y), y) ∈ ker θ1, for all x, y ∈ M, and θ1(M) is a system of representatives of M/ ker θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Finally, we collect some properties of idempotent solutions on arbitrary semigroups, that could be useful in a future study of these maps in the more general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Basics on solutions In this section, we give some basics on the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, we introduce some classes of solutions and provide several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' From now on, following the notation used in [3, Proposition 8], given a semigroup X, we will briefly call solution on X any map s : X × X → X × X given by s(x, y) = (xy, θx(y)), where θx is a map from X into itself, satisfying (P1) and (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' [17]) Let X be a set and f, g : X → X idempotent maps such that fg = gf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Set x · y = f(x), for all x, y ∈ X, one has that (X, ·) is a semigroup and the map s(x, y) = (x · y, g (y)) is a solution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Note that the previous example belongs to the class of P-QYBE solutions, namely the maps that are solutions both to the pentagon and the Yang-Baxter equations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let (X, ·) and (Y, ∗) be two semigroups and s(x, y) = (x · y, θx(y)) and r(a, b) = (a ∗ b, ηa(b)) two solutions on X and Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, s and r are isomorphic if there exists an isomorphism f : X → Y such that fθx(y) = ηf(x)f(y), (3) for all x, y ∈ X, or, equivalently, (f × f)s = r(f × f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 3 A complete description in the case of a group is given in [3, Theorem 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' For the sake of completeness, we recall such a result below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let G be a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Consider a normal subgroup K of G and a system of representatives R of G/K such that 1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If µ : G → R is a map such that µ(x) ∈ Kx, for every x ∈ G, then the map s(x, y) = � xy, µ (x)−1 µ (xy) � , for all x, y ∈ G, is a solution on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Conversely, if s is a solution on G, then the set K = {x ∈ G | θ1(x) = 1} is a normal subgroup of G for which Im θ1 is a system of representatives of G/K that contains 1, θ1(x) ∈ Kx, for every x ∈ G, and s(x, y) = � xy, θ1 (x)−1 θ1 (xy) � , for all x, y ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' By Theorem 2 and making explicit the condition (3), it is easy to check that two solutions s(x, y) = (xy, θx(y)) and r(x, y) = (xy, ηx (y)) on the same group G are isomorphic via f ∈ Aut(G) if and only if fθ1 = η1f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', f sends the system of representatives θ1(G) into the other one η1f(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' However, describing all the solutions, up to isomorphisms, on arbitrary semigroups turns out to be very hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Indeed, even in the case of semigroups of small order, there are a lot of solutions, as one can see in [16, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' A first step could be studying specific classes of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In this regard, one can find in [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='5] a complete description of all involutive solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let X be a semigroup and s(x, y) = (xy, θx(y)) a solution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' We say that the map s is non-degenerate if θx is bijective, for every x ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' involutive if s2 = idX×X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' idempotent if s2 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' [3, Examples 2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='] Let X be a semigroup and γ : X → X a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the map s(x, y) = (xy, γ (y)) , for all x, y ∈ X, is a solution if and only if γ ∈ End(X) and γ2 = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' One can easily check that such a solution s is non-degenerate if and only if γ = idX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' As a particular case of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', if X is a semigroup and e ∈ E(X), where E(X) denotes the set of the idempotents of X, the map s(x, y) = (xy, e) , for all x, y ∈ X, is an idempotent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let X be a semigroup belonging to the variety S := [abc = bc] (see [18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 370]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the map s (x, y) = (xy, xy) , for all x, y ∈ X, is an idempotent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Every Clifford semigroup X gives rise to the idempotent solution s given by s(x, y) = � xy, y−1y � , for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Recall that a Clifford semigroup X is a semigroup in which every x ∈ X admits a unique x−1 ∈ X such that xx−1x = x, x−1xx−1 = x−1, and xx−1 = x−1x (see [19, Exercise II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' [16, Appendix B] Let X = {0, a, b} and S the null semigroup on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', xy = 0, for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Consider the maps θ0 = idS and θa = θb such that θa(0) = 0, θa(a) = b, and θa(b) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the map s(x, y) = (0, θx(y)) is an idempotent and non-degenerate solution on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Other classes of solutions that can be studied are the commutative and the cocom- mutative ones (see [3, Definition 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' These kinds of solutions are in analogy to the commutative and the cocommutative multiplicative unitary operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', solutions of (1) defined on Hilbert spaces (see [1, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' A solution s : X × X → X × X is said to be commutative if s12s13 = s13s12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' cocommutative if s13s23 = s23s13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If X is a semigroup and s(x, y) = (xy, θx(y)) a solution on X, it is a routine computation to check that the map s is commutative if and only if xzy = xyz (C1) θx = θxy (C2) for all x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Instead, s is cocommutative if and only if xθy(z) = xz (CC1) θxθy = θyθx (CC2) for all x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' There exist solutions that are both commutative and cocommutative, such as the maps in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, according to [6, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='4], if s is an involutive solution, then s is both commutative and cocommutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Clearly, if M is a monoid, it follows by (CC1) that the unique cocommutative solution on M is given by s(x, y) = (xy, y), for all x, y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In the next result, we describe all the commutative solutions on a monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, a solution s(x, y) = (xy, θx(y)) on M is commutative if and only if M is a commutative monoid and θx = γ, with γ ∈ End(M), γ2 = γ, for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Initially, if the monoid M is commutative, then the map s(x, y) = (xy, γ (y)), with γ ∈ End(M), γ2 = γ, is a commutative solution on M (see Examples 2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Conversely, let us assume that s(x, y) = (xy, θx(y)) is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, by substitut- ing x = 1 in (C1), the monoid M is commutative and, by (C2), θ1 = θy, for every y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, by (P1) and (P2) the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Properties of the maps θx involving the idempotents In this section, we provide some properties of the maps θx which involve the idempotents of arbitrary semigroups and that will be used in the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Firstly, according to [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 69], among the idempotents in any semigroup X, there is a natural partial order relation by the rule that ∀ e, f ∈ E(X) e ≤ f ⇐⇒ ef = fe = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, we can collect the following easy properties for the maps θe on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let X be a semigroup, e, f ∈ E(X) such that e ≤ f, and s(x, y) = (xy, θx(y)) a solution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe(f) ∈ E(X), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe(e) ≤ θe(f), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θf = θθe(f)θe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' At first, we have that θe(f) = θe(f)θef(f) = θe(f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Besides, θe(e)θe(f) = θe(ef) = θe(e) = θe(f)θef(e) = θe(f)θe(e), thus θe(e) ≤ θe(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, θf = θθe(f)θef = θθe(f)θe, that is our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Now, following [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 22], given a semigroup X and e ∈ E(X), then e is a left identity (or right identity) if ex = x (or xe = x), for every x ∈ X, and the sets eX = {x ∈ X | ex = x} Xe = {x ∈ X | xe = x} are respectively the principal right and left ideals of X generated by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, we set eXe = eX ∩ Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' We can check the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let X be a semigroup, e ∈ E(X), and s(x, y) = (xy, θx(y)) a solution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If x ∈ Xe, then a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx(e) ∈ E(X), b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe = θθx(e)θx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' if x ∈ eX, then c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe(x) ∈ θe(e)X, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx = θθe(x)θx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Initially, assume that x ∈ Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, by (P1), θx(e) = θx(e)θxe(e) = θx(e)2 and, by (P2), θe = θθx(e)θxe = θθx(e)θx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Now, assume that x ∈ eX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, using (P1), we have that θe(x) = θe(ex) = θe(e)θe(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Hence, since by Lemma 6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe(e) ∈ E(X), we get θe(x) ∈ θe(e)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, by (P2), θx = θθe(x)θex = θθe(x)θx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 6 As a direct consequence of the previous lemma, we have the following properties for arbitrary solutions defined on a monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid with identity 1 and s(x, y) = (xy, θx(y)) a solution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx(1) ∈ E(M), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θ1 = θθx(1)θx, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θ1(x) ∈ θ1(1)M, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx = θθ1(x)θx, for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In particular, it follows by Lemma 8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' that the only non-degenerate solution on a monoid M is that for which θx = idM, for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' We conclude this section focusing on solutions on semigroups having central idem- potents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', it holds xe = ex, for all e ∈ E(X) and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Obviously, in this case, Xe = eX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The result we provide is consistent with Lemma 11 and the equation (4) of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let us first recall, that if e ∈ E(X), the set He = {x ∈ Xe | ∃ y ∈ Xe xy = yx = e} is a group with identity e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In particular, if e and f are distinct idempotents, then He and Hf are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If x ∈ He, let us denote by x− the inverse of x in He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let X be a semigroup having central idempotents, e ∈ E(X), x ∈ He, and s(x, y) = (xy, θx(y)) a solution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe(e) ≤ θx(e);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe(x) ∈ Hθe(e) and in particular θe(x)− = θx (x−);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' if f ∈ E(X) is such that f ≤ e and y ∈ Hf, then θx(y) = θe (x)− θe(xy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' At first, we have that, by Lemma 6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', θe(e) ∈ E(X), and, by Lemma 7-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', it holds that θx(e) ∈ E(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, θe(e)θx(e) = θe � xx−� θx(e) = θe(x)θex � x−� θx(e) = θe(x)θex(e)θex � x−� = θe(xe)θex � x−� = θe(x)θex � x−� = θe � xx−� = θe(e), and so 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Besides, by Lemma 7-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', θe(x) ∈ Xθe(e) and also θx (x−) ∈ Xθe(e), since θx (x−) θe(e) = θx (x−e) = θx (x−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Hence, we get θe(x)θx � x−� = θe(x)θex � x−� = θe � xx−� = θe(e) and θx � x−� θe(x) = θx � x−� θe(x)θe(e) θx (x−) ∈ Xθe(e) = θx � x−x � θe(e) = θx(e)θe(e) = θe(e) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 7 Finally, if f ∈ E(X) is such that f ≤ e and y ∈ Hf, then, by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', we obtain θx(y) = θx (efy) = θx � xx−fy � = θx � x−� θe(xy) = θe (x)− θe(xy), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Properties of the maps θx of idempotent solutions In this section, we collect some properties of the maps θx of idempotent solutions on arbitrary semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Initially, it is a routine computation to check that a solution s(x, y) = (xy, θx(y)) on a semigroup X is idempotent if and only if xyθx(y) = xy, (I1) θxyθx(y) = θx(y), (I2) for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In particular, by (I1), if X is a group the unique idempotent solution on X is the map s(x, y) = (xy, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' such a solution s belongs to the class of solutions discussed in Examples 2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='. On the other hand, considering this class of solutions on monoids, one can easily check the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid and γ : M → M a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, s(x, y) = (xy, γ(y)) is an idempotent solution on M if and only if γ ∈ End(M), γ2 = γ, and xγ(x) = x, for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' By the previous proposition, in particular, the solution s(x, y) = (xy, y) on M is idem- potent if and only if M is an idempotent monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' However, taking monoids even of small orders, one can note that among the solutions there are several of the idempotent type that do not belong to the class of solutions in Examples 2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='. The following is an easy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' [16, Appendix B] Let X = {1, a, b} and M be the commutative monoid on X with identity 1 and multiplication given by a2 = a, ab = a, b2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, there are three idempotent solutions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' s(x, y) = (xy, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' r(x, y) = (xy, γ(y)), with γ : M → M defined by γ(1) = γ(b) = 1 and γ(a) = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' t(x, y) = (xy, θx(y)), with θx : M → M the map given by θx(1) = 1, θx(a) = a, for every x ∈ X, and θ1(b) = θb(b) = 1 and θa(b) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Based on the above arguments, in the next, we will focus on idempotent solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' We first prove the following properties which hold, in general, for idempotent solutions on arbitrary semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 8 Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let X be a semigroup and s(x, y) = (xy, θx(y)) an idempotent solution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θθx(y) = θy, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θy = θyθxy, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx(yz) = θx(yz)θy(z), for all x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, we have θθx(y) = θθxyθx(y)θxyθx(y) by (P2) = θθx(y)θxy by (I2)-(I1) = θy by (P2), hence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, by (P2), θy = θθx(y)θxy = θyθxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Finally, θx(yz) = θx(y)θxy(z) by (P1) = θx(y)θxy(z)θθx(y)θxy(z) by (I1) = θx(yz)θy(z) by (P1)-(P2) and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let X be a semigroup, e ∈ E(X), and s(x, y) = (xy, θx(y)) an idem- potent solution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If x ∈ Xe, then a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' x ∈ Xθx(e), b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' ∀y ∈ X θy(x) ∈ Xθx(e), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe = θeθx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If x ∈ eS, then d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe(x) ∈ E(X), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' x ∈ X θe(x), f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' ∀y ∈ X θy(x) ∈ Xθe(x), g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx is an idempotent map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Initially, assume that x ∈ Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, by Lemma 7-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', θx(e) ∈ E(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, by (I1), we get x = xe = xeθx(e) = xθx(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Besides, if y ∈ X, by Proposition 11-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', we have that θy(x) = θy(xe) = θy(xe)θx(e) = θy(x)θx(e), and so b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Finally, by Proposition 11-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' , θe = θeθxe = θeθx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Now, suppose that x ∈ eX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' At first, by (I1), we obtain that x = ex = exθe(x) = xθe(x) and so θe(x) = θe (xθe (x)) = θe(x)θexθe(x) = θe(x)2, 9 where in the last equality we apply (I2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, θx = θθe(x) by Proposition 11-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' = θθexθe(x)θexθe(x) by (P2) = θθe(x)θx by (I2)-(I1) = θxθx by Proposition 11-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' hence θx is an idempotent map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Finally, if y ∈ X, by Proposition 11-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' , we have that θy(x) = θy(ex) = θy(ex)θe(x) = θy(x)θe(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Therefore, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' As a consequence of the previous proposition, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let X be a semigroup, e ∈ E(X), x ∈ eXe, and s(x, y) = (xy, θx(y)) an idempotent solution on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe(x) ∈ E(X), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' x ∈ Xθe(x) ∩ Xθx(e), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θe = θeθx, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' ∀y ∈ X θy(x) ∈ Xθe(x) ∩ Xθx(e), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx is idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Note that if s is an idempotent solution on a monoid M with identity 1, in general, θ1(E(M)) ̸= E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Indeed, if we consider the set X = {1, a, b} and the commutative monoid M on X with identity 1, E(M) = {1, a}, and such that ab = a, we have that the only idempotent solution is s(x, y) = (xy, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Idempotent solutions on monoids having central idempotents In this section, we focus only on idempotent solutions defined on monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In particular, we will give a description theorem for idempotent solutions defined on monoids having central idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In addition, we will show that specific idempotent solutions are strictly linked to the kernel congruence of an idempotent monoid homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Initially, given an idempotent solution s on M, all the statements of Corollary 13 hold for the identity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In particular, θ1(x) ∈ E(M) and x ∈ Mθ1(x), for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' As a consequence, we have the following: Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a cancellative monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, s(x, y) = (xy, 1) is the unique idempotent solution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Since the identity is the unique idempotent of M, then θ1(x) = 1, for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, by Proposition 11-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', θx = θθ1(x), for every x ∈ M, and so the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, from now on we will consider not cancellative monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' We have the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If M is a monoid and s(x, y) = (xy, θx(y)) an idempotent solution on M, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θ1(1) = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx = θθ1(x), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θ1(x) ≤ θx(1), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θx is idempotent, for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The first statement directly follows by (I1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The second one follows by Propo- sition 11-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' and the fourth one by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, if x ∈ M, then θ1(x) = θ1(x1) = θ1(x)θx(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' On the other hand, by (I2), it holds θxθ1(x) = θ1(x), and so we obtain θx(1)θ1(x) = θx(1)θxθ1(x) = θx(1θ1(x)) = θxθ1(x) = θ1(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', θ1(x) ≤ θx(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Given a monoid M, recall that a right unit is an element r of M for which there exists r′ ∈ M such that rr′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Analogously, l ∈ M is a left unit of M if there exists a left inverse l′ ∈ M such that l′l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Next, we prove some properties that hold for any right unit of a monoid M and that can be shown also for any left unit l ∈ M (exchanging the roles of r and l′ and of r′ and l, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid and s(x, y) = (xy, θx(y)) an idempotent solution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' If r ∈ M is a right unit of M, then the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θr (r′) = 1, where r′ ∈ M is such that rr′ = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θ1(r) = 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θ1 = θr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The equality θr (r′) = 1 follows by setting x = r and y = r′ in (I1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' As a consequence, we have that θ1(r) = θ1(r) · 1 = θ1(r)θr (r′) = θ1 (rr′) = θ1(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, by Lemma 8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', it follows that θr = θθ1(r) = θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In general, it is not true that θx (M ×) ⊆ M ×, where M X is the group of units of a monoid M, as we show in the next example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 11 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' [16, Appendix B] Let us consider the set X = {1, a, b} and the idempotent commutative monoid M on X with identity 1 and such that ab = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Clearly, M X = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, there exists an idempotent solution on M for which θb(1) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Such a solution is defined considering the maps θ1 = θa : M → M given by θ1(1) = θa(1) = 1 and θ1(b) = b and the map θb : M → M given by θb(1) = θb(a) = a and θb(b) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In the last part of this section, we will focus on idempotent solutions on monoids M for which θ1 is also a homomorphism from M to E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' This assumption is not restrictive, as we show in the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Indeed, it is a necessary condition for idempotent solutions defined on monoids in which the idempotents are central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In the following, let us denote by Z(M) the center of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid such that E(M) ⊆ Z(M) and s(x, y) = (xy, θx(y)) an idempotent solution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, the map θ1 is an idempotent monoid homomorphism from M to E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Initially, by Proposition 15-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', the map θ1 is idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Besides, recalling that θ1(x) ∈ E(M), for any x ∈ M, we have that θ1(xy) = θ1(x)θx(y)θ1(x) by Corollary 13-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' = θ1θ1(x)θθ1(x)(y)θ1(x) by Proposition 11-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' = θ1 (θ1(x)y) θ1(x) by (P1) = θ1 (yθ1(x)) θ1(x) = θ1(y)θyθ1(x)θ1(x) by (P1) = θ1(y)θ1(x)θyθ1(x) = θ1(y)θ1(x)θθ1(y)θ1(x) by Proposition 11-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' = θ1(y)θ1(x) by (I1) for all x, y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Finally, by Proposition 15-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', it holds that θ1(1) = 1, hence the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Note that the converse of Proposition 17 is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Indeed, the map s(x, y) = (xy, 1) is a solution in any monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid such that E(M) ⊆ Z(M) and s(x, y) = (xy, θx(y)) a solution on M for which θ1 is an idempotent monoid homomorphism from M to E(M) such that x = xθ1(x), for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, s is idempotent if and only if (I2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' It is enough to notice that (I1) holds since, by (P1), xy = xθ1(x)yθ1(y) = xyθ1(xy) = xyθ1(x)θx(y) = xθ1(x)yθx(y) = xyθx(y), for all x, y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 12 The next result is a description of all idempotent solutions on a monoid M having central idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid for which E(M) ⊆ Z(M) and µ an idempotent monoid homomorphism from M to E(M) such that, for every x ∈ M, µ(x) = ex, with ex ∈ E(M) a right identity for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, let {θe : M → M | e ∈ Im µ} be a family of maps such that θ1 = µ, θe(xy) = θe(x)θf(y), (4) for all x, y ∈ M and e ∈ Im µ, with f = µ(ex), and θe = θeθef, (5) for all e, f ∈ Im µ, and θefθe(x) = θe(x), (6) for all x ∈ M, e ∈ Im µ, with f = µ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, set θx = θµ(x), for every x ∈ M, one has that the map s : M × M → M × M given by s(x, y) = (xy, θx(y)) is an idempotent solution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Conversely, every idempotent solution on M can be so constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let x, y, z ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, using (4) we obtain (P1), since θx(y)θxy(z) = θµ(x)(y)θµ(µ(x)y)(z) = θµ(x)(yz) = θx(yz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, θθx(y)θxy(z) = θµθx(y)θµ(x)µ(y)(z) = θθ1θx(y)θθ1(x)θ1(y)(z) = θθ1(y)θθ1(x)θ1(y)(z) by (5) since θ1 = θ1θx = θy(z) by (5) and so (P2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, by Lemma 18, (I1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Finally, applying (6), we get θxyθx(y) = θµ(x)µ(y)θµ(x)(y) = θµ(x)(y) = θx(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Therefore, s(x, y) = (xy, θx(y)) is an idempotent solution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Vice versa, if we assume that s(x, y) = (xy, θx(y)) is an idempotent solution on M, then by Proposition 17, µ = θ1 is an idempotent monoid homomorphism, and, by Proposi- tion 12-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' (e), we have that x ∈ Mθ1(x), for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In addition, by Proposition 15- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', θx = θθ1(x), for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Hence, by(P1), we obtain (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Now, let e, f ∈ Im θ1, thus there exist x, y ∈ M such that e = θ1(x) and f = θ1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Besides, by (P2), θe = θθ1(x) = θx = θθy(x)θyx = θθ1(x)θθ1(y)θ1(x) = θeθfe, and so (6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Finally, by (I2), if e ∈ Im θ1, x ∈ M, and f = θ1(x), we obtain θefθe(x) = θθ1(xy)θθ1(x)(y) = θxyθx(y) = θx(y) = θθ1(x)(y) = θe(y), for every y ∈ M, hence (6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 13 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Unlike solutions defined on groups, in the case of idempotent solutions on monoids, it is not possible to find a way to write the maps θx by means of the map θ1 as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Indeed, if we look at the idempotent solutions on the monoid M in Example 4, one can see that there are three different solutions having the same map θ1 : M → M given by θ1(1) = θa(1) = 1 and θ1(b) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The next result narrows down that choice of the maps θe in Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Indeed, given an idempotent solution on a monoid M such that θ1 is a monoid homomorphism from M to E(M), one has that the kernel of θ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', the set ker θ1 = {(x, y) ∈ M × M | θ1(x) = θ1(y)} is a congruence relation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Thus, one can naturally consider the quotient monoid M/ ker θ1 (see [20] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Additionally, we have the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Theorem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid and s(x, y) = (xy, θx(y)) an idempotent solution on M such that θ1 is a monoid homomorphism from M to E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θ1(M) is a system of representatives of M/ ker θ1 that contains the identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' (θx(y), y) ∈ ker θ1, for all x, y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The first part is a consequence of the idempotence of the map θ1 by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' in Propo- sition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' In fact, Proposition 15-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', we have that 1 = θ1(1) ∈ θ1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Moreover, since by Proposition 15-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' θ1 is an idempotent map, we easily obtain that (θ1(x), x) ∈ ker θ1, for every x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Besides, if x, y ∈ M are distinct and such that (θ1(y), x) ∈ ker θ1, then we get θ1(y) = θ1(θ1(y)) = θ1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' The second part follows by Corollary 13-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', since θ1θx(y) = θ1(y), for all x, y ∈ M, and so (θx(y), y) ∈ ker θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Therefore, we get the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Corollary 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid for which E(M) ⊆ Z(M) and s(x, y) = (xy, θx(y)) an idempotent solution on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, θ1(M) is a system of representatives of M/ ker θ1 that contains the identity and (θx(y), y) ∈ ker θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' It is a consequence of Theorem 20, since, by Proposition 17, the map θ1 is a monoid homomorphism from M to E(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Let M be a monoid for which E(M) ⊆ Z(M) and s(x, y) = (xy, θx(y)) and r(x, y) = (xy, ηx(y)) two idempotent solutions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Then, by (3), if such solutions are isomorphic, there exists an isomorphism f of M such that fθ1 = η1f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=', f sends the system of representatives θ1(M) of M/ ker θ1 into the other one η1f(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' 14 References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Baaj, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Skandalis, Unitaires multiplicatifs et dualit´e pour les produits crois´es de C*-alg`ebres, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfrf0Z/content/2301.01643v1.pdf'} +page_content=' Sci.' metadata={'source': 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0000000000000000000000000000000000000000..f3f6619d4f8d9ea85d8e8f174e07ce0a9b02cfcf --- /dev/null +++ b/rdE1T4oBgHgl3EQfPwNi/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:62b9154c0bd3e46c071e0647abe4582ea30d2ff8d59396fe898dadf94094ad69 +size 124264 diff --git a/s9E3T4oBgHgl3EQf9gsi/content/tmp_files/2301.04816v1.pdf.txt b/s9E3T4oBgHgl3EQf9gsi/content/tmp_files/2301.04816v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..811aee7d80816db491e0a7337b5bb3e8cfd116ae --- /dev/null +++ b/s9E3T4oBgHgl3EQf9gsi/content/tmp_files/2301.04816v1.pdf.txt @@ -0,0 +1,1865 @@ +arXiv:2301.04816v1 [quant-ph] 12 Jan 2023 +Analytical Approximations for Generalized Landau-Zener Transitions in Multi-level +Non-Hermitian Systems +Chon-Fai Kam∗ and Yang Chen† +Department of Mathematics, Faculty of Science and Technology, +University of Macau, Avenida da Universidade, Taipa, Macau, China +We study the dynamics of non-adiabatic transitions in non-Hermitian multi-level parabolic models where the +separations of the diabatic energies are quadratic function of time. The model Hamiltonian has been used to +describe the non-Hermitian dynamics of two pairs of coupled cavities. In the absence of the coupling between +any two pairs of cavities, the wave amplitudes within each subsystem are described by the tri-confluent Heun +functions. When all the couplings between the cavities are present, we reduce the dynamics into a set of two +coupled tri-confluent Heun equations, from which we derive analytical approximations for the wave amplitudes +at different physical limits. +I. +INTRODUCTION +Since the dawn of the twentieth century, quantum mechan- +ics has been the foundation of modern technology from the +electronic computers to the most precise atomic clock. One of +the basic principles of quantum mechanics is that the spectra +of atoms are real and the time evolution of wave functions is +unitary, and thus the total probability is conserved. As such, +it was once widely believed that the Hamiltonian which de- +scribes the time evolution of any physical system has to be +self-adjoint or Hermitian [1]. However, over the years, people +started to realize that the Hermitian law is not unbreakable, as +the deviation of it does not directly implies complex-valued +spectra energies. Since Bender and Boettcher’s groundbreak- +ing work [2, 3], a new principle started to be affirmed is +that real energy spectra of a physical system are not ensured +by the Hermitian property, but rather ensured by the partity- +time (PT ) symmetry. The fundamental difference between +Hermitian system and non-Hermitian PT -symmetric system +is that the former one can only be used to describes closed +systems that have no exchange with the outer environment, +but the later one can be used to describe open systems with +two coupled subsystems, each of which is in contact with the +outer environment, but the probability in one subsystem with +gain compensates another subsystem with loss, so that the en- +tire system is in a dynamical equilibrium [4, 5]. Over the +decades, the new principle of pseudo-Hermiticity under PT - +symmetry has opened up countless new opportunities, and has +also revealed various application in modern technology which +ranges from optics [6] to +One of the most interesting effects of non-Hermitian sys- +tems is that the PT -symmetry leads to a new type of spectral +degeneracies, known as the exceptional points, at which not +only a finite numbers of eigenvalues coincide, but the asso- +ciated eigenstates also coincide [5]. In contrast to the spec- +tral degeneracies of Hermitian systems, at which the eigen- +states can be chosen to be orthogonal to each other, the spec- +tral degeneracies in PT -symmetric systems are peculiar as +∗ Email: dubussygauss@gmail.com +† Email: yangbrookchen@yahoo.co.uk +certain eigenstates are completely parallel and the Hamilto- +nian matrix becomes defective at the exceptional points [7]. +This intriguing properties of non-Hermitian physics give rise +to many counterintuitivefeatures. For example, a general PT - +symmetric Hamiltonian may undergo a spontaneous symme- +try breaking phase transition, beyond which complex eigen- +values emerge [8, 9]. In particular, when encircling an excep- +tional point, an unconventional level-crossing behavior will +appear, along with a phase change of one eigenstate but not of +the other [10, 11]. +In ordinary Hermitian quantum mechanics, there is a fun- +damental physical process called the Landau-Zener transition, +which describes the transition between two energy levels of +a quantum system directly driven through an avoided cross- +ing [12–14]. The Landau-Zener transition assumes a constant +coupling between bare states in the diabatic basis and a lin- +early varying separation of diabatic energies [15], which can +be exactly solved by the parabolic cylinder functions [16] or +the integral representation method [17]. Although the model +of Landau-Zener transition has achieved great success over +the years, there are indeed cases where the assumption of lin- +ear crossing between the diabatic states becomes no longer +valid. For the cases in which the crossing points merge to- +gether as a result of external fields, the Landau-Zener lin- +earization fails, and the linear-dependence of diabatic energies +has to be replaced by a parabolic or superparabolic one [18]. +Interestingly, the tunneling dynamics for the parabolic and cu- +bic models can still be exactly solved by the tri-confluent and +bi-confluent Heun functions [19, 20]. One can still express +the tunneling probability via the Stokes constants by using the +Zhu-Nakamura formula [21, 22]. +Compared to the two-state scenario, the research of gener- +alized Landau-Zener transition for complicated systems with +more than two states, even in ordinary Hermitian quantum +mechanics, has been arduous and in most cases inconclusive. +The main reason is that in conventional Landau-Zener tran- +sition, the coupling equations which govern the non-adiabatic +transition amplitudes between the two energy levels can be re- +duced to a single second-order differential equation, e.g., the +parabolic cylinder function for linear separation of diabatic +energies [16], or the confluent Heun functions for quadratic +and cubic separations of diabatic energies [23, 24]. In con- +trast, in the general multi-state scenario, if one attempts to re- + +2 +duce the coupling equations which govern the non-adiabatic +transition amplitudes between different energy levels in a sin- +gle equation, one would probably obtain an ordinary differen- +tial equation with order greater than two. Compared to those +of conventional second order differential equations, the ana- +lytic properties of solutions of differential equations with or- +der greater than two are harder to obtain by regular methods +like asymptotic analysis. The essential difficulty lies in the +fact that the Stokes curves for conventional second order dif- +ferential equations are straight lines which never cross each +other, but those for differential equations with order greater +than two are non-straight lines which always cross each other +unavoidably. This simple fact results in the breakdown of the +standard connection formula near the crossing points of the +Stokes curve. In this regard, the asymptotic WKB solutions of +generalized Landau-Zener non-adiabatic transitions for multi- +state systems are in general hard to obtain, if not totally im- +possible. +Despite its evident importance, the non-Hermitian general- +ization of the two-level Landau-Zener transition has only re- +cently been analyzed by Longstaff [25], the associated non- +Hermitian Landau-Zener-Stuckelberg interferometry was an- +alyzed by Shen [26], and the non-Hermitian generalization +of the parabolic and super-parabolic models, in which the +exceptional points are driven through at finite speeds which +are quadratic or cubic functions of time has been analyzed +by the authors [27]. +Compared to the two-state scenario, +the research of the non-Hermitian generalization of Landau- +Zener non-adiabatic transitions in the multi-state scenario is +still at its early stage. Recently, the three-state non-Hermitian +Landau-Zener model in the presence of an interaction with the +outer environment has been considered [28], and the Landau- +Zener transitions through a pair of higher order exceptional +points has been analyzed by Melanathuru [29]. In this work, +based on our analytical approximation methods used in pre- +vious researches [23, 24, 27], we will analyze the dynamics +of a four-state non-Hermitian PT -symmetric system which +directly passes through a collection of exceptional points. +II. +THE MODEL +To begin with, we consider a four-state non-Hermitian sys- +tem which has been used to describe the dynamics of four cou- +pled cavities with asymmetric losses [30]. The non-Hermitian +system consists of two pairs of coupled cavities, each of which +is described by a 2 × 2 non-Hermitian Hamiltonian +Hi = +� +ωi − iΓ0 +κ +κ +ωi − iΓ +� +, +(1) +where κ represents the coupling strength between the two cav- +ities, ωi (i = 1, 2) represents the same resonant frequency of +the two cavities, and Γ0 and Γ represent the intrinsic loss of +the each cavity. The whole non-Hermitian system consists of +two pairs of above-mentioned coupled cavities with the same +values of κ, Γ0 and Γ but different resonant frequencies ω1 +and ω2. The two pairs of cavities are coupled by connecting +each individual cavity of one pair with that of another pair by +a small tube by an inter-pair coupling strength η. The 4 × 4 +non-Hermitian Hamiltonian of the whole system becomes +H = + +ω2 − iΓ0 +κ +0 +η +κ +ω2 − iΓ +η +0 +0 +η +ω1 − iΓ0 +κ +η +0 +κ +ω1 − iΓ + +, +(2) +Evidently, the 2 × 2 Hamiltonian for each pair of coupled cav- +ities is PT-symmetric only when the intrinsic losses are sym- +metric, i.e., Γ0 = Γ, where P ≡ � 0 1 +1 0 +� denotes the parity op- +erator, and T denotes complex conjugation. From the Hamil- +tonian Eq. (2), one obtains the coupled equations for the four +wave amplitudes a1, a2, a3 and a4 +i˙a1 = (ω2 − iΓ0)a1 + κa2 + ηa4, +(3a) +i˙a2 = κa1 + (ω2 − iΓ)a2 + ηa3, +(3b) +i˙a3 = ηa2 + (ω1 − iΓ0)a3 + κa4, +(3c) +i˙a4 = ηa1 + κa3 + (ω1 − iΓ)a4. +(3d) +Using the change of variable bi ≡ e +� t +0 (¯Γ+i ¯ω)dsai to remove the +average resonant frequency ¯ω ≡ 1 +2(ω1 + ω2) and the average +intrinsic loss ¯Γ ≡ +1 +2(Γ + Γ0), one obtains the new coupled +equations for the wave amplitudes bi, which depend only on +the relative resonant frequency ∆ω ≡ ω1 − ω2 and the relative +intrinsic loss ∆Γ ≡ Γ − Γ0, as +i˙b1 = −Ω∗b1 + κb2 + ηb4, +(4a) +i˙b2 = κb1 − Ωb2 + ηb3, +(4b) +i˙b3 = ηb2 + Ωb3 + κb4, +(4c) +i˙b4 = ηb1 + κb3 + Ω∗b4, +(4d) +where Ω ≡ 1 +2(∆ω+i∆Γ). One may further define c1 ≡ b1+ib2, +c2 ≡ b1 − ib2, c3 ≡ b3 + ib4 and c4 ≡ b3 − ib4, and obtains +i˙c1 = −∆ωc1 + iγc2 + iηc4, +(5a) +i˙c2 = iγ′c1 − ∆ωc2 − iηc3, +(5b) +i˙c3 = iηc2 + ∆ωc3 + iγc4, +(5c) +i˙c4 = −iηc1 + iγ′c3 + ∆ωc4, +(5d) +where γ ≡ ∆Γ + κ and γ′ ≡ ∆Γ − κ. The Hamiltonian in the +diabatic basis then reads +H′ = + +−∆ω +iγ +0 +iη +iγ′ +−∆ω −iη +0 +0 +iη +∆ω +iγ +−iη +0 +iγ′ +∆ω + +. +(6) +From Eqs. (5a) and (5b), one immediately obtains +c3 = −1 +η +�˙c2 − i∆ωc2 − γ′c1 +� , +(7a) +c4 = 1 +η (˙c1 − i∆ωc1 − γc2) . +(7b) +Substitution of Eqs. (7a) and (7b) into Eqs. (5c) and (5d) +yields the following coupled equations +¨c1 + +� +η2 + ∆ω2 − γ′2 − i∆ ˙ω +� +c1 = 2κ˙c2 + 2i∆ω∆Γc2, +(8a) +¨c2 + +� +η2 + ∆ω2 − γ2 − i∆ ˙ω +� +c2 = −2κ˙c1 + 2i∆ω∆Γc1. (8b) + +3 +In particular, for the PT -symmetric case, i.e., ∆Γ = 0, one +immediately obtains +¨c1 + Q(t)c1 = 2κ˙c2, +(9a) +¨c2 + Q(t)c2 = −2κ˙c1, +(9b) +where Q(t) ≡ η2 − κ2 + ∆ω2(t) − i∆ ˙ω(t). A direct computation +yields +˙c1c2 − ˙c2c1 − κ(c2 +1 + c2 +2) = Const. +(10) +In particular, for a parabolic separation of diabatic energies, +i.e., ∆ω = α + βt2, one will obtain +Q(t) = η2 − κ2 + (α + βt2)2 − 2iβt += β2t4 + 2αβt2 − 2iβt + α2 + η2 − κ2. +(11) +One can see that as Q(t) is now a quartic function of time, +in the absence of the coupling κ within each pair of cavities, +Eqs. (9a) and (9b) are decoupled and solved by the superposi- +tion of the tri-confluent Heun functions T1 and T2. When the +coupling κ is nonzero, one can write c1 = �∞ +n=0 κnc(n) +1 and c2 = +�∞ +n=0 κnc(n) +2 , where c(0) +1 +≡ d1T1 + d2T2 and c(0) +2 +≡ e1T1 + e2T2, +and obtains the recursive relations +¨c(n) +1 + Q(t)c(n) +1 = 2˙c(n−1) +2 +, +(12a) +¨c(n) +2 + Q(t)c(n) +2 = −2˙c(n−1) +1 +. +(12b) +III. +INTEGRALS INVOLVING PRODUCTS OF HEUN +FUNCTIONS AND THEIR DERIVATIVES +To solve the recursive relations, one regards the right hand +side of Eqs. (12a) and (12b) as known functions of time, then +the n-th order terms c(n) +1 and c(n) +2 are integrals of the (n − 1)-th +order terms ˙c(n−1) +1 +and ˙c(n−1) +2 +, which may be expressed as +c(n) +1 = −2T1 +W +� +T2˙c(n−1) +2 +dt + 2T2 +W +� +T1˙c(n−1) +2 +dt, +(13a) +c(n) +2 = 2T1 +W +� +T2˙c(n−1) +1 +dt − 2T2 +W +� +T1˙c(n−1) +1 +dt. +(13b) +where W ≡ T1 ˙T2 − T2 ˙T1 is the Wronskian for T1 and T2, and +is a constant of time. Using integration by parts, Eqs. (13a) - +(13b) may be simplified as +c(n) +1 = 2T1 +W +� +˙T2c(n−1) +2 +dt − 2T2 +W +� +˙T1c(n−1) +2 +dt, +(14a) +c(n) +2 = −2T1 +W +� +˙T2c(n−1) +1 +dt + 2T2 +W +� +˙T1c(n−1) +1 +dt. +(14b) +To continue, one needs to evaluate integrals of the products of +Heun functions and their derivatives. To be more precise, let +us consider the following three kinds of indefinite integrals: +� +tny2dt, +� +tn˙yydt and tn˙y2dt, with y being any linear combi- +nation of the tri-confluent Heun functions T1 and T2, which +satisfies the tri-confluent Heun equation ¨y + Q(t)y = 0, where +Q(t) ≡ �4 +k=0 Aktk is a quartic function of time. Using the fol- +lowing relations +� +tn(y1˙y2 + y2˙y1)dt = tny1y2 − n +� +tn−1y1y2dt +and +� +tn(y1˙y2 − y2˙y1)dt = +W +n+1tn+1, one immediately obtains +� +tny1˙y2dt = 1 +2 +� +tny1y2 + Wtn+1 +n + 1 − n +� +tn−1y1y2dt +� +, +(15) +where y1 and y2 are two independent solutions of the tri- +confluent equation, and W ≡ y1˙y2 − y2˙y1 is the Wronskian of +them. The other two kinds of integrals will be more involved +to evaluate. Let us define +� +tny1y2dt ≡ Pny1y2 + Qn +2 (y1˙y2 + y2˙y1) + Rn˙y1˙y2, +(16a) +� +tn˙y1˙y2dt ≡ Lny1y2 + Mn +2 (y1˙y2 + y2˙y1) + Nn˙y1˙y2. +(16b) +The coefficients Pn, Qn and Rn may be determined by taking +derivative of Eq. (16a), which yields +tny1y2 = ( ˙Pn − QQn)y1y2 + 1 +2( ˙Qn + 2Pn − 2QRn)(y1˙y2 + y2˙y1) ++ ( ˙Rn + Qn)˙y1˙y2. +(17) +Hence, we obtain ˙Pn − QQn = tn, ˙Qn + 2Pn − 2QRn = 0 and +˙Rn + Qn = 0, which are solved by Qn = − ˙Rn and Pn = 1 +2 ¨Rn + +QRn, where Rn satisfies the third order differential equation +... +Rn + 4Q ˙Rn + 2 ˙QRn = 2tn. To evaluate the coefficients Ln, Mn +and Nn, one may use the following identity +� +tn˙y1˙y2dt = 1 +2tn(y1˙y2 + y2˙y1) − n +2tn−1y1y2 ++ +� +tnQy1y2dt + n(n − 1) +2 +� +tn−2y1y2dt. +(18) +Substitution of Eq. (16a) and Q(t) ≡ �4 +k=0 Aktk into Eq. (18) +yields +Ln = +4 +� +k=0 +AkPn+k + n(n − 1) +2 +Pn−2 − n +2tn−1, +(19a) +Mn = +4 +� +k=0 +AkQn+k + n(n − 1) +2 +Qn−2 + tn, +(19b) +Nn = +4 +� +k=0 +AkRn+k + n(n − 1) +2 +Rn−2. +(19c) +ACKNOWLEDGMENTS +This study was supported by the National Natural Science +Foundation of China (Grant nos. 12104524). +Appendix A: Formal solutions of c(n) +1 and c(n) +2 +From Eqs. (14a) and (14b), a direct computation will yield +c(1) +1 = tc(0) +2 , c(1) +2 = −tc(0) +1 . +(A1) + +4 +To proceed further, one can use the following integral identity +with respect to any functions y1 and y2 and their derivatives +� +ty1˙y2dt = t +2y1y2 + Wt2 +4 +− 1 +2 +� +y1y2dt, +(A2) +where W ≡ y1˙y2 − ˙y2y1 is the Wronskian with respect to y1 +and y2. From this, a direct computation yields +c(2) +1 = 1 +2(Q0 − t2)c(0) +1 + R0˙c(0) +1 , +(A3a) +c(2) +2 = 1 +2(Q0 − t2)c(0) +2 + R0˙c(0) +2 , +(A3b) +where Q0 ≡ − ˙R0 and R0 is the solution of the third order +differential equation +d3R0(t) +dt3 ++ 4Q(t)dR0(t) +dt ++ 2dQ(t) +dt +R0(t) = 2. +(A4) +In order to solve c(n) +1 and c(n) +2 , one needs the following integrals +LnTk = W +2 +�� tn+1 +n + 1 − n +2 Qn−1 +� +Tk − nRn−1 ˙Tk +� +≡ W +2 (EnTk + Fn ˙Tk), +(A5a) +Ln ˙Tk = W +2 (MnTk + 2Nn ˙Tk) = W +2 (GnTk + Hn ˙Tk), +(A5b) +where k = 1, 2, and Ln• ≡ T1 +� +tn ˙T2•dt−T2 +� +tn ˙T1•dt. Form +this, one may expand the functions En, Fn, Gn, and Hn (and +thus LnTk and Ln ˙Tk) in series of time as +LnTk ≡ W +2 + +∞ +� +l=0 +EnltlTk + +∞ +� +l=0 +Fnltl ˙Tk + , +(A6a) +Ln ˙Tk = W +2 + +∞ +� +l=0 +GnltlTk + +∞ +� +l=0 +Hnltl ˙Tk + . +(A6b) +In general, one can express the n-th order correction terms +c(n) +1 and c(n) +2 in terms of c(0) +1 , c(0) +2 , ˙c(0) +1 +and ˙c(0) +2 , and expand the +coefficients in series of time as +c(n) +1 ≡ +∞ +� +k=0 +� +α(n) +k c(0) +1 + β(n) +k c(0) +2 + γ(n) +k ˙c(0) +1 + δ(n) +k ˙c(0) +2 +� +tk, +(A7a) +c(n) +2 ≡ +∞ +� +k=0 +� +λ(n) +k c(0) +1 + µ(n) +k c(0) +2 + ν(n) +k ˙c(0) +1 + ξ(n) +k ˙c(0) +2 +� +tk. +(A7b) +Using Eqs. (A6a) - (A6b), one obtains the (n + 1)-th order +correction terms +c(n+1) +1 += 2 +W +∞ +� +k=0 +� +λ(n) +k Lkc(0) +1 + µ(n) +k Lkc(0) +2 + ν(n) +k Lk˙c(0) +1 + ξ(n) +k Lk˙c(0) +2 +� += +∞ +� +k,l=0 +� +(λ(n) +k Ekl + ν(n) +k Gkl)c(0) +1 + (µ(n) +k Ekl + ξ(n) +k Gkl)c(0) +2 ++ (λ(n) +k Fkl + ν(n) +k Hkl)˙c(0) +1 + (µ(n) +k Fkl + ξ(n) +k Hkl)˙c(0) +2 +� +tl. +(A8a) +c(n+1) +2 += −2 +W +∞ +� +k=0 +� +α(n) +k Lkc(0) +1 + β(n) +k Lkc(0) +2 + γ(n) +k Lk˙c(0) +1 + δ(n) +k Lk˙c(0) +2 +� += − +∞ +� +k,l=0 +� +(α(n) +k Ekl + γ(n) +k Gkl)c(0) +1 + (β(n) +k Ekl + δ(n) +k Gkl)c(0) +2 ++ (α(n) +k Fkl + γ(n) +k Hkl)˙c(0) +1 + (β(n) +k Fkl + δ(n) +k Hkl)˙c(0) +2 +� +tl. +(A8b) +A direct comparison between Eqs. (A7a) and (A8a) yields the +recursive relations +α(n+1) +k += +∞ +� +j=0 +(λ(n) +j E jk + ν(n) +j G jk), +(A9a) +β(n+1) +k += +∞ +� +j=0 +(µ(n) +j E jk + ξ(n) +j G jk), +(A9b) +γ(n+1) +k += +∞ +� +j=0 +(λ(n) +j F jk + ν(n) +j H jk), +(A9c) +δ(n+1) +k += +∞ +� +j=0 +(µ(n) +j F jk + ξ(n) +j H jk), +(A9d) +λ(n+1) +k += − +∞ +� +j=0 +(α(n) +j E jk + γ(n) +j G jk), +(A9e) +µ(n+1) +k += − +∞ +� +j=0 +(β(n) +j E jk + δ(n) +j G jk), +(A9f) +ν(n+1) +k += − +∞ +� +j=0 +(α(n) +j F jk + γ(n) +j H jk), +(A9g) +ξ(n+1) +k += − +∞ +� +j=0 +(β(n) +j F jk + δ(n) +j H jk). +(A9h) +Appendix B: Recursive relation for Rn +The coefficients Rn for different n are not independent. +From the identity +d +dt (˙y1˙y2 + Qy1y2) = ˙Qy1y2, we obtain the +indefinite integral +� ˙Qy1y2dt = ˙y1˙y2 + Qy1y2, which im- +plies that �4 +k=0 kAkRk−1 = 1. +Similarly, from the identity +d +dt[t(˙y1˙y2 + Qy1y2) − 1 +2(y1˙y2 + y2˙y1)] = (2Q + t ˙Q)y1y2, we +obtain the indefinite integral +� +(2Q + t ˙Q)y1y2dt = t(˙y1˙y2 + +Qy1y2) − 1 +2(y1˙y2 + y2˙y1), which yields �4 +k=0(k + 2)AkRk = t. + +5 +In general, one has the following identity +� +y1y2d(f Q) = +f(˙y1˙y2 + Qy1y2) − +� ˙f ˙y1˙y2dt for an arbitrary function f. In +particular, for f = tn, one obtains the following indefinite in- +tegral +� +(tn ˙Q + ntn−1Q)y1y2dt = tn(˙y1˙y2 + Qy1y2) − n +� +tn−1˙y1˙y2dt. +(B1) +Substitution of Eq. (18) into Eq. (B1) yields +� +(tn ˙Q + 2ntn−1Q)y1y2dt = tn(˙y1˙y2 + Qy1y2) − n +2tn−1(y1˙y2 + y2˙y1) ++n(n − 1) +2 +tn−2y1y2 − n(n − 1)(n − 2) +2 +� +tn−3y1y2dt. +(B2) +From Eq. (B2) and using tn ˙Q + 2ntn−1Q += +�4 +k=0(2n + +k)Aktn+k−1, one obtains the recursive relations for Rn with +n ≥ 3 +4 +� +k=0 +(2n + k)AkRn+k−1 = tn − n(n − 1)(n − 2) +2 +Rn−3. +(B3) +For example, for the Airy functions which satisfy ¨y+Q(t)y = 0 +with Q(t) = −t, one recovers the recursive relation obtained by +the authors previously [23] +Rn = +1 +2(2n + 1) +�n(n − 1)(n − 2)Rn−3 − 2tn� ; +(B4) +while for the Bessel functions which satisfy ¨y+λ2t4y = 0 with +Q(t) = λ2t4, one recovers the simple recursive relations [23] +Rn+3 = +−1 +4(n + 2)λ2 +�n(n − 1)(n − 2)Rn−3 − 2tn� . +(B5) +In general, when A4 � 0, one can obtain an explicit expression +for Rn in terms of R0, R1 and R2 by revising Eq. (B3) as +Rn+3 + +2 +� +k=−2 +gk+2 +n +Rn+k = Jn, +(B6) +where Jn ≡ +2tn−n(n−1)(n−2)Rn−3 +2(2n+4)A4 +, gk+1 +n +≡ +(2n+k) +(2n+4) +Ak +A4 , and g0 +n ≡ 0. +Clearly, all Rn may be expressed in terms of R0, R1 and R2. +Explicitly, the first few terms may be expressed as +R3 = J0 − +2 +� +k=0 +gk+2 +0 +Rk, +(B7a) +R4 = J1 − g4 +1J0 − +2 +� +k=0 +hk+1 +1 +Rk, +(B7b) +R5 = J2 − g4 +2J1 − h3 +2J0 − +2 +� +k=0 +wk +2Rk, +(B7c) +R6 = J3 − g4 +3J2 − h3 +3J1 − w2 +3J0 +− +2 +� +k=1 +uk−1 +3 +Rk + (g1 +1h3 +3 + g2 +0w2 +3)R0, +(B7d) +where hk +n ≡ gk +n−g4 +ngk+1 +n−1, wk +n ≡ hk +n−h3 +ngk+2 +n−2, and uk +n ≡ wk +n−w2 +ngk+3 +n−3. +A direct computation yields +R3 = +1 +4A4 +(1 − 3A3R2 − 2A2R1 − A1R0) , +(B8a) +R4 = +t +6A4 +− 5A3 +24A2 +4 +− + +2A2 +3A4 +− 5A2 +3 +8A2 +4 + R2 +− + +A1 +2A4 +− 5A2A3 +12A2 +4 + R1 − + +A0 +3A4 +− 5A1A3 +24A2 +4 + R0, +(B8b) +R5 = +t2 +8A4 +− 7A3t +48A2 +4 +− 3A2 +16A2 +4 ++ 35A2 +3 +192A3 +4 +− + +5A1 +8A4 +− 55A2A3 +48A2 +4 ++ +105A3 +3 +192A3 +4 + R2 +− + +A0 +2A4 +− 21A1A3 +48A2 +4 +− 3A2 +2 +8A2 +4 ++ 35A2A2 +3 +96A3 +4 + R1 ++ + +7A0A3 +24A2 +4 ++ 3A1A2 +16A2 +4 +− 35A1A2 +3 +192A3 +4 + R0, +(B8c) +R6 = +t3 +10A4 +− 9A3t2 +80A2 +4 +− + +2A2 +15A2 +4 +− 63A2 +3 +480A3 +4 + t +− 7A1 +40A2 +4 ++ 161A2A3 +160A3 +4 +− +21A3 +3 +128A4 +4 +− + +3A0 +5A4 +− 87A1A3 +80A2 +4 +− 8A2 +2 +15A2 +4 ++ 49A2A2 +3 +32A3 +4 +− 189A4 +3 +384A4 +4 + R2 ++ + +9A0A3 +20A2 +4 ++ 3A1A2 +4A2 +4 +− 63A1A2 +3 +160A3 +4 ++ A2 +2A3 +240A3 +4 +− 21A2A3 +3 +64A4 +4 + R1 +− + +3 +10A4 +− 4A0A2 +15A2 +4 +− 7A2 +1 +40A2 +4 ++ 21A0A2 +3 +80A3 +4 ++161A1A2A3 +480A3 +4 +− 315A1A3 +3 +1920A4 +4 + R0. +(B8d) +Appendix C: Explicit expressions for c(n) +1 (t) and c(n) +2 (t) for the +cases when |t| ≪ 1 and |t| → ∞ +1. +Cases for |t| ≪ 1 and Q(t) ≈ −2iβt + α2 + η2 − κ2 +When |t| ≪ 1, one can simply retain the linear teams in Q(t) +and obtains Q(t) ≈ A1t + A0 = −2iβt + α2 + η2 − κ2. After the +coordinate transformation z ≡ g(t + A0/A1), c(n) +1 +and c(n) +2 +are +determined by the recursive relations +d2c(n) +1 +dz2 +− zc(n) +1 = 2gdc(n−1) +2 +dz +, +(C1a) +d2c(n) +2 +dz2 +− zc(n) +2 = −2gdc(n−1) +1 +dz +, +(C1b) +where g ≡ eiπ/3A1/3 +1 . Here, the lowest-order terms c(0) +1 and c(0) +2 +are solved by the linear combinations of the Airy functions +Ai(z) and Bi(z), i.e., c(0) +1 (z) = d1 Ai(z) + d2 Bi(z) and c(0) +2 (z) = + +6 +e1 Ai(z) + e2 Bi(z). To proceed further, notice that the integral +of the product of any linear combinations of the Airy functions +has the form +� +zny1y2dz ≡ Pny1y2 + Qn +2 (y1y′ +2 + y2y′ +1) + Rny′ +1y′ +2, +(C2) +where Pn = 1 +2R′′ +n −zRn, Qn = −R′ +n and Rn is determined by the +third-order differential equation +d3Rn +dz3 − 4zdRn +dz − 2Rn = 2zn. +(C3) +A straightforward computation yields Rn, Qn and Pn for n ≤ 2 +R0 = −1, Q0 = 0, P0 = z, +(C4a) +R1 = − z +3, Q1 = 1 +3, P1 = z2 +3 , +(C4b) +R2 = −z2 +5 , Q2 = 2z +5 , P2 = z3 − 1 +5 +, +(C4c) +where Rn for n ≥ 3 is determined by the recursive relation +Rn = − +zn +2n + 1 + n(n − 1)(n − 2) +2(2n + 1) +Rn−3. +(C5) +Hence, Rn for n ≥ 3 is solved by +Rn = − +zn +2n + 1 − n(n − 1)(n − 2)zn−3 +2(2n + 1)(2n − 5) +− · · · − n(n − 1) · · ·(n − 3k + 1)zn−3k +2k(2n + 1) · · ·(2n + 1 − 6k) += − +zn +2n + 1 +k +� +j=0 +n!Γ( 2n+1 +6 )(12z3)− j +(n − 3 j)!Γ( 2n+1 +6 +− j), +(C6) +where k is the number such that n − 3k ∈ [0, 2]. Using the +relation Qn = − ˙Rn, a direct computation yields +Qn = nzn−1 +2n + 1 + n(n − 1)(n − 2)(n − 3)zn−4 +2(2n + 1)(2n − 5) ++ · · · + n(n − 1) · · ·(n − 3l)zn−3l−1 +2k(2n + 1) · · ·(2n + 1 − 6l) += nzn−1 +2n + 1 +l� +j=0 +(n − 1)!Γ( 2n+1 +6 )(12z3)− j +(n − 1 − 3 j)!Γ( 2n+1 +6 +− j), +(C7) +where l is the number such that n − 3l − 1 ∈ [0, 2]. Using +the above relations, Rn, Qn and Pn for n ≤ 6 can be explicitly +expressed as +R3 = −z3 + 3 +7 +, Q3 = 3z2 +7 , P3 = z4 +7 , +(C8a) +R4 = −z4 + 4z +9 +, Q4 = 4z3 + 4 +9 +, P4 = z5 − 2z2 +9 +, +(C8b) +R5 = −z5 + 6z2 +11 +, Q5 = 5z4 + 12z +11 +, P5 = z6 − 4z3 − 6 +11 +. (C8c) +2. +Cases for |t| ≫ 1 and Q(t) ≈ β2t4 +In the region |t| ≫ 1, one can only keep the highest order +term in Q(t) such that Q(t) ≈ β2t4. Then, c(n) +1 (t) and c(n) +2 (t) are +determined by the recursive equations +¨c(n) +1 + β2t4c(n) +1 = 2˙c(n−1) +2 +, +(C9a) +¨c(n) +2 + β2t4c(n) +2 = −2˙c(n−1) +1 +. +(C9b) +Here the zeroth order terms c(0) +1 (t) and c(0) +2 (t) are linear combi- +nation of ¯y1 ≡ √tJ1/6(βt3/3) and ¯y2 ≡ √tJ−1/6(βt3/3), where +Jν(z) are the Bessel function of the first kind defined by +Jν(z) ≡ +∞ +� +n=0 +(−1)n +Γ(ν + n + 1)n! +� z +2 +�ν+2n +. +(C10) +Hence, the fundamental solutions y1(t) and y2(t) of the equa- +tion ¨y + β2t4y = 0 and their derivatives have the series expan- +sions +y1 ≡ Γ +�7 +6 +� �β +6 +�− 1 +6 +¯y1 = +∞ +� +n=0 +Γ( 7 +6)(−1)n +Γ( 7 +6 + n)n! +�β +6 +�2n +t1+6n, +(C11a) +y2 ≡ Γ +�5 +6 +� �β +6 +� 1 +6 +¯y2 = +∞ +� +n=0 +Γ( 5 +6)(−1)n +Γ( 5 +6 + n)n! +�β +6 +�2n +t6n, +(C11b) +˙y1 = +∞ +� +n=0 +Γ( 1 +6)(−1)n +Γ( 1 +6 + n)n! +�β +6 +�2n +t6n, +(C11c) +˙y2 = −β2t5 +5 +∞ +� +n=0 +Γ( 11 +6 )(−1)n+1 +Γ( 11 +6 + n)n! +�β +6 +�2n +t6n. +(C11d) +The fundamental solutions y1(t) and y2(t) and their derivatives +can be expressed in terms of the generalized hypergeometric +functions pFq(a1, · · · , ap; b1, · · · , bq; z) as +y1(t) = 1F2 +� +1; 1, 7 +6; −β2t6 +36 +� +t, +(C12a) +y2(t) = 1F2 +� +1; 1, 5 +6; −β2t6 +36 +� +, +(C12b) +˙y1(t) = 1F2 +� +1; 1, 1 +6; −β2t6 +36 +� +, +(C12c) +˙y2(t) = −1F2 +� +1; 1, 11 +6 ; −β2t6 +36 +� β2t5 +5 , +(C12d) +which obey the initial conditions y1(0) = 0, y2(0) = 1, ˙y1(0) = +1 and ˙y2(0) = 0. Hence, the Wronskian with respect to y1 and +y2 is a constant, i.e., W(y1, y2) ≡ y1˙y2 − y2˙y1 = −1. A direct +computation yields the products of the fundamental solutions +and their derivatives +y1y2 = 1F2(1; 1, 5 +6; −β2t6 +36 )1F2(1; 1, 7 +6; −β2t6 +36 )t, +˙y1˙y2 = −1F2(1; 1, 1 +6; −β2t6 +36 )1F2(1; 1, 11 +6 ; −β2t6 +36 )β2t5 +5 , +y1˙y2 + y2˙y1 = 1F2(1; 1, 1 +6; −z2 +4 )1F2(1; 1, 5 +6; −z2 +4 ) +− 1F2(1; 1, 7 +6; −z2 +4 )1F2(1; 1, 11 +6 ; −z2 +4 )β2t6 +5 . +(C13) + +7 +Thus, the above products can be expanded in the Taylor series +as +y1y2 = t − 2 +35β2t7 + +6 +5005β4t13 + O(t19), +y1˙y2 + y2˙y1 = 1 − 2 +5β2t6 + +6 +385β4t12 + O(t18), +˙y1˙y2 = −1 +5β2t5 + 2 +55β4t11 + O(t17). +(C14) +Similar to the previous case, the integral of the product of the +fundamental solutions y1(t) and y2(t) with a power weight tn +can be written in the form +� +tny1y2dt ≡ Pny1y2 + Qn +2 (y1˙y2 + y2˙y1) + Rn˙y1˙y2, +(C15) +where Pn = 1 +2 ¨Rn + β2t4Rn, Qn = − ˙Rn, and Rn is determined by +the third-order differential equation +d3Rn +dt3 + 4β2t4 dRn +dt + 8β2t3Rn = 2tn, +(C16) +from which one can show that Rn obey the recursive relation +Rn+6 = +tn+3 +2β2(n + 5) − (n + 3)(n + 2)(n + 1) +4β2(n + 5) +Rn, +(C17) +A direct computation gives Rn, Qn and Pn for n = 3, 4, 5 +R3(t) = +1 +4β2 ,Q3(t) = 0, P3(t) = t4 +4 , +(C18a) +R4(t) = +t +6β2 ,Q4(t) = − 1 +6β2 , P4(t) = t5 +6 , +(C18b) +R5(t) = t2 +8β2 ,Q5(t) = − t +4β2 , P5(t) = t6 +8 + 1 +8β2 . +(C18c) +Then, one only need to compute Rn(t) for n = 0, 1 and +2. To proceed further, one can expand the function R0(t) as +R0(t) = �∞ +k=0 rktk. From Eq. (C16), one immediately obtains +the recursive relation +rk+6 = +−4β2(k + 2) +(k + 6)(k + 5)(k + 4)rk, +(C19) +For n = 0, the functions R0, Q0 and P0 can be expressed in +terms of the generalized hypergeometric function as +R0(t) = 2F3 +� +1, 5 +6; 7 +6, 8 +6, 9 +6; −β2t6 +9 +� t3 +3 , +(C20a) +Q0(t) = −2F3 +� +1, 5 +6; 3 +6, 7 +6, 8 +6; −β2t6 +9 +� +t2, +(C20b) +P0(t) = 2F3 +� +1, 5 +6; 2 +6, 3 +6, 7 +6; −β2t6 +9 +� +t ++ 2F3 +� +1, 5 +6; 7 +6, 8 +6, 9 +6; −β2t6 +9 +� β2t7 +3 . +(C20c) +These functions define different entire functions of t, which +has the following series expansions +R0(t) = 1 +3t3 − +5 +378β2t9 + +11 +51597β4t15 + O(t21), +Q0(t) = −t2 + 5 +42β2t8 − +55 +17199β4t14 + O(t20), +P0(t) = t − 1 +7β2t7 + +5 +546β4t13 + O(t19). +(C21) +For n = 1, the functions R1, Q1 and P1 can be expressed in +terms of the generalized hypergeometric function as +R1(t) = 2F3 +� +1, 1; 8 +6, 9 +6, 10 +6 ; −β2t6 +9 +� t4 +12, +(C22a) +Q1(t) = −2F3 +� +1, 1; 4 +6, 8 +6, 9 +6; −β2t6 +9 +� t3 +3 , +(C22b) +P1(t) = 2F3 +� +1, 1; 3 +6, 4 +6, 8 +6; −β2t6 +9 +� t2 +2 ++ 2F3 +� +1, 1; 8 +6, 9 +6, 10 +6 ; −β2t6 +9 +� β2t8 +12 . +(C22c) +These functions define different entire functions of t, which +has the following series expansions +R1(t) = 1 +12t4 − +1 +360β2t10 + +1 +25200β4t16 + O(t22), +Q1(t) = −t3 +3 + 1 +36β2t9 − +1 +1575β4t15 + O(t21), +P1(t) = t2 +2 − 1 +24β2t8 + +1 +504β4t14 + O(t20). +(C23) +For n = 2, the functions R2, Q2 and P2 can be expressed in +terms of the generalized hypergeometric function as +R2(t) = 2F3 +� +1, 7 +6; 9 +6, 10 +6 , 11 +6 ; −β2t6 +9 +� t5 +30, +(C24a) +Q2(t) = −2F3 +� +1, 7 +6; 5 +6, 9 +6, 10 +6 ; −β2t6 +9 +� t4 +6 , +(C24b) +P2(t) = 2F3 +� +1, 7 +6; 4 +6, 5 +6, 9 +6; −β2t6 +9 +� t3 +3 ++ 2F3 +� +1, 7 +6; 9 +6, 10 +6 , 11 +6 ; −β2t6 +9 +� β2t9 +30 . +(C24c) +These functions define different entire functions of t, which +has the following series expansions +R2(t) = 1 +30t5 − +7 +7425β2t11 + +91 +7573500β4t17 + O(t23), +Q2(t) = −t4 +6 + +7 +675β2t10 − +91 +445500β4t16 + O(t22), +P2(t) = t3 +3 − 1 +54β2t9 + +7 +10125β4t15 + O(t21). +(C25) +From the recursive relation Eq. (C17), one can derive Rn(t), +Qn(t) and Pn(t) for n = 6k + m with m = 0, 1, 2, and k being + +8 +any non-negative integer +Rn(t) = +2tn+32F3 +� +1, n+5 +6 ; n+7 +6 , n+8 +6 , n+9 +6 ; − β2t6 +9 +� +(n + 1)(n + 2)(n + 3) +, +Qn(t) = − +2tn+22F3 +� +1, n+5 +6 ; n+3 +6 , n+7 +6 , n+8 +6 ; − β2t6 +9 +� +(n + 1)(n + 2) +, +Pn(t) = +tn+12F3 +� +1, n+5 +6 ; n+2 +6 , n+3 +6 , n+7 +6 ; − β2t6 +9 +� +n + 1 ++ +2β2tn+72F3 +� +1, n+5 +6 ; n+7 +6 , n+8 +6 , n+9 +6 ; − β2t6 +9 +� +(n + 1)(n + 2)(n + 3) +. +(C26) +With the analytical expressions of Rn, Qn and Pn, one can +systematically derive c(n) +1 +and c(n) +2 . For example, for n = 2, +one obtains +c(2) +1 (t) = −t2 +2 +� +1 + 2F3 +� +1, 5 +6; 3 +6, 7 +6, 8 +6; −β2t6 +9 +�� +c(0) +1 (t) ++ t3 +3 2F3 +� +1, 5 +6; 7 +6, 8 +6, 9 +6; −β2t6 +9 +� +˙c(0) +1 (t), +(C27a) +c(2) +2 (t) = −t2 +2 +� +1 + 2F3 +� +1, 5 +6; 3 +6, 7 +6, 8 +6; −β2t6 +9 +�� +c(0) +2 (t) ++ t3 +3 2F3 +� +1, 5 +6; 7 +6, 8 +6, 9 +6; −β2t6 +9 +� +˙c(0) +2 (t). +(C27b) +[1] R. 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(Berlin) 533, 2000349 +(2021). +[28] B. Militello, Phys. Rev. A 99, 033415 (2019). +[29] R. Melanathuru, S. Malzard, and E. M. Graefe, Phys. Rev. A +106, 012208 (2022). +[30] K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, Phys. +Rev. X 6, 021007 (2016). + diff --git a/s9E3T4oBgHgl3EQf9gsi/content/tmp_files/load_file.txt b/s9E3T4oBgHgl3EQf9gsi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d44c6f295cf8dcd8e6757f8fb819b7c3c2dd29c1 --- /dev/null +++ b/s9E3T4oBgHgl3EQf9gsi/content/tmp_files/load_file.txt @@ -0,0 +1,568 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf,len=567 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='04816v1 [quant-ph] 12 Jan 2023 Analytical Approximations for Generalized Landau-Zener Transitions in Multi-level Non-Hermitian Systems Chon-Fai Kam∗ and Yang Chen† Department of Mathematics, Faculty of Science and Technology, University of Macau, Avenida da Universidade, Taipa, Macau, China We study the dynamics of non-adiabatic transitions in non-Hermitian multi-level parabolic models where the separations of the diabatic energies are quadratic function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The model Hamiltonian has been used to describe the non-Hermitian dynamics of two pairs of coupled cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' In the absence of the coupling between any two pairs of cavities, the wave amplitudes within each subsystem are described by the tri-confluent Heun functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' When all the couplings between the cavities are present, we reduce the dynamics into a set of two coupled tri-confluent Heun equations, from which we derive analytical approximations for the wave amplitudes at different physical limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' INTRODUCTION Since the dawn of the twentieth century, quantum mechan- ics has been the foundation of modern technology from the electronic computers to the most precise atomic clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' One of the basic principles of quantum mechanics is that the spectra of atoms are real and the time evolution of wave functions is unitary, and thus the total probability is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' As such, it was once widely believed that the Hamiltonian which de- scribes the time evolution of any physical system has to be self-adjoint or Hermitian [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' However, over the years, people started to realize that the Hermitian law is not unbreakable, as the deviation of it does not directly implies complex-valued spectra energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Since Bender and Boettcher’s groundbreak- ing work [2, 3], a new principle started to be affirmed is that real energy spectra of a physical system are not ensured by the Hermitian property, but rather ensured by the partity- time (PT ) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The fundamental difference between Hermitian system and non-Hermitian PT -symmetric system is that the former one can only be used to describes closed systems that have no exchange with the outer environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' but the later one can be used to describe open systems with two coupled subsystems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' each of which is in contact with the outer environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' but the probability in one subsystem with gain compensates another subsystem with loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' so that the en- tire system is in a dynamical equilibrium [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Over the decades,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' the new principle of pseudo-Hermiticity under PT - symmetry has opened up countless new opportunities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' and has also revealed various application in modern technology which ranges from optics [6] to One of the most interesting effects of non-Hermitian sys- tems is that the PT -symmetry leads to a new type of spectral degeneracies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' known as the exceptional points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' at which not only a finite numbers of eigenvalues coincide,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' but the asso- ciated eigenstates also coincide [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' In contrast to the spec- tral degeneracies of Hermitian systems, at which the eigen- states can be chosen to be orthogonal to each other, the spec- tral degeneracies in PT -symmetric systems are peculiar as ∗ Email: dubussygauss@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='com † Email: yangbrookchen@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='uk certain eigenstates are completely parallel and the Hamilto- nian matrix becomes defective at the exceptional points [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' This intriguing properties of non-Hermitian physics give rise to many counterintuitivefeatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' For example, a general PT - symmetric Hamiltonian may undergo a spontaneous symme- try breaking phase transition, beyond which complex eigen- values emerge [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' In particular, when encircling an excep- tional point, an unconventional level-crossing behavior will appear, along with a phase change of one eigenstate but not of the other [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' In ordinary Hermitian quantum mechanics, there is a fun- damental physical process called the Landau-Zener transition, which describes the transition between two energy levels of a quantum system directly driven through an avoided cross- ing [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The Landau-Zener transition assumes a constant coupling between bare states in the diabatic basis and a lin- early varying separation of diabatic energies [15], which can be exactly solved by the parabolic cylinder functions [16] or the integral representation method [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Although the model of Landau-Zener transition has achieved great success over the years, there are indeed cases where the assumption of lin- ear crossing between the diabatic states becomes no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' For the cases in which the crossing points merge to- gether as a result of external fields, the Landau-Zener lin- earization fails, and the linear-dependence of diabatic energies has to be replaced by a parabolic or superparabolic one [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Interestingly, the tunneling dynamics for the parabolic and cu- bic models can still be exactly solved by the tri-confluent and bi-confluent Heun functions [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' One can still express the tunneling probability via the Stokes constants by using the Zhu-Nakamura formula [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Compared to the two-state scenario, the research of gener- alized Landau-Zener transition for complicated systems with more than two states, even in ordinary Hermitian quantum mechanics, has been arduous and in most cases inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The main reason is that in conventional Landau-Zener tran- sition, the coupling equations which govern the non-adiabatic transition amplitudes between the two energy levels can be re- duced to a single second-order differential equation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=', the parabolic cylinder function for linear separation of diabatic energies [16], or the confluent Heun functions for quadratic and cubic separations of diabatic energies [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' In con- trast, in the general multi-state scenario, if one attempts to re- 2 duce the coupling equations which govern the non-adiabatic transition amplitudes between different energy levels in a sin- gle equation, one would probably obtain an ordinary differen- tial equation with order greater than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Compared to those of conventional second order differential equations, the ana- lytic properties of solutions of differential equations with or- der greater than two are harder to obtain by regular methods like asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The essential difficulty lies in the fact that the Stokes curves for conventional second order dif- ferential equations are straight lines which never cross each other, but those for differential equations with order greater than two are non-straight lines which always cross each other unavoidably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' This simple fact results in the breakdown of the standard connection formula near the crossing points of the Stokes curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' In this regard, the asymptotic WKB solutions of generalized Landau-Zener non-adiabatic transitions for multi- state systems are in general hard to obtain, if not totally im- possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Despite its evident importance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' the non-Hermitian general- ization of the two-level Landau-Zener transition has only re- cently been analyzed by Longstaff [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' the associated non- Hermitian Landau-Zener-Stuckelberg interferometry was an- alyzed by Shen [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' and the non-Hermitian generalization of the parabolic and super-parabolic models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' in which the exceptional points are driven through at finite speeds which are quadratic or cubic functions of time has been analyzed by the authors [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Compared to the two-state scenario, the research of the non-Hermitian generalization of Landau- Zener non-adiabatic transitions in the multi-state scenario is still at its early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Recently, the three-state non-Hermitian Landau-Zener model in the presence of an interaction with the outer environment has been considered [28], and the Landau- Zener transitions through a pair of higher order exceptional points has been analyzed by Melanathuru [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' In this work, based on our analytical approximation methods used in pre- vious researches [23, 24, 27], we will analyze the dynamics of a four-state non-Hermitian PT -symmetric system which directly passes through a collection of exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' THE MODEL To begin with, we consider a four-state non-Hermitian sys- tem which has been used to describe the dynamics of four cou- pled cavities with asymmetric losses [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The non-Hermitian system consists of two pairs of coupled cavities, each of which is described by a 2 × 2 non-Hermitian Hamiltonian Hi = � ωi − iΓ0 κ κ ωi − iΓ � , (1) where κ represents the coupling strength between the two cav- ities, ωi (i = 1, 2) represents the same resonant frequency of the two cavities, and Γ0 and Γ represent the intrinsic loss of the each cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The whole non-Hermitian system consists of two pairs of above-mentioned coupled cavities with the same values of κ, Γ0 and Γ but different resonant frequencies ω1 and ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The two pairs of cavities are coupled by connecting each individual cavity of one pair with that of another pair by a small tube by an inter-pair coupling strength η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The 4 × 4 non-Hermitian Hamiltonian of the whole system becomes H = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed ω2 − iΓ0 κ 0 η κ ω2 − iΓ η 0 0 η ω1 − iΓ0 κ η 0 κ ω1 − iΓ \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 , (2) Evidently, the 2 × 2 Hamiltonian for each pair of coupled cav- ities is PT-symmetric only when the intrinsic losses are sym- metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=', Γ0 = Γ, where P ≡ � 0 1 1 0 � denotes the parity op- erator, and T denotes complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' From the Hamil- tonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (2), one obtains the coupled equations for the four wave amplitudes a1, a2, a3 and a4 i˙a1 = (ω2 − iΓ0)a1 + κa2 + ηa4, (3a) i˙a2 = κa1 + (ω2 − iΓ)a2 + ηa3, (3b) i˙a3 = ηa2 + (ω1 − iΓ0)a3 + κa4, (3c) i˙a4 = ηa1 + κa3 + (ω1 − iΓ)a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (3d) Using the change of variable bi ≡ e � t 0 (¯Γ+i ¯ω)dsai to remove the average resonant frequency ¯ω ≡ 1 2(ω1 + ω2) and the average intrinsic loss ¯Γ ≡ 1 2(Γ + Γ0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' one obtains the new coupled equations for the wave amplitudes bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' which depend only on the relative resonant frequency ∆ω ≡ ω1 − ω2 and the relative intrinsic loss ∆Γ ≡ Γ − Γ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' as i˙b1 = −Ω∗b1 + κb2 + ηb4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (4a) i˙b2 = κb1 − Ωb2 + ηb3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (4b) i˙b3 = ηb2 + Ωb3 + κb4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (4c) i˙b4 = ηb1 + κb3 + Ω∗b4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (4d) where Ω ≡ 1 2(∆ω+i∆Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' One may further define c1 ≡ b1+ib2, c2 ≡ b1 − ib2, c3 ≡ b3 + ib4 and c4 ≡ b3 − ib4, and obtains i˙c1 = −∆ωc1 + iγc2 + iηc4, (5a) i˙c2 = iγ′c1 − ∆ωc2 − iηc3, (5b) i˙c3 = iηc2 + ∆ωc3 + iγc4, (5c) i˙c4 = −iηc1 + iγ′c3 + ∆ωc4, (5d) where γ ≡ ∆Γ + κ and γ′ ≡ ∆Γ − κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The Hamiltonian in the diabatic basis then reads H′ = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed −∆ω iγ 0 iη iγ′ −∆ω −iη 0 0 iη ∆ω iγ −iη 0 iγ′ ∆ω \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (6) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (5a) and (5b), one immediately obtains c3 = −1 η �˙c2 − i∆ωc2 − γ′c1 � , (7a) c4 = 1 η (˙c1 − i∆ωc1 − γc2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (7b) Substitution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (7a) and (7b) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (5c) and (5d) yields the following coupled equations ¨c1 + � η2 + ∆ω2 − γ′2 − i∆ ˙ω � c1 = 2κ˙c2 + 2i∆ω∆Γc2, (8a) ¨c2 + � η2 + ∆ω2 − γ2 − i∆ ˙ω � c2 = −2κ˙c1 + 2i∆ω∆Γc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (8b) 3 In particular, for the PT -symmetric case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=', ∆Γ = 0, one immediately obtains ¨c1 + Q(t)c1 = 2κ˙c2, (9a) ¨c2 + Q(t)c2 = −2κ˙c1, (9b) where Q(t) ≡ η2 − κ2 + ∆ω2(t) − i∆ ˙ω(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' A direct computation yields ˙c1c2 − ˙c2c1 − κ(c2 1 + c2 2) = Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (10) In particular, for a parabolic separation of diabatic energies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=', ∆ω = α + βt2, one will obtain Q(t) = η2 − κ2 + (α + βt2)2 − 2iβt = β2t4 + 2αβt2 − 2iβt + α2 + η2 − κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (11) One can see that as Q(t) is now a quartic function of time, in the absence of the coupling κ within each pair of cavities, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (9a) and (9b) are decoupled and solved by the superposi- tion of the tri-confluent Heun functions T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' When the coupling κ is nonzero, one can write c1 = �∞ n=0 κnc(n) 1 and c2 = �∞ n=0 κnc(n) 2 , where c(0) 1 ≡ d1T1 + d2T2 and c(0) 2 ≡ e1T1 + e2T2, and obtains the recursive relations ¨c(n) 1 + Q(t)c(n) 1 = 2˙c(n−1) 2 , (12a) ¨c(n) 2 + Q(t)c(n) 2 = −2˙c(n−1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (12b) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' INTEGRALS INVOLVING PRODUCTS OF HEUN FUNCTIONS AND THEIR DERIVATIVES To solve the recursive relations, one regards the right hand side of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (12a) and (12b) as known functions of time, then the n-th order terms c(n) 1 and c(n) 2 are integrals of the (n − 1)-th order terms ˙c(n−1) 1 and ˙c(n−1) 2 , which may be expressed as c(n) 1 = −2T1 W � T2˙c(n−1) 2 dt + 2T2 W � T1˙c(n−1) 2 dt, (13a) c(n) 2 = 2T1 W � T2˙c(n−1) 1 dt − 2T2 W � T1˙c(n−1) 1 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (13b) where W ≡ T1 ˙T2 − T2 ˙T1 is the Wronskian for T1 and T2, and is a constant of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Using integration by parts, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (13a) - (13b) may be simplified as c(n) 1 = 2T1 W � ˙T2c(n−1) 2 dt − 2T2 W � ˙T1c(n−1) 2 dt, (14a) c(n) 2 = −2T1 W � ˙T2c(n−1) 1 dt + 2T2 W � ˙T1c(n−1) 1 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (14b) To continue, one needs to evaluate integrals of the products of Heun functions and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' To be more precise, let us consider the following three kinds of indefinite integrals: � tny2dt, � tn˙yydt and tn˙y2dt, with y being any linear combi- nation of the tri-confluent Heun functions T1 and T2, which satisfies the tri-confluent Heun equation ¨y + Q(t)y = 0, where Q(t) ≡ �4 k=0 Aktk is a quartic function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Using the fol- lowing relations � tn(y1˙y2 + y2˙y1)dt = tny1y2 − n � tn−1y1y2dt and � tn(y1˙y2 − y2˙y1)dt = W n+1tn+1, one immediately obtains � tny1˙y2dt = 1 2 � tny1y2 + Wtn+1 n + 1 − n � tn−1y1y2dt � , (15) where y1 and y2 are two independent solutions of the tri- confluent equation, and W ≡ y1˙y2 − y2˙y1 is the Wronskian of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' The other two kinds of integrals will be more involved to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Let us define � tny1y2dt ≡ Pny1y2 + Qn 2 (y1˙y2 + y2˙y1) + Rn˙y1˙y2, (16a) � tn˙y1˙y2dt ≡ Lny1y2 + Mn 2 (y1˙y2 + y2˙y1) + Nn˙y1˙y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (16b) The coefficients Pn, Qn and Rn may be determined by taking derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (16a), which yields tny1y2 = ( ˙Pn − QQn)y1y2 + 1 2( ˙Qn + 2Pn − 2QRn)(y1˙y2 + y2˙y1) + ( ˙Rn + Qn)˙y1˙y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (17) Hence, we obtain ˙Pn − QQn = tn, ˙Qn + 2Pn − 2QRn = 0 and ˙Rn + Qn = 0, which are solved by Qn = − ˙Rn and Pn = 1 2 ¨Rn + QRn, where Rn satisfies the third order differential equation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Rn + 4Q ˙Rn + 2 ˙QRn = 2tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' To evaluate the coefficients Ln, Mn and Nn, one may use the following identity � tn˙y1˙y2dt = 1 2tn(y1˙y2 + y2˙y1) − n 2tn−1y1y2 + � tnQy1y2dt + n(n − 1) 2 � tn−2y1y2dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (18) Substitution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (16a) and Q(t) ≡ �4 k=0 Aktk into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (18) yields Ln = 4 � k=0 AkPn+k + n(n − 1) 2 Pn−2 − n 2tn−1, (19a) Mn = 4 � k=0 AkQn+k + n(n − 1) 2 Qn−2 + tn, (19b) Nn = 4 � k=0 AkRn+k + n(n − 1) 2 Rn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (19c) ACKNOWLEDGMENTS This study was supported by the National Natural Science Foundation of China (Grant nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 12104524).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Appendix A: Formal solutions of c(n) 1 and c(n) 2 From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (14a) and (14b), a direct computation will yield c(1) 1 = tc(0) 2 , c(1) 2 = −tc(0) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A1) 4 To proceed further, one can use the following integral identity with respect to any functions y1 and y2 and their derivatives � ty1˙y2dt = t 2y1y2 + Wt2 4 − 1 2 � y1y2dt, (A2) where W ≡ y1˙y2 − ˙y2y1 is the Wronskian with respect to y1 and y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' From this, a direct computation yields c(2) 1 = 1 2(Q0 − t2)c(0) 1 + R0˙c(0) 1 , (A3a) c(2) 2 = 1 2(Q0 − t2)c(0) 2 + R0˙c(0) 2 , (A3b) where Q0 ≡ − ˙R0 and R0 is the solution of the third order differential equation d3R0(t) dt3 + 4Q(t)dR0(t) dt + 2dQ(t) dt R0(t) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A4) In order to solve c(n) 1 and c(n) 2 , one needs the following integrals LnTk = W 2 �� tn+1 n + 1 − n 2 Qn−1 � Tk − nRn−1 ˙Tk � ≡ W 2 (EnTk + Fn ˙Tk), (A5a) Ln ˙Tk = W 2 (MnTk + 2Nn ˙Tk) = W 2 (GnTk + Hn ˙Tk), (A5b) where k = 1, 2, and Ln• ≡ T1 � tn ˙T2•dt−T2 � tn ˙T1•dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Form this, one may expand the functions En, Fn, Gn, and Hn (and thus LnTk and Ln ˙Tk) in series of time as LnTk ≡ W 2 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed ∞ � l=0 EnltlTk + ∞ � l=0 Fnltl ˙Tk \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 , (A6a) Ln ˙Tk = W 2 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed ∞ � l=0 GnltlTk + ∞ � l=0 Hnltl ˙Tk \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A6b) In general, one can express the n-th order correction terms c(n) 1 and c(n) 2 in terms of c(0) 1 , c(0) 2 , ˙c(0) 1 and ˙c(0) 2 , and expand the coefficients in series of time as c(n) 1 ≡ ∞ � k=0 � α(n) k c(0) 1 + β(n) k c(0) 2 + γ(n) k ˙c(0) 1 + δ(n) k ˙c(0) 2 � tk, (A7a) c(n) 2 ≡ ∞ � k=0 � λ(n) k c(0) 1 + µ(n) k c(0) 2 + ν(n) k ˙c(0) 1 + ξ(n) k ˙c(0) 2 � tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A7b) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A6a) - (A6b), one obtains the (n + 1)-th order correction terms c(n+1) 1 = 2 W ∞ � k=0 � λ(n) k Lkc(0) 1 + µ(n) k Lkc(0) 2 + ν(n) k Lk˙c(0) 1 + ξ(n) k Lk˙c(0) 2 � = ∞ � k,l=0 � (λ(n) k Ekl + ν(n) k Gkl)c(0) 1 + (µ(n) k Ekl + ξ(n) k Gkl)c(0) 2 + (λ(n) k Fkl + ν(n) k Hkl)˙c(0) 1 + (µ(n) k Fkl + ξ(n) k Hkl)˙c(0) 2 � tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A8a) c(n+1) 2 = −2 W ∞ � k=0 � α(n) k Lkc(0) 1 + β(n) k Lkc(0) 2 + γ(n) k Lk˙c(0) 1 + δ(n) k Lk˙c(0) 2 � = − ∞ � k,l=0 � (α(n) k Ekl + γ(n) k Gkl)c(0) 1 + (β(n) k Ekl + δ(n) k Gkl)c(0) 2 + (α(n) k Fkl + γ(n) k Hkl)˙c(0) 1 + (β(n) k Fkl + δ(n) k Hkl)˙c(0) 2 � tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A8b) A direct comparison between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A7a) and (A8a) yields the recursive relations α(n+1) k = ∞ � j=0 (λ(n) j E jk + ν(n) j G jk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A9a) β(n+1) k = ∞ � j=0 (µ(n) j E jk + ξ(n) j G jk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A9b) γ(n+1) k = ∞ � j=0 (λ(n) j F jk + ν(n) j H jk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A9c) δ(n+1) k = ∞ � j=0 (µ(n) j F jk + ξ(n) j H jk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A9d) λ(n+1) k = − ∞ � j=0 (α(n) j E jk + γ(n) j G jk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A9e) µ(n+1) k = − ∞ � j=0 (β(n) j E jk + δ(n) j G jk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A9f) ν(n+1) k = − ∞ � j=0 (α(n) j F jk + γ(n) j H jk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A9g) ξ(n+1) k = − ∞ � j=0 (β(n) j F jk + δ(n) j H jk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (A9h) Appendix B: Recursive relation for Rn The coefficients Rn for different n are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' From the identity d dt (˙y1˙y2 + Qy1y2) = ˙Qy1y2, we obtain the indefinite integral � ˙Qy1y2dt = ˙y1˙y2 + Qy1y2, which im- plies that �4 k=0 kAkRk−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Similarly, from the identity d dt[t(˙y1˙y2 + Qy1y2) − 1 2(y1˙y2 + y2˙y1)] = (2Q + t ˙Q)y1y2, we obtain the indefinite integral � (2Q + t ˙Q)y1y2dt = t(˙y1˙y2 + Qy1y2) − 1 2(y1˙y2 + y2˙y1), which yields �4 k=0(k + 2)AkRk = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 5 In general, one has the following identity � y1y2d(f Q) = f(˙y1˙y2 + Qy1y2) − � ˙f ˙y1˙y2dt for an arbitrary function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' In particular, for f = tn, one obtains the following indefinite in- tegral � (tn ˙Q + ntn−1Q)y1y2dt = tn(˙y1˙y2 + Qy1y2) − n � tn−1˙y1˙y2dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B1) Substitution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (18) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B1) yields � (tn ˙Q + 2ntn−1Q)y1y2dt = tn(˙y1˙y2 + Qy1y2) − n 2tn−1(y1˙y2 + y2˙y1) +n(n − 1) 2 tn−2y1y2 − n(n − 1)(n − 2) 2 � tn−3y1y2dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B2) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B2) and using tn ˙Q + 2ntn−1Q = �4 k=0(2n + k)Aktn+k−1, one obtains the recursive relations for Rn with n ≥ 3 4 � k=0 (2n + k)AkRn+k−1 = tn − n(n − 1)(n − 2) 2 Rn−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B3) For example, for the Airy functions which satisfy ¨y+Q(t)y = 0 with Q(t) = −t, one recovers the recursive relation obtained by the authors previously [23] Rn = 1 2(2n + 1) �n(n − 1)(n − 2)Rn−3 − 2tn� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B4) while for the Bessel functions which satisfy ¨y+λ2t4y = 0 with Q(t) = λ2t4, one recovers the simple recursive relations [23] Rn+3 = −1 4(n + 2)λ2 �n(n − 1)(n − 2)Rn−3 − 2tn� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B5) In general, when A4 � 0, one can obtain an explicit expression for Rn in terms of R0, R1 and R2 by revising Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B3) as Rn+3 + 2 � k=−2 gk+2 n Rn+k = Jn, (B6) where Jn ≡ 2tn−n(n−1)(n−2)Rn−3 2(2n+4)A4 , gk+1 n ≡ (2n+k) (2n+4) Ak A4 , and g0 n ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Clearly, all Rn may be expressed in terms of R0, R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Explicitly, the first few terms may be expressed as R3 = J0 − 2 � k=0 gk+2 0 Rk, (B7a) R4 = J1 − g4 1J0 − 2 � k=0 hk+1 1 Rk, (B7b) R5 = J2 − g4 2J1 − h3 2J0 − 2 � k=0 wk 2Rk, (B7c) R6 = J3 − g4 3J2 − h3 3J1 − w2 3J0 − 2 � k=1 uk−1 3 Rk + (g1 1h3 3 + g2 0w2 3)R0, (B7d) where hk n ≡ gk n−g4 ngk+1 n−1, wk n ≡ hk n−h3 ngk+2 n−2, and uk n ≡ wk n−w2 ngk+3 n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' A direct computation yields R3 = 1 4A4 (1 − 3A3R2 − 2A2R1 − A1R0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B8a) R4 = t 6A4 − 5A3 24A2 4 − \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 2A2 3A4 − 5A2 3 8A2 4 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 R2 − \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed A1 2A4 − 5A2A3 12A2 4 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 R1 − \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed A0 3A4 − 5A1A3 24A2 4 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B8b) R5 = t2 8A4 − 7A3t 48A2 4 − 3A2 16A2 4 + 35A2 3 192A3 4 − \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 5A1 8A4 − 55A2A3 48A2 4 + 105A3 3 192A3 4 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 R2 − \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed A0 2A4 − 21A1A3 48A2 4 − 3A2 2 8A2 4 + 35A2A2 3 96A3 4 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 R1 + \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 7A0A3 24A2 4 + 3A1A2 16A2 4 − 35A1A2 3 192A3 4 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 R0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='(B8c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='R6 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='t3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='10A4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='1920A4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (B8d) Appendix C: Explicit expressions for c(n) 1 (t) and c(n) 2 (t) for the cases when |t| ≪ 1 and |t| → ∞ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Cases for |t| ≪ 1 and Q(t) ≈ −2iβt + α2 + η2 − κ2 When |t| ≪ 1, one can simply retain the linear teams in Q(t) and obtains Q(t) ≈ A1t + A0 = −2iβt + α2 + η2 − κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' After the coordinate transformation z ≡ g(t + A0/A1), c(n) 1 and c(n) 2 are determined by the recursive relations d2c(n) 1 dz2 − zc(n) 1 = 2gdc(n−1) 2 dz , (C1a) d2c(n) 2 dz2 − zc(n) 2 = −2gdc(n−1) 1 dz , (C1b) where g ≡ eiπ/3A1/3 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Here, the lowest-order terms c(0) 1 and c(0) 2 are solved by the linear combinations of the Airy functions Ai(z) and Bi(z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=', c(0) 1 (z) = d1 Ai(z) + d2 Bi(z) and c(0) 2 (z) = 6 e1 Ai(z) + e2 Bi(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' To proceed further, notice that the integral of the product of any linear combinations of the Airy functions has the form � zny1y2dz ≡ Pny1y2 + Qn 2 (y1y′ 2 + y2y′ 1) + Rny′ 1y′ 2, (C2) where Pn = 1 2R′′ n −zRn, Qn = −R′ n and Rn is determined by the third-order differential equation d3Rn dz3 − 4zdRn dz − 2Rn = 2zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C3) A straightforward computation yields Rn, Qn and Pn for n ≤ 2 R0 = −1, Q0 = 0, P0 = z, (C4a) R1 = − z 3, Q1 = 1 3, P1 = z2 3 , (C4b) R2 = −z2 5 , Q2 = 2z 5 , P2 = z3 − 1 5 , (C4c) where Rn for n ≥ 3 is determined by the recursive relation Rn = − zn 2n + 1 + n(n − 1)(n − 2) 2(2n + 1) Rn−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C5) Hence, Rn for n ≥ 3 is solved by Rn = − zn 2n + 1 − n(n − 1)(n − 2)zn−3 2(2n + 1)(2n − 5) − · · · − n(n − 1) · · ·(n − 3k + 1)zn−3k 2k(2n + 1) · · ·(2n + 1 − 6k) = − zn 2n + 1 k � j=0 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='Γ( 2n+1 6 )(12z3)− j (n − 3 j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='Γ( 2n+1 6 − j), (C6) where k is the number such that n − 3k ∈ [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Using the relation Qn = − ˙Rn, a direct computation yields Qn = nzn−1 2n + 1 + n(n − 1)(n − 2)(n − 3)zn−4 2(2n + 1)(2n − 5) + · · · + n(n − 1) · · ·(n − 3l)zn−3l−1 2k(2n + 1) · · ·(2n + 1 − 6l) = nzn−1 2n + 1 l� j=0 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='Γ( 2n+1 6 )(12z3)− j (n − 1 − 3 j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='Γ( 2n+1 6 − j), (C7) where l is the number such that n − 3l − 1 ∈ [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Using the above relations, Rn, Qn and Pn for n ≤ 6 can be explicitly expressed as R3 = −z3 + 3 7 , Q3 = 3z2 7 , P3 = z4 7 , (C8a) R4 = −z4 + 4z 9 , Q4 = 4z3 + 4 9 , P4 = z5 − 2z2 9 , (C8b) R5 = −z5 + 6z2 11 , Q5 = 5z4 + 12z 11 , P5 = z6 − 4z3 − 6 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C8c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Cases for |t| ≫ 1 and Q(t) ≈ β2t4 In the region |t| ≫ 1, one can only keep the highest order term in Q(t) such that Q(t) ≈ β2t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Then, c(n) 1 (t) and c(n) 2 (t) are determined by the recursive equations ¨c(n) 1 + β2t4c(n) 1 = 2˙c(n−1) 2 , (C9a) ¨c(n) 2 + β2t4c(n) 2 = −2˙c(n−1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C9b) Here the zeroth order terms c(0) 1 (t) and c(0) 2 (t) are linear combi- nation of ¯y1 ≡ √tJ1/6(βt3/3) and ¯y2 ≡ √tJ−1/6(βt3/3), where Jν(z) are the Bessel function of the first kind defined by Jν(z) ≡ ∞ � n=0 (−1)n Γ(ν + n + 1)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' � z 2 �ν+2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C10) Hence, the fundamental solutions y1(t) and y2(t) of the equa- tion ¨y + β2t4y = 0 and their derivatives have the series expan- sions y1 ≡ Γ �7 6 � �β 6 �− 1 6 ¯y1 = ∞ � n=0 Γ( 7 6)(−1)n Γ( 7 6 + n)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' �β 6 �2n t1+6n, (C11a) y2 ≡ Γ �5 6 � �β 6 � 1 6 ¯y2 = ∞ � n=0 Γ( 5 6)(−1)n Γ( 5 6 + n)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' �β 6 �2n t6n, (C11b) ˙y1 = ∞ � n=0 Γ( 1 6)(−1)n Γ( 1 6 + n)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' �β 6 �2n t6n, (C11c) ˙y2 = −β2t5 5 ∞ � n=0 Γ( 11 6 )(−1)n+1 Γ( 11 6 + n)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' �β 6 �2n t6n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C11d) The fundamental solutions y1(t) and y2(t) and their derivatives can be expressed in terms of the generalized hypergeometric functions pFq(a1, · · · , ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' b1, · · · , bq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' z) as y1(t) = 1F2 � 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 7 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 36 � t, (C12a) y2(t) = 1F2 � 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 36 � , (C12b) ˙y1(t) = 1F2 � 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 1 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 36 � , (C12c) ˙y2(t) = −1F2 � 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 11 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 36 � β2t5 5 , (C12d) which obey the initial conditions y1(0) = 0, y2(0) = 1, ˙y1(0) = 1 and ˙y2(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Hence, the Wronskian with respect to y1 and y2 is a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=', W(y1, y2) ≡ y1˙y2 − y2˙y1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' A direct computation yields the products of the fundamental solutions and their derivatives y1y2 = 1F2(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 36 )1F2(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 7 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 36 )t, ˙y1˙y2 = −1F2(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 1 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 36 )1F2(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 11 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 36 )β2t5 5 , y1˙y2 + y2˙y1 = 1F2(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 1 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −z2 4 )1F2(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −z2 4 ) − 1F2(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 7 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −z2 4 )1F2(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 1, 11 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −z2 4 )β2t6 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C13) 7 Thus, the above products can be expanded in the Taylor series as y1y2 = t − 2 35β2t7 + 6 5005β4t13 + O(t19), y1˙y2 + y2˙y1 = 1 − 2 5β2t6 + 6 385β4t12 + O(t18), ˙y1˙y2 = −1 5β2t5 + 2 55β4t11 + O(t17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C14) Similar to the previous case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' the integral of the product of the fundamental solutions y1(t) and y2(t) with a power weight tn can be written in the form � tny1y2dt ≡ Pny1y2 + Qn 2 (y1˙y2 + y2˙y1) + Rn˙y1˙y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C15) where Pn = 1 2 ¨Rn + β2t4Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Qn = − ˙Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' and Rn is determined by the third-order differential equation d3Rn dt3 + 4β2t4 dRn dt + 8β2t3Rn = 2tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C16) from which one can show that Rn obey the recursive relation Rn+6 = tn+3 2β2(n + 5) − (n + 3)(n + 2)(n + 1) 4β2(n + 5) Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C17) A direct computation gives Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Qn and Pn for n = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 5 R3(t) = 1 4β2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='Q3(t) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' P3(t) = t4 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C18a) R4(t) = t 6β2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='Q4(t) = − 1 6β2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' P4(t) = t5 6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C18b) R5(t) = t2 8β2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content='Q5(t) = − t 4β2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' P5(t) = t6 8 + 1 8β2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C18c) Then, one only need to compute Rn(t) for n = 0, 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' To proceed further, one can expand the function R0(t) as R0(t) = �∞ k=0 rktk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C16), one immediately obtains the recursive relation rk+6 = −4β2(k + 2) (k + 6)(k + 5)(k + 4)rk, (C19) For n = 0, the functions R0, Q0 and P0 can be expressed in terms of the generalized hypergeometric function as R0(t) = 2F3 � 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 7 6, 8 6, 9 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t3 3 , (C20a) Q0(t) = −2F3 � 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 3 6, 7 6, 8 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t2, (C20b) P0(t) = 2F3 � 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 2 6, 3 6, 7 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t + 2F3 � 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 7 6, 8 6, 9 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � β2t7 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C20c) These functions define different entire functions of t, which has the following series expansions R0(t) = 1 3t3 − 5 378β2t9 + 11 51597β4t15 + O(t21), Q0(t) = −t2 + 5 42β2t8 − 55 17199β4t14 + O(t20), P0(t) = t − 1 7β2t7 + 5 546β4t13 + O(t19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C21) For n = 1, the functions R1, Q1 and P1 can be expressed in terms of the generalized hypergeometric function as R1(t) = 2F3 � 1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 8 6, 9 6, 10 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t4 12, (C22a) Q1(t) = −2F3 � 1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 4 6, 8 6, 9 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t3 3 , (C22b) P1(t) = 2F3 � 1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 3 6, 4 6, 8 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t2 2 + 2F3 � 1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 8 6, 9 6, 10 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � β2t8 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C22c) These functions define different entire functions of t, which has the following series expansions R1(t) = 1 12t4 − 1 360β2t10 + 1 25200β4t16 + O(t22), Q1(t) = −t3 3 + 1 36β2t9 − 1 1575β4t15 + O(t21), P1(t) = t2 2 − 1 24β2t8 + 1 504β4t14 + O(t20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C23) For n = 2, the functions R2, Q2 and P2 can be expressed in terms of the generalized hypergeometric function as R2(t) = 2F3 � 1, 7 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 9 6, 10 6 , 11 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t5 30, (C24a) Q2(t) = −2F3 � 1, 7 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 5 6, 9 6, 10 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t4 6 , (C24b) P2(t) = 2F3 � 1, 7 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 4 6, 5 6, 9 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � t3 3 + 2F3 � 1, 7 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 9 6, 10 6 , 11 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � β2t9 30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C24c) These functions define different entire functions of t, which has the following series expansions R2(t) = 1 30t5 − 7 7425β2t11 + 91 7573500β4t17 + O(t23), Q2(t) = −t4 6 + 7 675β2t10 − 91 445500β4t16 + O(t22), P2(t) = t3 3 − 1 54β2t9 + 7 10125β4t15 + O(t21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C25) From the recursive relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C17), one can derive Rn(t), Qn(t) and Pn(t) for n = 6k + m with m = 0, 1, 2, and k being 8 any non-negative integer Rn(t) = 2tn+32F3 � 1, n+5 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' n+7 6 , n+8 6 , n+9 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' − β2t6 9 � (n + 1)(n + 2)(n + 3) , Qn(t) = − 2tn+22F3 � 1, n+5 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' n+3 6 , n+7 6 , n+8 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' − β2t6 9 � (n + 1)(n + 2) , Pn(t) = tn+12F3 � 1, n+5 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' n+2 6 , n+3 6 , n+7 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' − β2t6 9 � n + 1 + 2β2tn+72F3 � 1, n+5 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' n+7 6 , n+8 6 , n+9 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' − β2t6 9 � (n + 1)(n + 2)(n + 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C26) With the analytical expressions of Rn, Qn and Pn, one can systematically derive c(n) 1 and c(n) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' For example, for n = 2, one obtains c(2) 1 (t) = −t2 2 � 1 + 2F3 � 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 3 6, 7 6, 8 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 �� c(0) 1 (t) + t3 3 2F3 � 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 7 6, 8 6, 9 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � ˙c(0) 1 (t), (C27a) c(2) 2 (t) = −t2 2 � 1 + 2F3 � 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 3 6, 7 6, 8 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 �� c(0) 2 (t) + t3 3 2F3 � 1, 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 7 6, 8 6, 9 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' −β2t6 9 � ˙c(0) 2 (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' (C27b) [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Shankar, Principles of quantum mechanics (Springer Science & Business Media, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Bender and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Boettcher, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 80, 5243 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' El-Ganainy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Makris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Khajavikhan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Mussli- mani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 18, 783 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Heiss and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Sannino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 23, 1167 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Heiss, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' E 61, 929 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Landau, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E3T4oBgHgl3EQf9gsi/content/2301.04816v1.pdf'} +page_content=' 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a/sNAyT4oBgHgl3EQfZve0/content/tmp_files/2301.00230v1.pdf.txt b/sNAyT4oBgHgl3EQfZve0/content/tmp_files/2301.00230v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ddc5dc0a3a8982afc60faa1273531b5fd417f6fc --- /dev/null +++ b/sNAyT4oBgHgl3EQfZve0/content/tmp_files/2301.00230v1.pdf.txt @@ -0,0 +1,1713 @@ +1 +Disjoint Masking with Joint Distillation for +Efficient Masked Image Modeling +Xin Ma, Chang Liu, Chunyu Xie, Long Ye, Yafeng Deng, and Xiangyang Ji +Abstract—Masked image modeling (MIM) has shown great +promise for self-supervised learning (SSL) yet been criticized +for learning inefficiency. We believe the insufficient utilization +of training signals should be responsible. To alleviate this issue, +we introduce a conceptually simple yet learning-efficient MIM +training scheme, termed Disjoint Masking with Joint Distillation +(DMJD). For disjoint masking (DM), we sequentially sample +multiple masked views per image in a mini-batch with the disjoint +regulation to raise the usage of tokens for reconstruction in +each image while keeping the masking rate of each view. For +joint distillation (JD), we adopt a dual branch architecture to +respectively predict invisible (masked) and visible (unmasked) +tokens with superior learning targets. Rooting in orthogonal +perspectives for training efficiency improvement, DM and JD +cooperatively accelerate the training convergence yet not sacri- +ficing the model generalization ability. Concretely, DM can train +ViT with half of the effective training epochs (3.7× less time- +consuming) to report competitive performance. With JD, our +DMJD clearly improves the linear probing classification accuracy +over ConvMAE by 5.8%. On fine-grained downstream tasks +like semantic segmentation, object detection, etc., our DMJD +also presents superior generalization compared with state-of-the- +art SSL methods. The code and model will be made public at +https://github.com/mx-mark/DMJD. +Index Terms—Disjoint masking, Joint distillation, Masked +image modeling, Self-supervised learning, and Training efficiency. +I. INTRODUCTION +W +ITH the surge of vision transformers [1]–[6], masked +image modeling (MIM) has recently prevailed on the +self-supervised representation learning (SSL) leaderboard [7]– +[9]. These approaches assimilate context semantics by pre- +dicting a portion of masked tokens. It has been justified that +discrete visual tokens [10]–[13], raw pixels [14], [15], and +hand-crafted features [16] are suitable targets to learn versatile +models for a broad spectrum of downstream tasks. +However, MIM methods typically require tremendous train- +ing costs, eg., thousands of pre-training epochs, which overbur- +dens academic and industrial communities. In general, an input +image is masked out with a high masking rate, also named +corruption rate (mcorr). While the rest unmasked patches are +ignored for self-supervision. We argue that a large proportion +of input signals is not fully exploited for model learning at each +X. Ma and L. Ye are with the School of Information and Communication +Engineering, Communication University of China, Beijing 100024, China, +Email: {mx mark, yelong}@cuc.edu.cn. +C. Liu and X. Ji are with the Department of Automation, Tsinghua University, +Beijing 100084, China, Email: {liuchang2022, xyji}@tsinghua.edu.cn. +C. Xie and Y. Deng are with 360 AI Research, Beijing, China, Email: +yuxie@buaa.edu.cn, dengyafeng@gmail.com. +X. Ji is the corresponding author. +0 +20 +40 +60 +80 +100 +Epochs +65.0 +67.5 +70.0 +72.5 +75.0 +77.5 +80.0 +82.5 +85.0 +Top 1 Acc. (Ft.) +baseline +raise both (0.9) +lower both (0.5) +(a) Varying both (mcorr = mpred). +0 +20 +40 +60 +80 +100 +Epochs +65.0 +67.5 +70.0 +72.5 +75.0 +77.5 +80.0 +82.5 +85.0 +Top 1 Acc. (Ft.) +baseline +raise mpred (1.0) +lower mpred (0.2) +(b) Varying mpred (mcorr = 0.75). +0 +20 +40 +60 +80 +100 +Epochs +65.0 +67.5 +70.0 +72.5 +75.0 +77.5 +80.0 +82.5 +85.0 +Top 1 Acc. (Ft.) +baseline +raise mcorr (0.9) +lower mcorr (0.5) +(c) Varying mcorr (mpred = 0.75). +0 +20 +40 +60 +80 +100 +Epochs +65.0 +67.5 +70.0 +72.5 +75.0 +77.5 +80.0 +82.5 +85.0 +Top 1 Acc. (Ft.) +baseline + +DM + +DMJD +(d) DMJD (Ours). +Fig. 1. MIM Training efficiency analysis. Specifically, we pre-train models +based on MAE [14] for 100 epochs on ImageNet-1k [17] with uniform masking +and plot the corresponding convergence curve on the validation set during +fine-tuning. The blue curve in each sub-figure denoted as the “baseline” is +reported under the optimal setting of mpred = mcorr = 75%. In sub-figures +(a), (b), and (c), no matter how mcorr and mpred change in each view of +an image, the optimal choice for fast convergence is the baseline setting. It +inspires us to develop a multi-view masking strategy, i.e., DM, to raise the +overall mpred in an image while keeping mpred equal to mcorr in each view. +In the spirit of increasing the utilization of input signals, we also introduce +JD with an additional visible distillation branch, which works cooperatively +with DM to expedite the training procedure, sub-figure (d). +loop mainly accounts for the training inefficiency. Actually, +the training efficiency of MIM is largely determined by the +prediction rate (mpred), the overall portion of input images +used to provide supervision for invisible region reconstruction. +To go deeper into how mcorr and mpred affect MIM training +efficiency, we quantitatively visualize the model convergence +curve with various settings in Fig. 1. Concretely, in Fig. 1a and +1c, one can see that only with a proper corruption rate, i.e., 0.75 +with uniform masking, MIM exhibits superior effectiveness and +efficiency. This is because since mcorr defines the difficulty +of the reconstruction task, the training efficiency intrinsically +depends on mcorr. With a too-high corruption rate, there is not +arXiv:2301.00230v1 [cs.CV] 31 Dec 2022 + +2 +Target +Input +Invisible Reconstruction +Supervision +(a) Vanilla MIM. +Invisible Reconstruction +Visible Distillation +Supervision +Multiple Masked Views +Target +Disjoint Masking + (DM) +Joint Distillation + (JD) +(b) DMJD (Ours). +Fig. 2. Pipeline illustration of vanilla MIM and our DMJD. Vanilla MIM +methods typically adopt a single-view masking strategy to perform invisible +reconstruction with mcorr = mpred. In contrast, our DMJD introduces a +multi-view masking strategy, i.e., DM, to increase the overall prediction rate of +each image while keeping mcorr = mpred in each view and a dual-branch +joint distillation architecture with an additional visible distillation branch to +take full use of the input signals with superior targets, eg. HOG. +enough context to correctly recover the masked patches [14], +making the model more intricate and unpredictable. And a low +corruption rate will degrade the model performance because the +model can trivially recover a missing patch from neighboring +patches without needing a high-level visual understanding. +What is worse, with the optimal setting of mcorr = 0.75, +naively varying mpred also hampers the model training, Fig. 1b. +These facts reveal the challenges of improving MIM training +efficiency and inspire us to think differently to increase the +utilization of input signals. +In this paper, we develop a conceptually simple yet learning- +efficient MIM training scheme, termed Disjoint Masking with +Joint Distillation (DMJD), Fig. 2. Disjoint masking (DM) is +a multiple-view sampling strategy targeting flexibly raising +the prediction rate of each input while keeping the corruption +rate of each view. Concretely, we sequentially sample a series +of masked views of an image from its disjoint portions, i.e., +overall have been previously unmasked/masked patches, for +wider coverage of the invisible regions in a training loop +for reconstruction. Since the increasing utilization of training +signals in a batch may reduce the gradient variance and hamper +the generalization of learned models [18]–[20], we introduce an +adaptive learning rate scale rule taking the relative prediction +rate increment as a scale factor to enhance the model training. +Joint distillation (JD) performs distillation on visible and +invisible regions with a dual branch architecture to further +increase training efficiency. In addition to the original masked +prediction branch (MPB), we add a visible distillation branch +(VDB) to provide semantic guidance through parametric or +non-parametric tokenizers for visible token learning. With +increased prediction rates, visible distillation, and superior +targets, the proposed DMJD ensures higher learning efficiency +from orthogonal perspectives yet works cooperatively. +It is worth noting that our DMJD accelerates training conver- +gence but not sacrificing model generalizability. Specifically, +ViT/ConViT [2], [21] equipped with DMJD typically improves +performances with improved training efficiency not only on +ImageNet-1K classification but also on fine-grained downstream +tasks, eg. semantic/instance segmentation, object detection, +etc. Take an example, for linear probing classification on +MaskFeat [16] and ConvMAE [21] baselines, DMJD achieves +performance gains of 3.4% and 5.8% with 1.8× and 3× +acceleration. We hope these observations shed new light on +MIM training efficiency research in the community. +Our contributions can be summarized as follows: +• We propose a conceptually simple yet learning-efficient +MIM training scheme, termed disjoint masking with +joint distillation (DMJD), which targets increasing the +utilization of per image at each training loop. +• We devise a multi-view generation strategy, i.e., disjoint +masking (DM), to increase the prediction rate while +keeping the corruption rate for efficient MIM and introduce +the adaptive learning rate scale rule for better model +generalization with augmented training batches. +• We develop a dual-branch architecture for joint distillation +(JD), effectively pursuing representation learning on both +visible and invisible regions with superior targets. +• We conduct sufficient evaluations justifying our DMJD can +significantly accelerate model convergence and achieve +outstanding performances on standard benchmarks. +II. RELATED WORK +A. Masked Image Modeling +Since ViT [2] overcomes the architectural obstacle of +applying masked signal modeling for visual pre-training, MIM +has achieved great success recently. The increasing attention +can be roughly categorized into three aspects, including learning +targets, backbone extensions, and masking strategies. +Learning targets are critical since they provide explicit +guidance for visual learning through reconstruction and their +characteristics can be injected into the learned model. MAE [14] +and SimMIM [15] reveal that raw pixels as reconstruction +targets are adequate for effective visual learning. MaskFeat [16] +introduces local invariant features, such as HOG [22], to avoid +modeling high-frequency low-level details in the pixel space +and concentrate meaningful semantic abstraction. BEiT [10] +and PeCo [11] further apply a discrete visual codebook + +3 +Masked Prediction Branch +Visible Distillation Branch +Joint Distillation +(JD) +Encoder +… +Disjoint Masking + (DM) +Decoder +⁝ +⁝ +⁝ +Projector +& +Predictor +⁝ +⁝ +tkn +: Concatenate +: Visible Token Features +: Masked Token +: Target +i + +k +i + +k +i + +i + +i + +mim +L +vis +L +1 +) +( +i +i +k + + + +X +Fig. 3. The pipeline of the proposed DMJD. During pre-training, K masked views of each image are randomly sampled in a mini-batch with DM. Then, they +will be fed to the encoder and the dual branch decoder for invisible reconstruction and visible distillation with targets extracted by the tokenizer tkn(·). +produced by dVAE [23] with an additional pre-training stage for +high-level semantic abstraction. To get rid of extra pre-training, +iBOT [13] and data2vec [24] jointly optimize the model and +the target tokenizer. With the fast development of multi-modal +foundation models, CLIP [25] is actively exploited yet not +limited as an effective target tokenizer for MIM [26]–[30]. +As advanced backbones take advantage of more discrimina- +tive representations, researchers are inspired to extend MIM to +multi-scale hybrid convolution-transformer architectures [15], +[21]. Also, to explore what kind of information in images +to be predicted benefits the model learning, several masking +strategies [31]–[34] have been proposed to improve the vanilla +random masking. However, tremendous training costs for +convergence limit MIM methods for application and research. +The training efficiency issue becomes urgent to be attended. +B. MIM Training Efficiency +Powerful computation resources have grown rapidly to +facilitate fast training with lower numerical precision and +larger batch sizes for large-scale models [19], [20], [35]– +[41]. However, the high computation and time costs are +still an obstacle to the practical applications of MIM. A +feasible attempt is to apply efficient hierarchical transformers. +Concretely, UM-MAE [42] designs a masking strategy to +preserve the relative position relationship between visible +patches. GreenMIM [43] proposes a uniform partition to ensure +that visible patches within each local window can be grouped +with equal size. MixMIM [44] creates a mixed view that +the masked tokens from one image are replaced with visible +tokens from another image. An alternate way for MIM training +efficiency is reducing the computational complexity of invisible +reconstruction. Specifically, LoMaR [45] performs masked +reconstruction within small 7×7 windows. FastMIM [46] +directly pre-trains model with low-resolution inputs. +Differently, inspired by MLM works [47], [48] which attempt +to supervise a larger portion of input signals for learning +efficiency, our DMJD is devised to increase the utilization of +each image with a multi-view masking strategy [49]–[52] and +dual-branch distillation using high-level learning targets. +III. METHOD +A. Overview +MIM methods are typically built on ViTs [2], where an input +image Xi will be split into a set of non-overlapping patches +and linearly projected to a sequence of tokens. This process can +be formulated as Xi = {(xij, pij)N +j=1}, where pij denotes the +positional embedding of the j-th token xij. To yield masked +views for reconstruction, specific masking strategies can be +defined as +M(Xi, mpatt, mcorr) = Mi, +(1) +where Mi = (mij)N +j=1, mij equals 1 if the j-th token xij +needs to be masked, otherwise 0. mcorr and mpatt are the +corruption rate and the masking pattern, eg. random, block-wise, +etc. The reconstruction loss function is defined as +Lmim = − +B +� +i=1 +N +� +j=1 +mij · ℓ(x′ +ij, tij), +(2) +where x′ +ij and tij are the network prediction and the corre- +sponding reconstruction target of xij. B is the batch size. ℓ(·) +is typically the mean squared error (MSE). +Since gradients are only back-propagated on masked tokens, +a small subset of an image, it may cause the training inefficiency +problem [47], [48]. And it is not wise to simply increase +the masking rate or recover unmasked patches, which even + +4 +1 +1 +1 +1 +1 +1 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +1 +1 +1 +1 +0 +0 +1 +1 +0 +0 +1 +1 +0 +0 +0 +0 +1 +1 +1 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +Previous masks +⁝ +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +0 +0 +0 +0 +1 +1 +1 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +K +i + +M +1 +i + +1 +K +i + + +1 +i +i +K  + +X M +1) +(1 +K +i +i + + + +X +M +K +i + +i +X +���′������������ > ��� +Disjoint Regulation +Fig. 4. Illustration of disjoint masking. With DM, we sequentially sample the K-th view from disjoint regions of the currently have not been masked token set +Xi · MK−1 +i +and the rest part Xi · (1 − MK−1 +i +) with the disjoint regulation, i.e., keeping the corruption rate m +′ +corr in Xi · (1 − MK−1 +i +) positive. The +regulation ensures the overall prediction rate of each image is larger than the pre-defined corruption rate. +slows down the convergence, Fig. 1. Thus, we propose +disjoint masking with joint distillation (DMJD), a dual-branch +distillation framework with a multiple-masked view sampling +approach to increase the prediction rate with a fixed corruption +rate for efficient MIM, Fig. 3. +Concretely, we impose a masking regulation to generate +multiple complementary views facilitating more invisible tokens +of each image to be reconstructed in the masked prediction +branch (MPB). In the visible distillation branch (VDB), we +bridge the semantic gap between the visible token features and +their corresponding learning targets through distillation, further +enhancing the training efficiency [16], [24], [53]. +B. Disjoint Masking +DM targets sequentially sampling multiple masked views +under a regulation to increase the overall portions of tokens +used for reconstruction. Since augmented views reduce the +samples with a given batch size, We modify the learning rate +scale rule to optimize the model training for generalization. +1) Disjoint Regulation: Suppose M1 +i = M(Xi, mpatt, +mcorr). The set of tokens that have been masked in the previous +K-1 masks is defined as MK−1 +i += 1 − +K−1 +� +k=1 +(1 − M k +i ), where +K ≥ 2. Thereafter, DM (Mdm) with the disjoint regulation +can be formulated based on Eq. 1 as, +Mdm(Xi, mpatt, mcorr, MK−1 +i +) += M K +i , +s.t. +M K +i +· (1 − MK−1 +i +) +̸= 0. +(3) +It is not hard to find that only if M K +i +can mask tokens that +have never been masked once, Eq. 3 can hold. Specifically, to +sample the K-th mask under the regulation, we first split the +image tokens into two disjoint subsets: Xi · MK−1 +i +(previously +masked tokens) and Xi · (1 − MK−1 +i +) (currently unmasked +tokens), Fig. 4. Then, we partially sample tokens to be masked +from the unmasked token set with a positive corruption rate +m +′ +corr > 0 and the rest from the masked set. In this way, DM +guarantees that each masked view will include new tokens +beyond previous views with a fixed corruption rate while +increasing the overall prediction rate of each image. +2) Adaptive Learning Rate Scale: Since gradients are +averaged in a mini-batch, the learning rate scale rule is proposed +to alleviate the generalization degradation caused by the over- +smoothed gradient variance with large batch sizes [35]. It +normally scales up the base learning rate ηbase with the batch +size b as η = ηbase · b/256, where 256 is a scale factor. +With DM, a mini-batch only contains bu unique samples, +where bu = b/K. Since repetitively utilizing each image has +a similar effect of enlarging the batch size on the gradient +variance, we intend to scale up the learning rate by the relative +utilization improvement of tokens used for reconstruction, i.e., +mpred +mcorr . Thus, the modified adaptive learning rate scale can be +defined as +η = ηbase · +� +bu · mpred +mcorr +� +/ 256. +(4) +C. Joint Distillation +As shown in Fig. 3, we add an additional visible distillation +branch (VDB) to bridge the visible encoding features and its +corresponding targets [16], [24], [53]. +1) Model Architecture: +a) Projector (g) & Predictor (q): We introduce a nonlin- +ear projector to reduce the information loss [54]. Specifically, it +has a 3-layer MLP-like structure with a LayerNorm (LN) [55] +applied to each fully-connected (FC) layer following Sim- +CLR [54]. The output dimension of all MLP hidden layers is +512. Only the visible token features are fed to the projector. +To align the output dimensions of the projector and the target +tokenizer, we insert a single FC layer as the predictor. +b) Target Encoding (tkn(·)): We evaluate two types of +targets to introduce direct guidance for visible token learning: i) +representative local invariant features, eg. HOG [22]; ii) features +extracted by deep models which extract context information +well, eg. CLIP [25]. The target encoding process can be +expressed as Ti = tkn(Xi). +2) Learning Objectives: We distill the target through Smooth +L1 loss in VDB, as +Lvis = +� +1 +2D2/β, +if |D| ≤ β, +(|D| − 1 +2β), +otherwise, +(5) + +5 +40 +50 +60 +70 +80 +Corruption Rates. +81.0 +81.5 +82.0 +82.5 +83.0 +Top-1 Acc. (Ft.) +82.3 +82.5 +82.6 +82.4 +82.1 +82.4 +82.5 +82.4 +block-wise masking +uniform masking +Fig. 5. Ablation study about corruption rates under different masking patterns. +Fixing the prediction rate at 100%, different masking patterns perform a similar +arc-shaped trend after fine-tuning. +where D = q(g(Zk +i )) − norm(Ti · (1 − M k +i )); norm(·) is a +normalization function, eg. LN, which enhances the patch-level +local contrast for better performance [14]; Zk +i and Ti·(1−M k +i ) +are the encoder outputs of visible tokens and the corresponding +targets, respectively; β is experimentally set to 2.0. +Combined with the reconstruction loss Lmim in MPB, Eq. 2, +the overall learning objective is updated as, +L = Lvis + λ · Lmim, +(6) +where λ is set to 1 by default. +IV. EXPERIMENT +A. Experimental Settings +Datasets. Following a standard SSL evaluation recipe [14], +[21], we conduct end-to-end fine-tuning or linear probing for +classification on ImageNet-1K and transfer learning for object +detection/instance segmentation on COCO [56] and semantic +segmentation on ADE20k [57]. Specifically, ImageNet-1K [17] +is organized according to the WordNet hierarchy. It has 1000 +classes and roughly 1.28M images, which is normally leveraged +to evaluate the progress in image recognition across a wider +variety of objects. MS-COCO 2017 [56] contains ∼118k images +for training, 5k for validation, and ∼20k for testing. ADE20K, +as a common semantic segmentation dataset, contains 20k +images of 150 fine-grained categories. +Architectures. Following prior arts [10], [14], [15], we use +ViT [2] with different scales as backbones, i.e., ViT-B/16 and +ViT-L/16, where /16 denotes the patch size. We pre-train and +fine-tune the model with the image size of 224 × 224. To +further unleash the potential of the proposed DMJD, we extend +evaluations to hybrid convolution-transformer architectures, eg. +ConViT [21] with a multi-scale decoder. For segmentation, +we build FPN on the 3rd, 5th, 7th, and 11th layers of output +features following UperNet [58] and resize images to 512×512. +The multi-scale features from ConViT are also fed into the +MaskRCNN [59] head for object detection. +Hyper-parameters. We pre-train models on ImageNet- +1K [17] following the settings of MAE [14], where the decoder +TABLE I +ABLATION STUDY ON THE ADAPTIVE LEARNING RATE SCALE RULE. +Learning Rate Scale Rule +ηbase · b/256 +ηbase · (bu +mpred +mcorr )/256 +Fine-tuning +82.1 +82.4 +TABLE II +TRAINING EFFICIENCY OF DIFFERENT MASKING PATTERNS. “Gh.” RECORDS +GPU HOURS TAKEN TO REACH THE FINE-TUNING ACCURACY (“FT.”) OF +MAE WITH 1600 EPOCHS (i.e., 83.6 %). +Method +mpatt +Epoch +ETE +Ft. +Gh. +Speedup +MAE [14] +Uniform +1600 +1600 +83.6 +227 +1× ++DM +Uniform +800 +1600 +83.6 +127 +1.8× ++DM +Block-wise +400 +800 +83.6 +70 +3.2× +has a depth of 8 and a width of 512. Concretely, our DMJD is +optimized by AdamW [60] with a batch size of 1024, a learning +rate of 1.5e-4, and a weight decay of 0.05. We pre-train our +model for 800 epochs and 20 epochs warmup with the cosine +decay learning rate schedule [61]. Only random resize crop and +horizontal flipping are employed for data augmentation in the +pre-training stage. For DM, the number of masked views K is +set to 2 since the prediction rate can easily saturate to 1 under +a corruption rate larger than 0.5. The block-wise masking with +a corruption rate of 0.6 is adopted. +Evaluation Metrics. As methods with DM actually have +more views with the same pre-training epochs, we adopt the +metric of effective training epochs (ETE) [13] for fair training +efficiency comparisons, where ETE = K × Epoch. The +training efficiency is quantified by GPU hours (Gh.) achieving +comparable performance on 8× Nvidia A100 (40GB). +B. Ablation Study +1) Disjoint Masking: We evaluate DM based on MAE [14] +with normalized pixels as reconstruction targets and only pre- +train 100 epochs for fast ablation study on ImageNet-1K. The +uniform masking is applied with mcorr of 0.75. +Learning rate scale. As shown in Table I, the learned model +achieves 0.3% fine-tuning classification accuracy gain with the +newly proposed learning rate scale rule, which illustrates the +efficacy of the modification. +Masking pattern mpatt. We figure out the optimal corrup- +tion rates for specific masking patterns with DM in Fig. 5. +Concretely, block-wise masking [15] tends to remove large +patch blocks and makes the reconstruction more difficult, that +it prefers a lower optimal corruption rate around 60% rather +than 70% in uniform masking. +In Table II, we further compare the convergence perfor- +mances of different masking patterns with sufficient training. +Equipped with DM of uniform masking, MAE achieves +1.8× convergence speedup. Notably, block-wise masking [15] +reaches the same accuracy using only half of the ETEs and +3.2× lower Gh. This is because masking patterns with smaller +optimal masking rate (e.g. block-wise) have more room for +prediction rate increment which may take more advantages of + +6 +0 +20 +40 +60 +80 +100 +Epochs +70 +72 +74 +76 +78 +80 +82 +84 +Top 1 Acc. (Ft.) +baseline +mpred = 0.85 +mpred = 0.95 +mpred = 1.0 +(a) Prediction rates of DM. +0 +20 +40 +60 +80 +100 +Epochs +76 +78 +80 +82 +84 +86 +Top 1 Acc. (Ft.) +RGB +HOG +CLIP +(b) Learning targets of JD. +0 +20 +40 +60 +80 +Epochs +52 +54 +56 +58 +60 +62 +64 +66 +68 +Top 1 Acc. (Lin.) +w/o VDB dec2 +w VDB dec2 +w VDB dec4 +w VDB dec8 +(c) Architecture settings of JD. +0 +20 +40 +60 +80 +Epochs +51 +54 +57 +60 +63 +Top 1 Acc. (Lin.) +59.6 +61.8 +60.6 +62.5 +baseline + +DM + +JD + +DMJD +(d) Main components of DMJD. +Fig. 6. Ablation study on training efficiency of our DMJD: (a) higher prediction rates with fixed mcorr of 0.75 boosts the model training; (b) learning target +types account for convergence speed; (c) MPB with a deep decoder and VDB both benefit learning efficiency; (d) DM and JD work cooperatively. +TABLE III +TRAINING EFFICIENCY COMPARISONS WITH OUR DM ON IMAGENET-1K +FOR CLASSIFICATION AND ADE20K FOR SEMANTIC SEGMENTATION +(MIOU). † DENOTES THE BLOCK-WISE MASKING WITH A OPTIMAL +CORRUPTION RATE OF 0.6 IS ADOPTED. +Models +ETE +Gh. +Fine-tuning +mIoU +Speedup +ViT-Base +BEiT [10] +800 +184 +83.4 +45.6 +1× ++DM +600 +89 +83.5 +45.6 +2.1× +SimMIM [15] +800 +120 +83.8 +- +1× ++DM +800 +86 +83.8 +- +1.4× +MAE [14] +1600 +227 +83.6 +48.1 +1× ++DM +1600 +127 +83.6 +48.3 +1.8× +MaskFeat [16] +1600 +240 +84.0 +- +1× ++DM† +800 +66 +84.0 +- +3.6× +ViT-Large +MAE [14] +1600 +272 +85.9 +53.6 +1× ++DM† +800 +100 +85.9 +53.5 +2.7× +MaskFeat [16] +1600 +280 +85.7 +- +1× ++DM† +800 +76 +85.9 +- +3.7× +DM. These facts clearly justified the training efficiency of our +DM on MAE. +Prediction rate mpred. With fixed mcorr, a larger prediction +rate ensures a higher training efficiency which saturates at +around 95%, Fig. 6a. It justifies that DM facilitates flexible +controlling of the prediction rate to make full use of training +signals which leads to superior model convergence. +Efficiency Effects. In Table III, we evaluate the training +efficiency of DM based on several MIM methods with their +original pre-training recipes, including MAE [14], BEiT [10], +SimMIM [15], and MaskFeat [16]. Specifically, we compare the +end-to-end fine-tuning classification accuracy on ImageNet-1K +and the transfer learning performance of semantic segmentation +(without multi-scale testing) on ADE20K [57]. +On the one hand, DM significantly reduces the training +time with equal or fewer ETEs to achieve similar (sometimes +better) classification accuracy, i.e., 1.4× ∼ 3.7× speedup. This +is because DM requires fewer disk I/O, demonstrating it as +a hardware-friendly strategy. On the other hand, with block- +wise masking, DM further accelerates model convergence. For +example, MAE-L+DM achieves 85.9% classification accuracy +with only 800 ETEs, half of the original budgets. And so does +TABLE IV +ABLATION STUDY OF LEARNING TARGETS IN THE PROPOSED JD. +Learning Targets +RGB +RGB +HOG +HOG +CLIP +norm(·) +× +✓ +✓ +LayerNorm +LayerNorm +Fine-tuning +83.1 +83.9 +84.2 +84.4 +85.2 +TABLE V +ABLATION STUDY ON ARCHITECTURE SETTINGS IN JD, i.e., THE DEPTH OF +THE DECODER IN MPB AND THE PROJECTOR IN VDB. “-” DENOTES VDB +IS NOT USED. +Decoder Depth +2 +2 +2 +4 +8 +Projector +- +Linear +Nonlinear +Nonlinear +Nonlinear +Linear Probing +61.8 +60.4 +62.5 +65.0 +67.2 +MaskFeat. These facts indicate that our DM remarkably relieves +the long pre-training requirement for MIM and accelerates +model convergence greatly. +For segmentation, DM also achieves faster convergence and +performs on par with the baselines. For example, MAE-L+DM +pre-trained with only 800 ETEs is competitive to the model with +1600 ETEs, speeding 2.7×. These results evidence that DM +keeps the model generalizability during training acceleration. +2) Joint Distillation: We ablate JD based on ConvMAE- +B [21] with DM for faster convergence and pre-train 50 +epochs with a batch size of 1024, the learning target as HOG +features [22] normalized by a non-parametric LN layer [55]. +Learning Targets are critical for MIM since they provide +explicit guidance for representation abstraction and their +characteristics can be directly injected into the learned model. +In Table IV, we can conclude that the target normalization +is important for good performance, which aligns with the +observations in MAE [14]. Specifically, non-parametric layer +normalization [55] is slightly more suitable for the HOG [22] +target than the original L2 normalization (84.4% vs. 84.2%). +With the CLIP target, layer normalization still works well and +reports superior accuracy of 85.2 %. +In Fig. 6b, we illustrate the effects on training efficiency +with different targets and reveal that targets with high-level +semantics may achieve superior learning efficiency. Concretely, +compared to HOG, raw pixels come with more sensitivity +and ambiguity in the masked prediction with wrong color and + +7 +TABLE VI +PERFORMANCE COMPARISONS WITH STATE-OF-THE-ART SSL METHODS ON IMAGENET-1K. THE EXTRA ETES FOR PRE-TRAINING TOKENIZERS ARE +FORMULATED BY NORMALIZED IMAGENET-1K EPOCHS [16]. GPU HOURS OF METHODS THAT DO NOT RELEASE THEIR CODE ARE SKIPPED BY “-”. +MODELS TRAINED WITH HUGE EFFECTIVE EPOCHS ARE DE-EMPHASIZED WITH LIGHT GREY (DITTO FOR OTHER TABLES). +Method +Publication +Backbone +ETE +Gh. +Learning Target +Fine-tuning +Linear Probing +non-parametric tokenizer +SplitMask [12] +Arxiv2021 +ViT-B +600 +- +Random Patches +83.6 +46.5 +MAE [14] +CVPR22 +ViT-B +1600 +227 +RGB +83.6 +68.0 +SimMIM [15] +CVPR22 +ViT-B +800 +184 +RGB +83.8 +56.7 +MaskFeat [16] +CVPR22 +ViT-B +1600 +240 +HOG +84.0 +68.5 ++DMJD +- +ViT-B +1600 +132 (1.8×) +HOG +84.1 (+0.1) +71.9 (+3.4) +ConvMAE [21] +NeurIPS22 +ConViT-B +1600 +300 +RGB +85.0 +70.9 ++DMJD +- +ConViT-B +800 +101 (3×) +HOG +85.2 (+0.2) +76.7 (+5.8) +ConvMAE [21] +NeurIPS22 +ConViT-L +800 +480 +RGB +86.2 +- ++DMJD +- +ConViT-L +800 +267 (1.8×) +HOG +86.3 (+0.1) +79.7 +parametric tokenizer +MoCov3 [8] +ICCV21 +ViT-B +1200 +- +Momentum +83.2 +76.7 +DINO [7] +ICCV21 +ViT-B +1600 +- +Momentum +83.6 +78.2 +BEiT [10] +ICLR22 +ViT-B +800+1199 +- +DALL-E +83.4 +37.6 +CAE [62] +Arxiv2022 +ViT-B +1600+1199 +- +DALL-E +83.9 +70.4 +PeCo [11] +CVPR22 +ViT-B +800+100 +- +Perceptual Codebook +84.5 +- +mc-BEiT [63] +ECCV22 +ViT-B +800+1199 +- +Perceptual Codebook +84.1 +- +iBOT [13] +ICLR22 +ViT-B +1600 +233 +Momentum +84.0 +79.5 +SdAE [64] +ECCV22 +ViT-B +1200 +- +Momentum +84.1 +64.9 +data2vec [24] +ICML22 +ViT-B +800 +- +Momentum +84.2 +- +MVP [26] +Arxiv2022 +ViT-B +300+10000 +- +CLIP +84.4 +75.4 +ConvMAE [21] +NeurIPS22 +ConViT-B +400+10000 +- +CLIP +85.2 +- ++DMJD +- +ConViT-B +400+10000 +- +CLIP +85.4 (+0.2) +80.1 ++DMJD +- +ConViT-L +400+10000 +- +CLIP +86.8 (+1.6) +81.0 +texture [16]. The corresponding high loss penalty will mislead +the model to overfit local statistics, which is insignificant +for visual understanding, resulting in training inefficiency. As +high-level semantic abstractions, targets extracted by CLIP +significantly boost the model convergence compared with HOG +features and raw pixels. +Masked Prediction Branch. Since specialized in recon- +struction, the decoders in an autoencoder with different depths +report sensitive linear probing performances. Specifically, as +shown in Table V and Fig. 6c, compared with a depth of 2, the +decoder with a depth of 8 gains 4.7% linear probing accuracy +with higher training efficiency. It is because a deep decoder +can reconstruct more low-level details, maintaining the latent +representations with more semantic abstraction. +Visible Distillation Branch. In Table V, we study the effect +of varying projector configurations in VDB. With the decoder +depth of 2, the nonlinear projector setting reports 2.1% higher +accuracy than the linear projector and outperforms the model +learned without VDB by 0.7%. This is because a powerful +projector can benefit representation learning by relieving the +backbone’s burden of target fitting, similar to the depth effect +of the decoder in MDP. The convergence curves of whether or +not VDB is used are plotted in Fig. 6c, which comprehensively +proves the advantages of our JD for MIM training efficiency. +3) Disjoint Masking & Joint Distillation: Following the +settings in Sec. IV-B2, we conduct evaluations with ConvMAE +to justify the training efficiency effects of each component +of our DMJD. In Fig. 6d, one can see that DM and JD both +improve the training efficiency significantly and their combi- +nation further accelerates the model convergence remarkably. +This is because they are devised from orthogonal perspectives +and thus work cooperatively. +TABLE VII +PERFORMANCE COMPARISONS OF SEMANTIC SEGMENTATION ON ADE20K +(MIOU) AND OBJECT DETECTION AND INSTANCE SEGMENTATION ON +COCO (APb/APm) WITH LEADING MIM METHODS. +Method +Backbone +ETE +mIoU +APb/APm +non-parametric tkn(·) +SplitMask [12] +ViT-B +600 +45.7 +46.8/42.1 +MAE [14] +ViT-B +1600 +48.1 +50.3/44.9 +SimMIM [15] +Swin-B +800 +52.8 +52.3/- +ConvMAE [21] +ConViT-B +1600 +51.7 +53.2/47.1 ++DMJD +ConViT-B +800 +53.3 +53.4/47.6 +parametric tkn(·) +DINO [7] +ViT-B +1600 +46.8 +50.1/43.4 +BEiT [10] +ViT-B +1999 +47.1 +50.1/43.5 +mc-BEiT [63] +ViT-B +1999 +50.8 +49.2/44.0 +SdAE [64] +ViT-B +1200 +48.6 +48.9/43.0 +PeCo [11] +ViT-B +900 +48.5 +44.9/40.4 +CAE [62] +ViT-B +2799 +50.2 +50.0/44.0 +iBOT [13] +ViT-B +1600 +50.0 +51.2/44.2 +ConvMAE [21] +ConViT-B +10400 +52.8 +53.3/47.3 +C. Performance Comparisons +In Table VI, we roughly categorize the leading baseline +methods by whether or not the supervision signal stems from +a parametric tokenizer. Non-parametric tokenizers include +pixels or hand-crafted features, while parametric ones are deep +features from an online or pre-trained teacher network. For a +fair efficiency comparison, the additional cost of pre-training +tokenizer, eg. CLIP [25], needs to be counted on. Following +MaskFeat [16], we normalize the external training epochs by +the cost of each epoch on ImageNet-1K training set with the +view size of 224 × 224 for unified evaluation. +Classification. With a non-parametric tokenizer of HOG, our + +8 +40 +20 +0 +20 +40 +40 +20 +0 +20 +40 +TSNE +(a) MAE ViT-B [14] +40 +20 +0 +20 +40 +40 +20 +0 +20 +40 +TSNE +(b) Maskfeat ViT-B [16] +40 +20 +0 +20 +40 +40 +20 +0 +20 +40 +60 +TSNE +(c) DMJD HOG ViT-B (Ours) +40 +20 +0 +20 +40 +40 +30 +20 +10 +0 +10 +20 +30 +40 +TSNE +(d) ConvMAE ConViT-B [21] +40 +20 +0 +20 +40 +40 +20 +0 +20 +40 +TSNE +(e) DMJD HOG ConViT-B (Ours) +40 +20 +0 +20 +40 +40 +20 +0 +20 +40 +TSNE +(f) DMJD HOG ConViT-L (Ours) +40 +20 +0 +20 +40 +40 +20 +0 +20 +40 +TSNE +(g) DMJD CLIP ConViT-B (Ours) +40 +20 +0 +20 +40 +60 +40 +20 +0 +20 +40 +60 +TSNE +(h) DMJD CLIP ConViT-L (Ours) +Fig. 7. t-SNE visualization of representation spaces learned by state-of-the-art methods (fully pre-trained with ETEs of 1600, 1600, 1600, 1600, 800, 800, 400, +and 400 respectively), where test images from 20 classes are randomly sampled in the ImageNet-1K validation set. +DMJD improves the fine-tuning and linear probing accuracy by +0.2%, 5.8% respectively, compared with ConvMAE (ConViT-B) +of the RGB target and notably consumes only half of training +ETEs (800ep vs. 1600ep) and 1 +3 Gh. (300 vs. 101). With the +same targets and a ViT-B backbone, our DMJD consistently +improves the linear probing performance with MaskFeat by +a large margin (71.9 vs. 68.5) with less training cost of Gh. +Overall, it is clear that our DMJD framework can significantly +improve the linear probing results in a plug-and-play fashion +while decreasing training costs. +As MIM is obsessed with patch-level semantic relation +reasoning, it is typically inferior to contrastive learning-based +pre-training models under the linear probing evaluation, where +image-level discriminative representations are critical. However, +our DMJD with ConvMAE (ConViT-B) improves the baseline +of 5.8% and achieves comparable performance with state-of- +the-art contrastive learning-based SSL methods, e.g., iBOT [13] +(76.7 vs. 79.5). When scaling up the model size from ConViT-B +to ConViT-L, our DMJD further reports a higher liner probing +accuracy of 79.7. With CLIP targets, DMJD further pushes the +accuracy to another height of 81.0 with ConVit-L. +In Fig. 7, we present qualitative evaluations of the learned +models’ representations. The t-SNE visualization presents a +trend consistent with the linear probing performance that +our DMJD with high-level learning targets produces more +discriminative embedding spaces, which again justifies the +superiority of our designs. +Semantic Segmentation and Object Detection. We con- +duct fine-grained transfer learning evaluations to verify the +generalization ability of learned representations by the proposed +DMJD, Table VII. Since multi-scale features benefit down- +stream tasks, we adopt UperNet [58] and Mask R-CNN [59] +as segmentation and object detection heads, respectively. +Concretely, with a non-parametric target tokenizer, DMJD +surpasses the ConvMAE baseline by 1.6 mIoU (53.3 vs. 51.7) +on ADE20K for semantic segmentation and achieves 0.2 AP box +(53.4 vs. 53.2) and 0.5 AP mask (47.6 vs. 47.1) gains on COCO +for object detection and instance segmentation. It is worth not- +ing that our DMJD outperforms models learned with parametric +target tokenizers, eg. CLIP [25], and only consumes half of +ETEs. These results justify that the learned representations +of DMJD generalize well for several downstream tasks while +keeping efficiency. +V. CONCLUSION +In this paper, we propose a conceptually simple yet training- +efficient MIM framework, termed disjoint masking with joint +distillation (DMJD). With the multi-view disjoint masking +strategy, our approach improves the utilization of training +signals accelerating the model convergence significantly. In- +troducing explicit semantic guidance on the visible parts via +joint distillation, the proposed DMJD cooperatively boosts the +training efficiency while keeping the learned representations +to generalize well to downstream tasks. We hope these +achievements shed new light on training efficiency research in +the MIM community. + +9 +REFERENCES +[1] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, +L. Kaiser, and I. Polosukhin, “Attention Is All You Need,” arXiv e-prints, +p. arXiv:1706.03762, Jun. 2017. +[2] G. Sharir, A. Noy, and L. Zelnik-Manor, “An Image is Worth 16x16 +Words, What is a Video Worth?” arXiv e-prints, p. arXiv:2103.13915, +Mar. 2021. +[3] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, and B. 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Tian, “SdAE: Self-distillated Masked Autoencoder,” arXiv e-prints, p. +arXiv:2208.00449, Jul. 2022. + diff --git a/sNAyT4oBgHgl3EQfZve0/content/tmp_files/load_file.txt b/sNAyT4oBgHgl3EQfZve0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8fe6e2ad8bf6a6768797e6c579eb184ed016047b --- /dev/null +++ b/sNAyT4oBgHgl3EQfZve0/content/tmp_files/load_file.txt @@ -0,0 +1,1355 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf,len=1354 +page_content='1 Disjoint Masking with Joint Distillation for Efficient Masked Image Modeling Xin Ma, Chang Liu, Chunyu Xie, Long Ye, Yafeng Deng, and Xiangyang Ji Abstract—Masked image modeling (MIM) has shown great promise for self-supervised learning (SSL) yet been criticized for learning inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We believe the insufficient utilization of training signals should be responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' To alleviate this issue, we introduce a conceptually simple yet learning-efficient MIM training scheme, termed Disjoint Masking with Joint Distillation (DMJD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' For disjoint masking (DM), we sequentially sample multiple masked views per image in a mini-batch with the disjoint regulation to raise the usage of tokens for reconstruction in each image while keeping the masking rate of each view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' For joint distillation (JD), we adopt a dual branch architecture to respectively predict invisible (masked) and visible (unmasked) tokens with superior learning targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Rooting in orthogonal perspectives for training efficiency improvement, DM and JD cooperatively accelerate the training convergence yet not sacri- ficing the model generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, DM can train ViT with half of the effective training epochs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7× less time- consuming) to report competitive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With JD, our DMJD clearly improves the linear probing classification accuracy over ConvMAE by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' On fine-grained downstream tasks like semantic segmentation, object detection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', our DMJD also presents superior generalization compared with state-of-the- art SSL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The code and model will be made public at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='com/mx-mark/DMJD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Index Terms—Disjoint masking, Joint distillation, Masked image modeling, Self-supervised learning, and Training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' INTRODUCTION W ITH the surge of vision transformers [1]–[6], masked image modeling (MIM) has recently prevailed on the self-supervised representation learning (SSL) leaderboard [7]– [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' These approaches assimilate context semantics by pre- dicting a portion of masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' It has been justified that discrete visual tokens [10]–[13], raw pixels [14], [15], and hand-crafted features [16] are suitable targets to learn versatile models for a broad spectrum of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' However, MIM methods typically require tremendous train- ing costs, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', thousands of pre-training epochs, which overbur- dens academic and industrial communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In general, an input image is masked out with a high masking rate, also named corruption rate (mcorr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' While the rest unmasked patches are ignored for self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We argue that a large proportion of input signals is not fully exploited for model learning at each X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Ma and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Ye are with the School of Information and Communication Engineering, Communication University of China, Beijing 100024, China, Email: {mx mark, yelong}@cuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Liu and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Ji are with the Department of Automation, Tsinghua University, Beijing 100084, China, Email: {liuchang2022, xyji}@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Xie and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Deng are with 360 AI Research, Beijing, China, Email: yuxie@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='cn, dengyafeng@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Ji is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 0 20 40 60 80 100 Epochs 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 Top 1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') baseline raise both (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9) lower both (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5) (a) Varying both (mcorr = mpred).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 0 20 40 60 80 100 Epochs 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 Top 1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') baseline raise mpred (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0) lower mpred (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2) (b) Varying mpred (mcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 0 20 40 60 80 100 Epochs 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 Top 1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') baseline raise mcorr (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9) lower mcorr (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5) (c) Varying mcorr (mpred = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 0 20 40 60 80 100 Epochs 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 Top 1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') baseline +DM +DMJD (d) DMJD (Ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' MIM Training efficiency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, we pre-train models based on MAE [14] for 100 epochs on ImageNet-1k [17] with uniform masking and plot the corresponding convergence curve on the validation set during fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The blue curve in each sub-figure denoted as the “baseline” is reported under the optimal setting of mpred = mcorr = 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In sub-figures (a), (b), and (c), no matter how mcorr and mpred change in each view of an image, the optimal choice for fast convergence is the baseline setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' It inspires us to develop a multi-view masking strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', DM, to raise the overall mpred in an image while keeping mpred equal to mcorr in each view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In the spirit of increasing the utilization of input signals, we also introduce JD with an additional visible distillation branch, which works cooperatively with DM to expedite the training procedure, sub-figure (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' loop mainly accounts for the training inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Actually, the training efficiency of MIM is largely determined by the prediction rate (mpred), the overall portion of input images used to provide supervision for invisible region reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' To go deeper into how mcorr and mpred affect MIM training efficiency, we quantitatively visualize the model convergence curve with various settings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1a and 1c, one can see that only with a proper corruption rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='75 with uniform masking, MIM exhibits superior effectiveness and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' This is because since mcorr defines the difficulty of the reconstruction task, the training efficiency intrinsically depends on mcorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With a too-high corruption rate, there is not arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='00230v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='CV] 31 Dec 2022 2 Target Input Invisible Reconstruction Supervision (a) Vanilla MIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Invisible Reconstruction Visible Distillation Supervision Multiple Masked Views Target Disjoint Masking (DM) Joint Distillation (JD) (b) DMJD (Ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Pipeline illustration of vanilla MIM and our DMJD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Vanilla MIM methods typically adopt a single-view masking strategy to perform invisible reconstruction with mcorr = mpred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In contrast, our DMJD introduces a multi-view masking strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', DM, to increase the overall prediction rate of each image while keeping mcorr = mpred in each view and a dual-branch joint distillation architecture with an additional visible distillation branch to take full use of the input signals with superior targets, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' HOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' enough context to correctly recover the masked patches [14], making the model more intricate and unpredictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' And a low corruption rate will degrade the model performance because the model can trivially recover a missing patch from neighboring patches without needing a high-level visual understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' What is worse, with the optimal setting of mcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='75, naively varying mpred also hampers the model training, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' These facts reveal the challenges of improving MIM training efficiency and inspire us to think differently to increase the utilization of input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In this paper, we develop a conceptually simple yet learning- efficient MIM training scheme, termed Disjoint Masking with Joint Distillation (DMJD), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Disjoint masking (DM) is a multiple-view sampling strategy targeting flexibly raising the prediction rate of each input while keeping the corruption rate of each view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, we sequentially sample a series of masked views of an image from its disjoint portions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', overall have been previously unmasked/masked patches, for wider coverage of the invisible regions in a training loop for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Since the increasing utilization of training signals in a batch may reduce the gradient variance and hamper the generalization of learned models [18]–[20], we introduce an adaptive learning rate scale rule taking the relative prediction rate increment as a scale factor to enhance the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Joint distillation (JD) performs distillation on visible and invisible regions with a dual branch architecture to further increase training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In addition to the original masked prediction branch (MPB), we add a visible distillation branch (VDB) to provide semantic guidance through parametric or non-parametric tokenizers for visible token learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With increased prediction rates, visible distillation, and superior targets, the proposed DMJD ensures higher learning efficiency from orthogonal perspectives yet works cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' It is worth noting that our DMJD accelerates training conver- gence but not sacrificing model generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, ViT/ConViT [2], [21] equipped with DMJD typically improves performances with improved training efficiency not only on ImageNet-1K classification but also on fine-grained downstream tasks, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' semantic/instance segmentation, object detection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Take an example, for linear probing classification on MaskFeat [16] and ConvMAE [21] baselines, DMJD achieves performance gains of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8% with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8× and 3× acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We hope these observations shed new light on MIM training efficiency research in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Our contributions can be summarized as follows: We propose a conceptually simple yet learning-efficient MIM training scheme, termed disjoint masking with joint distillation (DMJD), which targets increasing the utilization of per image at each training loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We devise a multi-view generation strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', disjoint masking (DM), to increase the prediction rate while keeping the corruption rate for efficient MIM and introduce the adaptive learning rate scale rule for better model generalization with augmented training batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We develop a dual-branch architecture for joint distillation (JD), effectively pursuing representation learning on both visible and invisible regions with superior targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We conduct sufficient evaluations justifying our DMJD can significantly accelerate model convergence and achieve outstanding performances on standard benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Masked Image Modeling Since ViT [2] overcomes the architectural obstacle of applying masked signal modeling for visual pre-training, MIM has achieved great success recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The increasing attention can be roughly categorized into three aspects, including learning targets, backbone extensions, and masking strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Learning targets are critical since they provide explicit guidance for visual learning through reconstruction and their characteristics can be injected into the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' MAE [14] and SimMIM [15] reveal that raw pixels as reconstruction targets are adequate for effective visual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' MaskFeat [16] introduces local invariant features, such as HOG [22], to avoid modeling high-frequency low-level details in the pixel space and concentrate meaningful semantic abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' BEiT [10] and PeCo [11] further apply a discrete visual codebook 3 Masked Prediction Branch Visible Distillation Branch Joint Distillation (JD) Encoder … Disjoint Masking (DM) Decoder ⁝ ⁝ ⁝ Projector & Predictor ⁝ ⁝ tkn : Concatenate : Visible Token Features : Masked Token : Target i \uf043 k i \uf05a k i \uf05a i \uf054 i \uf054 mim L vis L 1 ) ( i i k \uf04d \uf0d7 \uf02d X Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The pipeline of the proposed DMJD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' During pre-training, K masked views of each image are randomly sampled in a mini-batch with DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Then, they will be fed to the encoder and the dual branch decoder for invisible reconstruction and visible distillation with targets extracted by the tokenizer tkn(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' produced by dVAE [23] with an additional pre-training stage for high-level semantic abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' To get rid of extra pre-training, iBOT [13] and data2vec [24] jointly optimize the model and the target tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With the fast development of multi-modal foundation models, CLIP [25] is actively exploited yet not limited as an effective target tokenizer for MIM [26]–[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' As advanced backbones take advantage of more discrimina- tive representations, researchers are inspired to extend MIM to multi-scale hybrid convolution-transformer architectures [15], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Also, to explore what kind of information in images to be predicted benefits the model learning, several masking strategies [31]–[34] have been proposed to improve the vanilla random masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' However, tremendous training costs for convergence limit MIM methods for application and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The training efficiency issue becomes urgent to be attended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' MIM Training Efficiency Powerful computation resources have grown rapidly to facilitate fast training with lower numerical precision and larger batch sizes for large-scale models [19], [20], [35]– [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' However, the high computation and time costs are still an obstacle to the practical applications of MIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' A feasible attempt is to apply efficient hierarchical transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, UM-MAE [42] designs a masking strategy to preserve the relative position relationship between visible patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' GreenMIM [43] proposes a uniform partition to ensure that visible patches within each local window can be grouped with equal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' MixMIM [44] creates a mixed view that the masked tokens from one image are replaced with visible tokens from another image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' An alternate way for MIM training efficiency is reducing the computational complexity of invisible reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, LoMaR [45] performs masked reconstruction within small 7×7 windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' FastMIM [46] directly pre-trains model with low-resolution inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Differently, inspired by MLM works [47], [48] which attempt to supervise a larger portion of input signals for learning efficiency, our DMJD is devised to increase the utilization of each image with a multi-view masking strategy [49]–[52] and dual-branch distillation using high-level learning targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Overview MIM methods are typically built on ViTs [2], where an input image Xi will be split into a set of non-overlapping patches and linearly projected to a sequence of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' This process can be formulated as Xi = {(xij, pij)N j=1}, where pij denotes the positional embedding of the j-th token xij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' To yield masked views for reconstruction, specific masking strategies can be defined as M(Xi, mpatt, mcorr) = Mi, (1) where Mi = (mij)N j=1, mij equals 1 if the j-th token xij needs to be masked, otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' mcorr and mpatt are the corruption rate and the masking pattern, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' random, block-wise, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The reconstruction loss function is defined as Lmim = − B � i=1 N � j=1 mij · ℓ(x′ ij, tij), (2) where x′ ij and tij are the network prediction and the corre- sponding reconstruction target of xij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' B is the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' ℓ(·) is typically the mean squared error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Since gradients are only back-propagated on masked tokens, a small subset of an image, it may cause the training inefficiency problem [47], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' And it is not wise to simply increase the masking rate or recover unmasked patches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' which even 4 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 Previous masks ⁝ 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 K i \uf02d M 1 i \uf04d 1 K i \uf04d \uf02d 1 i i K \uf02d \uf0d7 X M 1) (1 K i i \uf02d \uf0d7 \uf02d X M K i \uf04d i X ���′������������ > ��� Disjoint Regulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Illustration of disjoint masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With DM, we sequentially sample the K-th view from disjoint regions of the currently have not been masked token set Xi · MK−1 i and the rest part Xi · (1 − MK−1 i ) with the disjoint regulation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', keeping the corruption rate m ′ corr in Xi · (1 − MK−1 i ) positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The regulation ensures the overall prediction rate of each image is larger than the pre-defined corruption rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' slows down the convergence, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Thus, we propose disjoint masking with joint distillation (DMJD), a dual-branch distillation framework with a multiple-masked view sampling approach to increase the prediction rate with a fixed corruption rate for efficient MIM, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, we impose a masking regulation to generate multiple complementary views facilitating more invisible tokens of each image to be reconstructed in the masked prediction branch (MPB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In the visible distillation branch (VDB), we bridge the semantic gap between the visible token features and their corresponding learning targets through distillation, further enhancing the training efficiency [16], [24], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Disjoint Masking DM targets sequentially sampling multiple masked views under a regulation to increase the overall portions of tokens used for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Since augmented views reduce the samples with a given batch size, We modify the learning rate scale rule to optimize the model training for generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1) Disjoint Regulation: Suppose M1 i = M(Xi, mpatt, mcorr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The set of tokens that have been masked in the previous K-1 masks is defined as MK−1 i = 1 − K−1 � k=1 (1 − M k i ), where K ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Thereafter, DM (Mdm) with the disjoint regulation can be formulated based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1 as, Mdm(Xi, mpatt, mcorr, MK−1 i ) = M K i , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' M K i (1 − MK−1 i ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (3) It is not hard to find that only if M K i can mask tokens that have never been masked once, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 3 can hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, to sample the K-th mask under the regulation, we first split the image tokens into two disjoint subsets: Xi · MK−1 i (previously masked tokens) and Xi · (1 − MK−1 i ) (currently unmasked tokens), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Then, we partially sample tokens to be masked from the unmasked token set with a positive corruption rate m ′ corr > 0 and the rest from the masked set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In this way, DM guarantees that each masked view will include new tokens beyond previous views with a fixed corruption rate while increasing the overall prediction rate of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 2) Adaptive Learning Rate Scale: Since gradients are averaged in a mini-batch, the learning rate scale rule is proposed to alleviate the generalization degradation caused by the over- smoothed gradient variance with large batch sizes [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' It normally scales up the base learning rate ηbase with the batch size b as η = ηbase · b/256, where 256 is a scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With DM, a mini-batch only contains bu unique samples, where bu = b/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Since repetitively utilizing each image has a similar effect of enlarging the batch size on the gradient variance, we intend to scale up the learning rate by the relative utilization improvement of tokens used for reconstruction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', mpred mcorr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Thus, the modified adaptive learning rate scale can be defined as η = ηbase · � bu · mpred mcorr � / 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (4) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Joint Distillation As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 3, we add an additional visible distillation branch (VDB) to bridge the visible encoding features and its corresponding targets [16], [24], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1) Model Architecture: a) Projector (g) & Predictor (q): We introduce a nonlin- ear projector to reduce the information loss [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, it has a 3-layer MLP-like structure with a LayerNorm (LN) [55] applied to each fully-connected (FC) layer following Sim- CLR [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The output dimension of all MLP hidden layers is 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Only the visible token features are fed to the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' To align the output dimensions of the projector and the target tokenizer, we insert a single FC layer as the predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' b) Target Encoding (tkn(·)): We evaluate two types of targets to introduce direct guidance for visible token learning: i) representative local invariant features, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' HOG [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' ii) features extracted by deep models which extract context information well, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' CLIP [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The target encoding process can be expressed as Ti = tkn(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 2) Learning Objectives: We distill the target through Smooth L1 loss in VDB, as Lvis = � 1 2D2/β, if |D| ≤ β, (|D| − 1 2β), otherwise, (5) 5 40 50 60 70 80 Corruption Rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 Top-1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 block-wise masking uniform masking Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Ablation study about corruption rates under different masking patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Fixing the prediction rate at 100%, different masking patterns perform a similar arc-shaped trend after fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' where D = q(g(Zk i )) − norm(Ti · (1 − M k i ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' norm(·) is a normalization function, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' LN, which enhances the patch-level local contrast for better performance [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Zk i and Ti·(1−M k i ) are the encoder outputs of visible tokens and the corresponding targets, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' β is experimentally set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Combined with the reconstruction loss Lmim in MPB, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 2, the overall learning objective is updated as, L = Lvis + λ · Lmim, (6) where λ is set to 1 by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' EXPERIMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Experimental Settings Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Following a standard SSL evaluation recipe [14], [21], we conduct end-to-end fine-tuning or linear probing for classification on ImageNet-1K and transfer learning for object detection/instance segmentation on COCO [56] and semantic segmentation on ADE20k [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, ImageNet-1K [17] is organized according to the WordNet hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' It has 1000 classes and roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='28M images, which is normally leveraged to evaluate the progress in image recognition across a wider variety of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' MS-COCO 2017 [56] contains ∼118k images for training, 5k for validation, and ∼20k for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' ADE20K, as a common semantic segmentation dataset, contains 20k images of 150 fine-grained categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Following prior arts [10], [14], [15], we use ViT [2] with different scales as backbones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', ViT-B/16 and ViT-L/16, where /16 denotes the patch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We pre-train and fine-tune the model with the image size of 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' To further unleash the potential of the proposed DMJD, we extend evaluations to hybrid convolution-transformer architectures, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' ConViT [21] with a multi-scale decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' For segmentation, we build FPN on the 3rd, 5th, 7th, and 11th layers of output features following UperNet [58] and resize images to 512×512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The multi-scale features from ConViT are also fed into the MaskRCNN [59] head for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We pre-train models on ImageNet- 1K [17] following the settings of MAE [14], where the decoder TABLE I ABLATION STUDY ON THE ADAPTIVE LEARNING RATE SCALE RULE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Learning Rate Scale Rule ηbase · b/256 ηbase · (bu mpred mcorr )/256 Fine-tuning 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 TABLE II TRAINING EFFICIENCY OF DIFFERENT MASKING PATTERNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' “Gh.” RECORDS GPU HOURS TAKEN TO REACH THE FINE-TUNING ACCURACY (“FT.”) OF MAE WITH 1600 EPOCHS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Method mpatt Epoch ETE Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Speedup MAE [14] Uniform 1600 1600 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 227 1× +DM Uniform 800 1600 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 127 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8× +DM Block-wise 400 800 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2× has a depth of 8 and a width of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, our DMJD is optimized by AdamW [60] with a batch size of 1024, a learning rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5e-4, and a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We pre-train our model for 800 epochs and 20 epochs warmup with the cosine decay learning rate schedule [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Only random resize crop and horizontal flipping are employed for data augmentation in the pre-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' For DM, the number of masked views K is set to 2 since the prediction rate can easily saturate to 1 under a corruption rate larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The block-wise masking with a corruption rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' As methods with DM actually have more views with the same pre-training epochs, we adopt the metric of effective training epochs (ETE) [13] for fair training efficiency comparisons, where ETE = K × Epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The training efficiency is quantified by GPU hours (Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') achieving comparable performance on 8× Nvidia A100 (40GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Ablation Study 1) Disjoint Masking: We evaluate DM based on MAE [14] with normalized pixels as reconstruction targets and only pre- train 100 epochs for fast ablation study on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The uniform masking is applied with mcorr of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Learning rate scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' As shown in Table I, the learned model achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3% fine-tuning classification accuracy gain with the newly proposed learning rate scale rule, which illustrates the efficacy of the modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Masking pattern mpatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We figure out the optimal corrup- tion rates for specific masking patterns with DM in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, block-wise masking [15] tends to remove large patch blocks and makes the reconstruction more difficult, that it prefers a lower optimal corruption rate around 60% rather than 70% in uniform masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In Table II, we further compare the convergence perfor- mances of different masking patterns with sufficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Equipped with DM of uniform masking, MAE achieves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8× convergence speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Notably, block-wise masking [15] reaches the same accuracy using only half of the ETEs and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2× lower Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' This is because masking patterns with smaller optimal masking rate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' block-wise) have more room for prediction rate increment which may take more advantages of 6 0 20 40 60 80 100 Epochs 70 72 74 76 78 80 82 84 Top 1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') baseline mpred = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='85 mpred = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='95 mpred = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 (a) Prediction rates of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 0 20 40 60 80 100 Epochs 76 78 80 82 84 86 Top 1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') RGB HOG CLIP (b) Learning targets of JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 0 20 40 60 80 Epochs 52 54 56 58 60 62 64 66 68 Top 1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') w/o VDB dec2 w VDB dec2 w VDB dec4 w VDB dec8 (c) Architecture settings of JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 0 20 40 60 80 Epochs 51 54 57 60 63 Top 1 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=') 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 baseline +DM +JD +DMJD (d) Main components of DMJD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Ablation study on training efficiency of our DMJD: (a) higher prediction rates with fixed mcorr of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='75 boosts the model training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (b) learning target types account for convergence speed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (c) MPB with a deep decoder and VDB both benefit learning efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (d) DM and JD work cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' TABLE III TRAINING EFFICIENCY COMPARISONS WITH OUR DM ON IMAGENET-1K FOR CLASSIFICATION AND ADE20K FOR SEMANTIC SEGMENTATION (MIOU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' † DENOTES THE BLOCK-WISE MASKING WITH A OPTIMAL CORRUPTION RATE OF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 IS ADOPTED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Models ETE Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Fine-tuning mIoU Speedup ViT-Base BEiT [10] 800 184 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 1× +DM 600 89 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1× SimMIM [15] 800 120 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 1× +DM 800 86 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4× MAE [14] 1600 227 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 1× +DM 1600 127 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8× MaskFeat [16] 1600 240 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 1× +DM† 800 66 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6× ViT-Large MAE [14] 1600 272 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 1× +DM† 800 100 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7× MaskFeat [16] 1600 280 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7 1× +DM† 800 76 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7× DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' These facts clearly justified the training efficiency of our DM on MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Prediction rate mpred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With fixed mcorr, a larger prediction rate ensures a higher training efficiency which saturates at around 95%, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' It justifies that DM facilitates flexible controlling of the prediction rate to make full use of training signals which leads to superior model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Efficiency Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In Table III, we evaluate the training efficiency of DM based on several MIM methods with their original pre-training recipes, including MAE [14], BEiT [10], SimMIM [15], and MaskFeat [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, we compare the end-to-end fine-tuning classification accuracy on ImageNet-1K and the transfer learning performance of semantic segmentation (without multi-scale testing) on ADE20K [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' On the one hand, DM significantly reduces the training time with equal or fewer ETEs to achieve similar (sometimes better) classification accuracy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4× ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7× speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' This is because DM requires fewer disk I/O, demonstrating it as a hardware-friendly strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' On the other hand, with block- wise masking, DM further accelerates model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' For example, MAE-L+DM achieves 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9% classification accuracy with only 800 ETEs, half of the original budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' And so does TABLE IV ABLATION STUDY OF LEARNING TARGETS IN THE PROPOSED JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Learning Targets RGB RGB HOG HOG CLIP norm(·) × ✓ ✓ LayerNorm LayerNorm Fine-tuning 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 TABLE V ABLATION STUDY ON ARCHITECTURE SETTINGS IN JD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', THE DEPTH OF THE DECODER IN MPB AND THE PROJECTOR IN VDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' “-” DENOTES VDB IS NOT USED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Decoder Depth 2 2 2 4 8 Projector Linear Nonlinear Nonlinear Nonlinear Linear Probing 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 MaskFeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' These facts indicate that our DM remarkably relieves the long pre-training requirement for MIM and accelerates model convergence greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' For segmentation, DM also achieves faster convergence and performs on par with the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' For example, MAE-L+DM pre-trained with only 800 ETEs is competitive to the model with 1600 ETEs, speeding 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' These results evidence that DM keeps the model generalizability during training acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 2) Joint Distillation: We ablate JD based on ConvMAE- B [21] with DM for faster convergence and pre-train 50 epochs with a batch size of 1024, the learning target as HOG features [22] normalized by a non-parametric LN layer [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Learning Targets are critical for MIM since they provide explicit guidance for representation abstraction and their characteristics can be directly injected into the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In Table IV, we can conclude that the target normalization is important for good performance, which aligns with the observations in MAE [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, non-parametric layer normalization [55] is slightly more suitable for the HOG [22] target than the original L2 normalization (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With the CLIP target, layer normalization still works well and reports superior accuracy of 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 6b, we illustrate the effects on training efficiency with different targets and reveal that targets with high-level semantics may achieve superior learning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, compared to HOG, raw pixels come with more sensitivity and ambiguity in the masked prediction with wrong color and 7 TABLE VI PERFORMANCE COMPARISONS WITH STATE-OF-THE-ART SSL METHODS ON IMAGENET-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' THE EXTRA ETES FOR PRE-TRAINING TOKENIZERS ARE FORMULATED BY NORMALIZED IMAGENET-1K EPOCHS [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' GPU HOURS OF METHODS THAT DO NOT RELEASE THEIR CODE ARE SKIPPED BY “-”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' MODELS TRAINED WITH HUGE EFFECTIVE EPOCHS ARE DE-EMPHASIZED WITH LIGHT GREY (DITTO FOR OTHER TABLES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Method Publication Backbone ETE Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Learning Target Fine-tuning Linear Probing non-parametric tokenizer SplitMask [12] Arxiv2021 ViT-B 600 Random Patches 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 MAE [14] CVPR22 ViT-B 1600 227 RGB 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 SimMIM [15] CVPR22 ViT-B 800 184 RGB 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7 MaskFeat [16] CVPR22 ViT-B 1600 240 HOG 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 +DMJD ViT-B 1600 132 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8×) HOG 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 (+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4) ConvMAE [21] NeurIPS22 ConViT-B 1600 300 RGB 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 +DMJD ConViT-B 800 101 (3×) HOG 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7 (+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8) ConvMAE [21] NeurIPS22 ConViT-L 800 480 RGB 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 +DMJD ConViT-L 800 267 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8×) HOG 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7 parametric tokenizer MoCov3 [8] ICCV21 ViT-B 1200 Momentum 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7 DINO [7] ICCV21 ViT-B 1600 Momentum 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 BEiT [10] ICLR22 ViT-B 800+1199 DALL-E 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 CAE [62] Arxiv2022 ViT-B 1600+1199 DALL-E 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 PeCo [11] CVPR22 ViT-B 800+100 Perceptual Codebook 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 mc-BEiT [63] ECCV22 ViT-B 800+1199 Perceptual Codebook 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 iBOT [13] ICLR22 ViT-B 1600 233 Momentum 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 SdAE [64] ECCV22 ViT-B 1200 Momentum 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 data2vec [24] ICML22 ViT-B 800 Momentum 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 MVP [26] Arxiv2022 ViT-B 300+10000 CLIP 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 ConvMAE [21] NeurIPS22 ConViT-B 400+10000 CLIP 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 +DMJD ConViT-B 400+10000 CLIP 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 +DMJD ConViT-L 400+10000 CLIP 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 texture [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The corresponding high loss penalty will mislead the model to overfit local statistics, which is insignificant for visual understanding, resulting in training inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' As high-level semantic abstractions, targets extracted by CLIP significantly boost the model convergence compared with HOG features and raw pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Masked Prediction Branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Since specialized in recon- struction, the decoders in an autoencoder with different depths report sensitive linear probing performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Specifically, as shown in Table V and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 6c, compared with a depth of 2, the decoder with a depth of 8 gains 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7% linear probing accuracy with higher training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' It is because a deep decoder can reconstruct more low-level details, maintaining the latent representations with more semantic abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Visible Distillation Branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In Table V, we study the effect of varying projector configurations in VDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With the decoder depth of 2, the nonlinear projector setting reports 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1% higher accuracy than the linear projector and outperforms the model learned without VDB by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' This is because a powerful projector can benefit representation learning by relieving the backbone’s burden of target fitting, similar to the depth effect of the decoder in MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The convergence curves of whether or not VDB is used are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 6c, which comprehensively proves the advantages of our JD for MIM training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 3) Disjoint Masking & Joint Distillation: Following the settings in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' IV-B2, we conduct evaluations with ConvMAE to justify the training efficiency effects of each component of our DMJD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 6d, one can see that DM and JD both improve the training efficiency significantly and their combi- nation further accelerates the model convergence remarkably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' This is because they are devised from orthogonal perspectives and thus work cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' TABLE VII PERFORMANCE COMPARISONS OF SEMANTIC SEGMENTATION ON ADE20K (MIOU) AND OBJECT DETECTION AND INSTANCE SEGMENTATION ON COCO (APb/APm) WITH LEADING MIM METHODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Method Backbone ETE mIoU APb/APm non-parametric tkn(·) SplitMask [12] ViT-B 600 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8/42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 MAE [14] ViT-B 1600 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 SimMIM [15] Swin-B 800 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3/- ConvMAE [21] ConViT-B 1600 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2/47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 +DMJD ConViT-B 800 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4/47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 parametric tkn(·) DINO [7] ViT-B 1600 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 BEiT [10] ViT-B 1999 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 mc-BEiT [63] ViT-B 1999 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 SdAE [64] ViT-B 1200 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 PeCo [11] ViT-B 900 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9/40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 CAE [62] ViT-B 2799 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 iBOT [13] ViT-B 1600 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2/44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 ConvMAE [21] ConViT-B 10400 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3/47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Performance Comparisons In Table VI, we roughly categorize the leading baseline methods by whether or not the supervision signal stems from a parametric tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Non-parametric tokenizers include pixels or hand-crafted features, while parametric ones are deep features from an online or pre-trained teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' For a fair efficiency comparison, the additional cost of pre-training tokenizer, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' CLIP [25], needs to be counted on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Following MaskFeat [16], we normalize the external training epochs by the cost of each epoch on ImageNet-1K training set with the view size of 224 × 224 for unified evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With a non-parametric tokenizer of HOG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' our ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='TSNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='(a) MAE ViT-B [14] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='TSNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='(b) Maskfeat ViT-B [16] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='TSNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='(c) DMJD HOG ViT-B (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='TSNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='(d) ConvMAE ConViT-B [21] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='TSNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='(e) DMJD HOG ConViT-B (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='TSNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='(f) DMJD HOG ConViT-L (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='TSNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='(g) DMJD CLIP ConViT-B (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='TSNE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='(h) DMJD CLIP ConViT-L (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' t-SNE visualization of representation spaces learned by state-of-the-art methods (fully pre-trained with ETEs of 1600, 1600, 1600, 1600, 800, 800, 400, and 400 respectively), where test images from 20 classes are randomly sampled in the ImageNet-1K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' DMJD improves the fine-tuning and linear probing accuracy by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8% respectively, compared with ConvMAE (ConViT-B) of the RGB target and notably consumes only half of training ETEs (800ep vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 1600ep) and 1 3 Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' (300 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With the same targets and a ViT-B backbone, our DMJD consistently improves the linear probing performance with MaskFeat by a large margin (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='9 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5) with less training cost of Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Overall, it is clear that our DMJD framework can significantly improve the linear probing results in a plug-and-play fashion while decreasing training costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' As MIM is obsessed with patch-level semantic relation reasoning, it is typically inferior to contrastive learning-based pre-training models under the linear probing evaluation, where image-level discriminative representations are critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' However, our DMJD with ConvMAE (ConViT-B) improves the baseline of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='8% and achieves comparable performance with state-of- the-art contrastive learning-based SSL methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=', iBOT [13] (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' When scaling up the model size from ConViT-B to ConViT-L, our DMJD further reports a higher liner probing accuracy of 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With CLIP targets, DMJD further pushes the accuracy to another height of 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='0 with ConVit-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 7, we present qualitative evaluations of the learned models’ representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' The t-SNE visualization presents a trend consistent with the linear probing performance that our DMJD with high-level learning targets produces more discriminative embedding spaces, which again justifies the superiority of our designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Semantic Segmentation and Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We con- duct fine-grained transfer learning evaluations to verify the generalization ability of learned representations by the proposed DMJD, Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Since multi-scale features benefit down- stream tasks, we adopt UperNet [58] and Mask R-CNN [59] as segmentation and object detection heads, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Concretely, with a non-parametric target tokenizer, DMJD surpasses the ConvMAE baseline by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 mIoU (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='3 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='7) on ADE20K for semantic segmentation and achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2 AP box (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='4 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='2) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='5 AP mask (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='6 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content='1) gains on COCO for object detection and instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' It is worth not- ing that our DMJD outperforms models learned with parametric target tokenizers, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' CLIP [25], and only consumes half of ETEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' These results justify that the learned representations of DMJD generalize well for several downstream tasks while keeping efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' CONCLUSION In this paper, we propose a conceptually simple yet training- efficient MIM framework, termed disjoint masking with joint distillation (DMJD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' With the multi-view disjoint masking strategy, our approach improves the utilization of training signals accelerating the model convergence significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' In- troducing explicit semantic guidance on the visible parts via joint distillation, the proposed DMJD cooperatively boosts the training efficiency while keeping the learned representations to generalize well to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' We hope these achievements shed new light on training efficiency research in the MIM community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' 9 REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfZve0/content/2301.00230v1.pdf'} +page_content=' Vaswani, N.' metadata={'source': 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transformers +Dmitry Nikolaev +Sebastian Padó +IMS, University of Stuttgart +dnikolaev@fastmail.com +pado@ims.uni-stuttgart.de +Abstract +Variants of the BERT architecture specialised +for producing full-sentence representations of- +ten achieve better performance on downstream +tasks than sentence embeddings extracted +from vanilla BERT. However, there is still little +understanding of what properties of inputs de- +termine the properties of such representations. +In this study, we construct several sets of sen- +tences with pre-defined lexical and syntactic +structures and show that SOTA sentence trans- +formers have a strong nominal-participant-set +bias: cosine similarities between pairs of sen- +tences are more strongly determined by the +overlap in the set of their noun participants +than by having the same predicates, lengthy +nominal modifiers, or adjuncts. At the same +time, the precise syntactic-thematic functions +of the participants are largely irrelevant. +1 +Introduction +Transformer-based encoder-only models derived +from the BERT architecture and pre-trained us- +ing similar objective and training regimens (De- +vlin et al., 2019; Liu et al., 2019) have become +the standard tool for downstream tasks at the level +of individual tokens and token sequences (Tenney +et al., 2019; Wang et al., 2021). Whole-sentence +representations can also be easily extracted from +the outputs of these models by either using the +embedding of the special [CLS] token, in cases +where the model was trained on the next-sentence- +prediction task, or averaging or max-pooling the +embeddings of all tokens produced by the model +(Zhelezniak et al., 2019). While both approaches +are widely used in practice, it has been argued +that these representations are not well suited for +sentence-level downstream tasks. Several modifica- +tions to the architecture and training regime were +proposed, which are known collectively as sentence +transformers (STs; Reimers and Gurevych, 2019). +STs have achieved state-of-the-art performance +on downstream tasks such as semantic search and +question answering (Santander-Cruz et al., 2022; +Ha et al., 2021). Their analysis, however, has re- +ceived considerably less attention than the analysis +of the vanilla BERT model and its variants (Rogers +et al., 2020; Conia and Navigli, 2022). In fact, these +models are often considered to be uninterpretable +(Minaee et al., 2021). +A common feature of STs is that they are fine- +tuned to produce similar vector-space representa- +tions for semantically similar sentences. This ob- +jective induces a complex loss landscape shaped by +the available training data. The original Sentence- +BERT model (Reimers and Gurevych, 2019) was +trained on natural language inference data, and sen- +tences were considered to be semantically similar +if their NLI label was that of entailment. SOTA +models were trained on a much larger web-crawled +corpus including more than 1 billion sentence pairs +mined from sources such as Reddit conversations, +duplicate question pairs from WikiAnswers, etc.1 +The richness and variability of this dataset begs the +question of what notion of semantic similarity is +implicitly learned by the models trained on it. +In this study, we begin addressing this question +through analysis of natural-looking synthetic sen- +tences with controlled syntactic and lexical content. +We concentrate on three questions. +First, we test if STs have part-of-speech biases. +We show that, all other things being equal, informa- +tion provided by nouns plays more important role +than the information provided by verbs, both in +simple sentences and in sentences with coordinated +verbal phrases. +Second, we compare the relative importance of +the overlap in the sets of participants in two sen- +tences with that of how many participants have +identical syntactic functions. We show that raw +lexical overlap is relatively more important than +having the same nouns in the same syntactic slots. +1See +the +list +at +https://huggingface.co/ +sentence-transformers/all-mpnet-base-v2 +arXiv:2301.13039v1 [cs.CL] 30 Jan 2023 + +Third, we check how strongly sentence represen- +tations are affected by other sentential elements, +such as adverbials and nominal modifiers of differ- +ent types and lengths. We show that, unlike BERT +with token averaging, STs seem to largely disregard +these components in favor of nominal participants. +The paper is structured as follows: § 2 presents +the methodology that we follow in our analyses +and the models we employ; § 3 presents the case +studies and their results; § 4 provides an overall +discussion; § 5 surveys related work; § 6 concludes +the paper. +2 +Methods and Experimental Setup +We experiment with representations produced +by three models. +Two are SOTA STs: all- +mpnet-base-v2 (MPNET) is an instance of +mpnet-base (Song et al., 2020) fine-tuned on +the 1B sentence-pair corpus using the training ar- +chitecture from Reimers and Gurevych (2019); +all-distilroberta-v1 (DistilRoberta) is +a distilled instance of roberta-base (Sanh +et +al., +2019) +fine-tuned +in +the +same +way. +The third model is the vanilla pre-trained +bert-large-uncased (BERT), as a point of +comparison for the first two. +All models were downloaded from HuggingFace. +Standard APIs from the Sentence Transformers +library2 were used to compute embeddings using +MPNET and DistilRoberta; for the vanilla BERT +model, we averaged the embeddings of all sentence +tokens, including [CLS] and [SEP].3 +We structure the presentation as a series of case +studies. For each case study, we construct a set of +sentences controlled for lexical content and syntac- +tic structure. Sentences are created in such a way as +to be grammatically correct, look naturalistic, and +as far as possible not bias the analysis.4 They are +arguably less complex and variable than examples +sampled from real-word corpora; however, we be- +lieve that an analysis based on simple sentences is +a reasonable first step towards a better understand- +ing of model representations, as previous work has +2https://www.sbert.net/index.html, +Reimers and Gurevych (2019). +3We experimented with omitting the special tokens, but +this led to sentence representations dominated by punctuation +signs and other undesired effects. In line with previous work +(Ma et al., 2019), we also found that using [CLS] embeddings +leads to bad results due to their high redundancy, and we do +not discuss them. +4Sentence-generating and model-fitting scripts can be +found in the Supplementary Materials. +shown for sentiment analysis (Kiritchenko and Mo- +hammad, 2018) and syntactic analysis (Marvin and +Linzen, 2018). +For each case study, we compute embeddings +for all sentences, together with cosine similarities +between embeddings of sentence pairs. We analyze +the similarities by means of regression modelling. +More precisely, we regress cosine similarities, z- +scored to improve comparability between encoders, +on the properties of sentence pairs, such as lexical +overlap, presence of identical participants in identi- +cal syntactic positions, or POS tags of participants. +We inspect the coefficients of the resulting regres- +sion fits to assess the relative importance of these +properties. Since (almost) all properties are coded +as binary variables, their magnitudes are directly +comparable in terms of importance. +For terminological clarity, we will use the term +models to refer to the regression models we use to +analyse the impact of sentence properties on rep- +resentational similarity. We call the transformers +computing these embeddings encoders. +Where the features of sentence pairs can be +straightforwardly related to simple properties of +individual sentences (e.g., in case when we are +testing if they have the same subject or direct ob- +ject), we also project sentence embeddings on a +2-D surface using UMAP (McInnes et al., 2018)5 +and check if the spatial organisation of the points +is in line with our observations. +Lexical choice +A potential confound of our ex- +perimental setup is lexical choice, which is never +completely neutral. For example, by taking a se- +mantically close pair of verbs, we can considerably +reduce the effect of predicate mismatch between +two sentences. Moreover, encoders can react id- +iosyncratically to particular words and word com- +binations. Including all combinations of words and +their positions in sentence pairs as predictor vari- +ables is not a solution, however, as it defeats the +purpose of identifying structural patterns and, in +the limit, amounts to replicating the encoders. We +address this confound in three ways. +First, we select nouns to be always at least as +interchangeable as words of other parts of speech +in terms of belonging to similar mid-to-high fre- +quency bands and referring to conceptually simple, +concrete objects. This follows from our working +hypothesis that encoders give preferential treatment +5We use the default settings and pairwise cosine dissimi- +larities as distance measure. + +to nominal elements, whose (generally entity re- +ferring) semantics is arguably easier to capture +than, for example, that of (generally event refer- +ring) verbs (Baroni and Lenci, 2011). +Second, we compare the analysis of the ST en- +coders against the analysis of the vanilla BERT en- +coder. As they are derived from averaging, vanilla +BERT embeddings treat all words equally, so if our +sentences, e.g., undersell differences in adverbs +because we chose two nearly synonymous ones, +this should be visible in the small coefficient track- +ing the impact of adverbs in the regression model +based on BERT embeddings. As will be shown +below, however, the hierarchy of coefficients for +regression models of STs is very different from that +for vanilla BERT, which arguably indicates that the +role of lexical effects is minor. +Third, we re-run all reported models on sen- +tences of the same structure with different lexical +content; see the Appendix for details. We observe +high stability of coefficients across replications, +higher for STs than for vanilla BERT. This further +corroborates the validity of our generalisations. +3 +Case Studies +This section presents a series of case studies testing +the sensitivity of embeddings produced by sentence +transformers and BERT token averages to proper- +ties of input sentences. We start with analysing +simple intransitive sentences (§ 3.1) and simple +transitive sentences (§ 3.2). We then make specific +aspects of the structure more complex, analysing +the effect of lengthy NPs (§ 3.3) and coordinated +VPs (§ 3.4). Finally, we look more closely at the +syntax-semantics interface by inverting the proto- +typical alignment of POS tags and syntactic func- +tions (predicative nominals and gerund subjects, +§ 3.5) and by testing the degree to which encoders +track particular syntactic functions of verb argu- +ments (§ 3.6). +3.1 +Simple Intransitive Sentences +Data +The main goal of the analysis of simple +intransitive sentences is to check the relative con- +tribution of their components to their embeddings. +We study a nearly-minimal sentence template with +a nominal subject, an adverbial adjunct, and an in- +transitive verb. We construct a set of 256 sentences +of the form ‘[det] [subj] [adverb] [verb][punct]’, +where det ranges over {a, the}; subj ranges over +mpnet +distilroberta +bert +SameDet +0.07 +0.07 +0.37 +SameAdv +0.33 +0.31 +0.45 +SamePred +0.74 +0.61 +0.58 +SamePunct +0.24 +0.24 +0.84 +SameSubj +2.26 +2.40 +1.27 +R-squared +0.67 +0.71 +0.48 +Table 1: A summary of the models predicting z-scored +pairwise cosine similarities between embeddings of +sentences with intransitive verbs. All coefficients are +significant with p < 0.001. +a set of nouns,6 adverb ranges over {quickly, +slowly}, verb ranges over {appears, vanishes, +stops, moves}, and punct, over {., !}. Here and in +subsequent experiments, the generation procedure +assures that all sentence features are statistically +independent, which is a crucial prerequisite for +linear-regression modelling. +Model +The regression model matrix is based on +32,640 pairs of generated sentences, which differ in +the value of at least one feature, with predictor vari- +ables SameDeterminer, SameAdverb, SameVerb, +SamePunct, and SameSubj. We regress z-score- +transformed cosine similarities between sentence +embeddings computed by three different encoders +on these predictor variables. The coefficients of the +fitted models are shown in Table 1.7 +Results +Three observations from Table 1 hold for +all subsequent analyses. +(i) The coefficients are positive for all models +and all features. This means that sentence pairs +which agree in some constituent are always more +similar than sentence pairs that do not – as ex- +pected. +(ii) The coefficient of determination (R2) is +larger for ST-focused linear models. This means +that the embeddings computed by the ST encoders +are more dependent on the features of the sentences +we track and less dependent on identities of lexical +units. (It can be noted that the fact that we achieve +R2 ≈ 0.7 using only a few structural properties is +remarkable in itself.) +(iii) The differences among coefficients of the +ST-focused linear models are in general larger than +6{cat, dog, artist, teacher, planet, star, wind, rain} +7Replication models, fitted on sentences with the same +structure but different lexical content, are shown in Table 8 in +the Appendix. + +10 +0 +10 +0 +10 +mpnet +0.0 +2.5 +5.0 +7.5 +2.5 +5.0 +7.5 +bert +subj +cat +dog +artist +teacher +planet +star +wind +rain +10 +0 +10 +0 +10 +mpnet +0.0 +2.5 +5.0 +7.5 +2.5 +5.0 +7.5 +bert +adv +quickly +slowly +10 +0 +10 +0 +10 +mpnet +0.0 +2.5 +5.0 +7.5 +2.5 +5.0 +7.5 +bert +pred +appears +vanishes +stops +moves +10 +0 +10 +0 +10 +mpnet +0.0 +2.5 +5.0 +7.5 +2.5 +5.0 +7.5 +bert +punct +. +! +Figure 1: UMAP projections of embeddings of sen- +tences with intransitive verbs (left: +sentence trans- +former, right: BERT). +those of the linear model analysing BERT: in the +latter, the biggest coefficient (1.27 for SameSubj) +is only ≈ 3.5 times higher than the smallest one +(0.37 for SameDet), while for the ST models this +ratio is above 30. This is connected to the fact that +BERT-derived sentence representations are more +dependent on semantically impoverished elements, +such as determiners and punctuation signs, which +dampen the effect of other constituents. For the +sake of brevity, we do not analyse determiners and +punctuation in subsequent experiments and keep +them constant as the and . respectively. +Turning to the comparison of coefficients inside +models, we see that STs pay considerably more +attention to subjects than to predicates: all things +being equal, sentences with different predicates and +adverbs but the same subject will be more similar +than sentences with the same predicate and adverb +and different subjects. The influence of punctuation +is surprisingly strong, being comparable to that of +adverbs, while the effect of determiners is very +weak, albeit statistically significant. +A plot of UMAP projections of sentence em- +beddings produced by MPNET and BERT, shown +in Figure 1, underlines that while averaged BERT +embeddings distinguish punctuation signs but do +not distinguish subjects, the situation is reversed +for the sentence transformer: it distinguishes sub- +jects cleanly but largely abstracts away from other +structural properties. +3.2 +Transitive Sentences +Data +The transitive sentences used in the anal- +ysis are generated using the following template: +‘The [subj] [adverb] [verb] the [obj].’ The range of +nouns was slightly extended;8 the same adverbs as +in the previous experiment were used, while verb +ranged over {sees, chases, draws, meets, remem- +bers, pokes}. This produces 672 different sentences +and 225,456 sentence pairs. +Model +The coding for SameAdv and SamePred +remains as above. The main focus in this study is +on whether sentence similarities are dominated by +the sentences having the same subject, the same +direct object, or the same words in these two po- +sitions even if their order were reversed. To test +for this, we added a categorical variable with the +following values: +00 no overlap in subject and object (the baseline); +A0 same subject, different objects; +0B same object, different subjects; +0A the subject of the first sentence is the object +of the second; +B0 the object of the first sentence is the subject +of the second; +BA subject and object are swapped; +AB the same subject and object. +Results +A summary of the fitted models is given +in Table 2.9 It demonstrates that when it comes to +simple transitive sentences, our understanding of +their embeddings produced by sentence transform- +ers remains high, despite the sentences being more +complex (R2 ≈ 0.7), while BERT embeddings +become more unpredictable (R2 ≈ 0.31). Fur- +thermore, while BERT again essentially treats all +tokens more or less equally, with adverbs slightly +discounted, STs prioritise participants (even B0 has +higher coefficients than SamePred). +On the other hand, neither BERT nor STs priori- +tise the exact syntactic function of the participants: +coefficients for A0 vs. 0A, 0B vs. B0, and AB vs. +8To {cat, dog, teacher, artist, robot, machine, tree, bush, +planet, star, wind, rain}. +9A summary of the replication model fits is provided in +Table 9 in the Appendix. + +mpnet +distilroberta +bert +SameAdv +0.49 +0.36 +0.56 +SamePred +0.73 +0.42 +0.78 +SubjObj_0A +1.27 +1.40 +0.65 +SubjObj_0B +1.31 +1.45 +0.69 +SubjObj_A0 +1.44 +1.45 +0.75 +SubjObj_AB +2.98 +3.08 +1.60 +SubjObj_B0 +1.37 +1.42 +0.58 +SubjObj_BA +2.85 +2.98 +1.39 +R-squared +0.74 +0.73 +0.31 +Table 2: A summary of the models predicting z-scored +pairwise cosine similarities between embeddings of +sentences with transitive verbs. All coefficients are sig- +nificant with p < 0.001. +BA are largely comparable across all models with +BA ≈ A0 + 0B. That is, the effects of subjects +and objects are largely independent of one another. +A UMAP plot with the embeddings for the tran- +sitive sentences is shown in Figure 2 in the Ap- +pendix. It demonstrates that STs arrive at a much +more fine-grained clustering of sentences, largely +dominated by subjects and objects. They largely +discount predicates and adverbs which are quite +prominent in averaged BERT embeddings. +3.3 +Transitive Sentences with Long NP +Modifiers +The previous analyses showed that representations +computed by STs are highly attuned to verb par- +ticipants but not to their particular syntactic roles. +This may mean that ST may be potentially misled +by nouns in other positions in the sentence, which +have less relevance to the described situation. This +study explores this possibility. +Data +We repeat the analysis from § 3.2 using +the template of the form ‘The [subj] [modifier] +[adverb] [verb] the [obj]’, with a smaller set of sub- +jects,10 and the modifier ranging over {with big +shiny eyes, that my brother saw yesterday, whose +photo was in the papers, worth a great deal of +money}. Altogether this gives 1,440 sentences and +1,036,080 sentence pairs. The modifiers have inter- +nal syntactic structure and contain a non-negligible +amount of lexical material that the models have to +‘skip over’ if their representations were focused on +the participant structure of the matrix clause. +10{cat, dog, rat, giraffe, wombat, hippo} +mpnet +distilroberta +bert +SameMod +1.01 +1.02 +1.62 +SameAdv +0.40 +0.42 +0.27 +SamePred +0.89 +0.67 +0.40 +SubjObj_0A +0.83 +1.06 +0.32 +SubjObj_0B +0.97 +1.27 +0.42 +SubjObj_A0 +1.11 +1.14 +0.53 +SubjObj_AB +2.14 +2.44 +1.00 +SubjObj_B0 +1.20 +1.30 +0.54 +SubjObj_BA +2.09 +2.40 +0.91 +R-squared +0.73 +0.81 +0.61 +Table 3: A summary of the models predicting z-scored +pairwise cosine similarities between embeddings of +sentences with transitive verbs and lengthy subject +modifiers. +All coefficients are significant with p < +0.001. +Model +The same coding strategy as in the preced- +ing section is used, augmented by a new binary vari- +able, SameMod, tracking whether two sentences +have the same modifier for the subject. +Results +Both the model coefficients, shown in +Table 3, and the UMAP plot, shown in Figure 3 +in the Appendix, indicate that BERT embeddings +are highly sensitive to lengthy modifiers:11 the +SameMod coefficient in the linear model is larger +than the coefficients for the same predicate and the +same subject-object combination added together. +The situation is very different for STs: SameMod +is more important than SamePred, especially for +DistilRoberta, but, with one exception, not more +important than even a partial overlap in participants. +Having the same participants, in either the same or +swapped syntactic functions, is more than twice as +important. We take this as evidence that STs have +a specific bias towards matrix-clause participant +sets, that is, the nouns that fill a thematic role of the +main predicate, while their precise functions and +nouns found in other positions in the sentence are +less important. +3.4 +Coordinated Verbal Phrases +The analyses presented above show that the main +predicate of the sentence has only a limited influ- +ence on the representations computed by STs, com- +pared to its subjects and objects. Here, we show +that this effect still holds if there is more than one +11The results of the replication fits are shown in Table 10 in +the Appendix. + +mpnet +distilroberta +bert +V1Same +0.41 +0.26 +0.21 +V2Same +0.13 +0.08 +0.23 +V3Same +0.36 +0.34 +0.41 +N1Same +0.33 +0.35 +0.23 +N2Same +0.12 +0.22 +0.30 +N3Same +0.56 +0.57 +0.41 +R-squared +0.11 +0.1 +0.09 +Table 4: A summary of the models predicting z-scored +pairwise cosine similarities between embeddings of +sentences with coordinated VPs from binary predictors. +All coefficients are significant with p < 0.001. +main predicate. +Data +Using the same sets of nouns and transi- +tive verbs as in the previous experiment, we con- +struct sentences of the form ‘The man [verb1] +the [noun1], [verb2] the [noun2], and [verb3] the +[noun3]’, where triples of verbs and nouns are +taken from the Cartesian product of the sets of +all noun and verb combinations of size 3 without +replacement. To alleviate a possible ordering bias, +all verb and noun triples are shuffled for each sen- +tence. This results in 400 sentences and 79,800 +sentence pairs. +Models and results +The analysis proceeds in +three stages. First, we check if positions 1, 2, and +3 have different importance by regressing the nor- +malised cosine similarity on six binary variables +N[oun]1Same, V[erb]1Same, N2Same, etc. The +models, summarised in Table 4,12 show low coeffi- +cients of determination (with R2 around 0.1), but +they indicate that positions are of unequal impor- +tance: BERT gives more weight to the last noun +and the last verb, while STs focus on the first and +the last N-V pair and largely ignore the second one. +A significantly better fit can be achieved by re- +placing binary predictors with overlap scores for +nouns and verbs. As Table 513 shows, this type +of model, even though it contains only 2 variables +instead of 6, obtains R2 ≈ 0.65 for STs. It is also +evident that all three models place more weight +on noun overlap than on verb overlap, with Dis- +tilRoberta showing the biggest difference between +the two. +12A summary of the replication fits is given in Table 11 in +the Appendix. +13See Table 12 in the Appendix for the replication fits. +mpnet +distilroberta +bert +VerbOverlap +0.78 +0.59 +0.64 +NounOverlap +0.93 +1.09 +0.88 +R-squared +0.65 +0.68 +0.52 +Table 5: A summary of the models predicting z-scored +pairwise cosine similarities between embeddings of +sentences with coordinated VPs from overlap scores. +All coefficients are significant with p < 0.001. +This raises the question of whether particular +verb-noun collocations play a noticeable role, i.e., +if a sentence containing chases the wombat will +be considerably more similar to another sentence +containing the exact phrase compared to a sentence +containing chases and wombat but not as a trigram. +Simply adding n-gram overlap scores to the model +is not possible, however, because it is highly cor- +related with both noun overlap and verb overlap. +In order to obviate this obstacle, we first construct +an auxiliary linear model predicting trigram over- +lap from noun and verb overlap and then use the +residuals of this regression in the main model. +The results are ambiguous: on one hand, the +coefficient for residualised trigram overlap is sta- +tistically significant with p < 0.001. On the other +hand, the effect is very weak (more than ten times +weaker than that of either noun overlap or verb +overlap), and the addition of trigram overlap to the +model improves R2 by less than 0.001. This seems +to indicate that trigram overlap is not important for +practical purposes. +3.5 +Predicative Nominals with Gerund +Subjects +A potential weak point of our analysis is that parts +of speech and syntactic functions are not decoupled: +it is not yet clear whether the encoders pay attention +to nouns or to subjects and objects. +Data +To address this issue, we construct another +set of sentences where the subject is a gerund and +the predicate is nominal. The template is ‘[gerund] +[object] [copula] a [adjective] [predicate]’, where +gerund ranges over {continuing, abandoning, +starting, completing}, object ranges over {it, them, +the project, the plan}, copula is one of {is, was, will +be, is going to be}, adjectives are {big, real, negli- +gible, insignificant}, and the predicative nominal +ranges over {solution, mistake, failure, triumph}. +This gives 1024 sentences and 523,776 sentence + +mpnet +distilroberta +bert +SameSubj +0.82 +0.70 +0.31 +SameCop +0.35 +0.30 +0.55 +SameAdj +0.58 +0.79 +0.50 +SamePred +0.99 +1.01 +0.52 +SameObjNoun +1.01 +1.04 +0.60 +SameObjPron +0.44 +0.50 +0.42 +R-squared +0.50 +0.54 +0.22 +Table 6: A summary of the models predicting z-scored +pairwise cosine similarities between embeddings of +sentences with gerund subjects and nominal predicates. +All coefficients are significant with p < 0.001. +pairs. A variable copula provides an additional test +as to whether the sentence encoders can recognise +multi-word sequences with low semantic content. +Model +The sentence pair encoding includes four +binary variables (SameSubj, SameCop, SameAdj, +SamePred) and a nominal variable for the direct ob- +ject, indicating whether objects are different (base- +line), are identical and pronominal (SamePron), or +are identical and nominal (SameNoun). +Results +The results in Table 614 demonstrate that +all models treat both nominal predicates and nom- +inal direct objects as more important than gerund +subjects. STs, moreover, pay less attention to iden- +tical pronominal objects and discount multi-word +copula forms. R2 values for the ST model are +lower than in the previous experiments (in the 0.50– +0.55 range), which may potentially indicate a poor +choice of lexical items; however, replication ex- +periments with a different set of words (except for +copula forms) achieved comparable results. This +suggests that embeddings of sentences of this type +are less easily explainable as additive combinations +of individual words compared to the sentence types +surveyed previously. +3.6 +Revisiting Participant Sets: Ditransitive +Sentences +Our final experiment revisits the opposition be- +tween lexical overlap in verbal phrases and exact +argument-predicate matching. In this case, we fo- +cus on ditransitive verbs with two arguments: a +direct object and an oblique object which is an +14See an overview of replication fits in Table 13 in the +Appendix. +mpnet +distilroberta +bert +SameAdv +1.05 +1.07 +0.64 +SamePred +0.93 +0.64 +0.83 +Overlap +0.90 +1.00 +0.91 +SPCRes +0.03 +0.02 +0.10 +R-squared +0.745 +0.738 +0.57 +R-squared +(w/o SPCRes) +0.744 +0.737 +0.56 +Table 7: A summary of the models predicting z-scored +pairwise cosine similarities between embeddings of +sentences with ditransitive verbs. SPCRes stands for +SamePosCountRes, i.e. the residuals of the number +of identical words in identical positions regressed on +lexical overlap. +All coefficients are significant with +p < 0.001. +integral part of the situation.15 +Data +All permutations of the triple of basic +nouns {cat, dog, rat} are generated. For each per- +mutation, all three nouns are, in turn, replaced with +one of the members of the set of extra nouns {gi- +raffe, wombat, hippo}; the original permutations +are also used. This provides a set of unique triples +of nouns where each pair of triples has from one +to three nouns in common. The Cartesian product +of this set of triples with a set of ditransitive verbs +({describes, sells, shows}) and a set of adverbs +({happily, quickly, secretly) is used to fill the tem- +plate ‘The [noun1] [adverb] [verb] the [noun2] to +the [noun3].’ This procedures gives 540 sentences +and 145,530 sentence pairs. +Model +The sentence pairs are coded for same +adverb, same predicate, the number of matching +nouns in matching positions (SamePosCount), and +lexical overlap minus 1 (the baseline value of 0 +corresponds to overlap of 1; each successive value +corresponds to increase in overlap). As with over- +lapping words and trigrams above, these predictors +are correlated. Therefore, we residualise Same- +PosCount after regressing it on lexical overlap. +Results +Table 7 is inconclusive in a similar way +to results from § 3.5. The coefficients for residu- +alised SamePosCount are significant; however, in +the ST models, their size is very small, and Same- +15Many English ditransitive verbs can undergo the ‘dative +alternation’, which swaps the oblique object with a preposi- +tional phrase: Give the book to me/John vs. Give me/John a +book (Levin, 1993). Of the verbs we use, show and sell partic- +ipate in it, and the status of describe varies across speakers. + +PosCount does not materially improve the predic- +tive power. We conclude, therefore, that syntactic +positions do not matter a great deal, in line with +our ‘participant set’ interpretation from § 3.4. +4 +Discussion +Our analysis arguably goes some way towards ex- +plaining why sentence transformers beat vanilla +BERT-based models with token averaging on +sentence-modelling tasks. Token averaging makes +it impossible to distinguish between semantically +rich and impoverished sentence elements, nor be- +tween syntactically central vs. peripheral elements: +punctuation signs and determiners contribute on +the same level as the matrix-clause predicate and +main participants, while lengthy modifiers, such +as relative clauses, and multi-word copula forms +dominate the representation. +Sentence transformers, on the other hand, learn +to discount elements that only serve a grammatical +function or present background information and fo- +cus instead on the semantic kernel of the sentence. +The latter is in effect largely synonymous with the +set of nominal elements in the main clause, first +of all participants, but also predicative nominals. +Importantly, despite their evident syntactic-analytic +capabilities (e.g., in our setting they can distinguish +between participants of main and relatives clauses +and between main and auxiliary verbs), STs seem +to not pay much attention to the distinction between +subjects and direct or indirect objects. Instead they +prioritise raw overlap in the set of nominal partic- +ipants of the matrix clause. This can be seen, by +slightly abusing terminology of theoretical linguis- +tics, as a focus on the aboutness/topic of sentences, +what things they describe, and not on their predica- +tion/comment, what they actually say about those +things (Hu and Pan, 2009). +We believe that this focus is not inherent to the +architecture of sentence transformers but reflects +the nature of the datasets used for fine-tuning STs. +The size of these datasets makes it impossible to +convincingly reason about their contents, but their +genres (QA pairs, Reddit threads, etc.) makes it +plausible to expect a high degree of topic-based +overlap: questions and conversations tend to re- +volve around entities (persons and things), with +their actions and properties repeating less often. +This naturally leads to a focus on nouns referring +to prominent entities, which are known to appear +preferentially as subjects or objects for reasons of +coherence (Barzilay and Lapata, 2008), arguably a +good match to the patterns we observe. +5 +Related Work +Analysis of transformer-based models for sentence- +level tasks, such as NLI, question answering, or +text classification, has largely followed the same +approaches as found in the general BERTology +(Rogers et al., 2020): probing, analysis of the ge- +ometry of the embedding space, extraction of parts +of input that are particularly important for model +performance, and behavioural analysis. In this vein, +Liu et al. (2021) and Peyrard et al. (2021) analyse +the attention patterns powering the performance of +transformer models on different types of sentence +classification, and Li et al. (2020) show that embed- +dings of sentences computed by BERT-based mod- +els, including siamese-fine-tuned sentence trans- +formers, are anisotropic and can be improved via +normalisation. Chrysostomou and Aletras (2021) +survey the existing methods for extracting ratio- +nales from input sentences in the context of text +classification and propose an improved approach, +while Luo et al. (2021) demonstrate that sentence +embeddings derived by averaging BERT token rep- +resentations suffer from artefacts arising from po- +sitional embeddings. Zhelezniak et al. (2019) ar- +gue that averaging should be replaced with max- +pooling. +Very similar to ours is the approach adopted by +MacAvaney et al. (2022), who construct a series of +probes to analyse the performance of several mod- +els on the task of information retrieval. While their +methodology relies on high-level document statis- +tics and wholistic document manipulation (word +and sentence shuffling, token-frequency similar- +ity between the document and the query, textual +fluency, etc.), our study analyses the role of lin- +guistically motivated structural factors and thus +complements their findings. +Opitz and Frank (2022) aim at directly decom- +posing the representations produced by sentence +transformers into several parts capturing different +properties of sentences reflected in AMR annota- +tions (presence of negation, concepts included in +the sentence, etc.). While our study tries to as- +certain what meaning components dominate the +representations, Opitz and Frank assume that these +components are known in advance and are equally +important: sentence embeddings in their modified +SBERT model are split into 15 segments, each of + +which corresponds to one AMR-based meaning +component, plus a residual part to capture every- +thing not covered by AMR annotations. +6 +Conclusion +This paper aims at making a contribution towards +a better understanding of sentence transformers, +which are often seen as black boxes. We have +demonstrated that we can make surprisingly precise +inferences about sentence-pair similarities using +simple linguistic features such as lexical overlap. +The crucial difference between bag-of-words dis- +tributional models and current encoders is that STs +have became quite adept at disregarding ‘irrelevant’ +parts of the sentence and concentrating on its key +elements. Unlike vanilla BERT sentence embed- +dings obtained by token averaging, STs yield more +structured embeddings that focus on the matrix +clause and are less tied to individual lexical items +and strings of function words. +This progress, however, comes with a particu- +lar type of bias: the structures that lead to high +sentence similarity in STs, i.e. the overlap in nomi- +nal ‘participant sets’, seem to mirror the dominant +type of paraphrases found in the data the STs were +tuned on, and STs are not compelled to look at +finer structures of input sentences. At least without +further fine tuning, this would appear to make them +unsuitable for downstream tasks that require knowl- +edge about more fine-grained aspects of sentence +structure, such as semantic roles (Conia and Nav- +igli, 2022), or extra-propositional aspects, such as +monotonicity, negation, or modality (Yanaka et al., +2021; Nakov, 2016). +An interesting direction for future research +would be to explore the ways of decomposing sen- +tence representations into additive aspects such as +participant structure, main predication, etc. The +additional challenge here is that while theoretical +semantics has a lot to say about aspects of sentence +meaning (Pagin, 2016), there remains a lack of +analysis linking the notion of one-dimensional se- +mantic similarity (Agirre et al., 2012) that underlies +the optimisation of current sentence transformers +with theoretically more substantial concepts. +Limitations +The limitations of the proposed analysis are the +following: +1. The analysis is based on synthetic data. This +allows us to fully control the sentence struc- +ture and use balanced lexical material, but it +does not necessarily reflect the performance +of models on real-world data, especially when +sentences or text fragments are much longer. +However, synthetic data have generally shown +to be a good first step toward understanding +the behaviour of complex models. +2. The analysis does not cover graded distinc- +tions between words, i.e. we did not experi- +ment with filling the slots with synonymous +words, as opposed to completely unrelated +words. This makes it impossible to decide if +the models are sensitive to word identities or +to their actual semantics, as long as these two +notions are distinguishable. +3. The outputs of the models are interpreted +using linear regression analysis anchored to +the properties of synthetic sentences. This +kind of analysis makes it possible to disentan- +gle additive effects of different components +of sentence structure and provides statistical- +significance estimates, while high R2 values +indicate that our findings have some valid- +ity. However, it cannot fully account for the +lexical effects (which we tried to safeguard +against by carefully selecting template fillers), +non-linear effects, and hidden collinearity pat- +terns (beyond those we addressed using resid- +ualised analysis). +4. The range of models analysed in the paper is +restricted. It covers some amount of variabil- +ity (sentence transformers vs. vanilla BERT; +two different variants of a base model for STs, +one of them distilled), but other combinations +of model architecture and training/fine-tuning +regime can lead to different outcomes. +References +Eneko Agirre, Daniel Cer, Mona Diab, and Aitor +Gonzalez-Agirre. 2012. +SemEval-2012 task 6: A +pilot on semantic textual similarity. 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Ham- +merla. 2019. +Don’t settle for average, go for the +max: +Fuzzy sets and max-pooled word vectors. +CoRR, abs/1904.13264. + +mpnet +distilroberta +bert +SameDet +0.08 +0.11 +0.26 +SameAdv +0.38 +0.38 +0.96 +SamePred +1.02 +0.95 +0.49 +SamePunct +0.18 +0.26 +0.64 +SameSubj +2.15 +2.17 +0.65 +R-squared +0.71 +0.71 +0.43 +Table 8: A summary of the replication models predict- +ing z-scored pairwise cosine similarities between em- +beddings of sentences with intransitive verbs. All coef- +ficients are significant with p < 0.001. +A +Appendix +A.1 +Dimensionality-reduction plots +A.1.1 +Simple transitive sentences +A UMAP plot of embeddings of simple transitive +sentences encoded accordings to their properties is +shown in Figure 2. +A.1.2 +Transitive sentences with long NP +modifiers +A UMAP plot of embeddings of transitive sen- +tences with lengthy subject modifiers encoded ac- +cordings to their properties is shown in Figure 3. +A.2 +Replication-model fits +A.2.1 +Simple intransitive sentences +The following lexical items were used for the repli- +cation experiment: +• Nouns: wolf, bear, fruit, vegetable, building, +car, lightning, wave +• Verbs: stabilizes, bursts, grows, shrinks +• Adverbs: suddenly, predictably +A summary of the replication models is shown in +Table 8. +A.2.2 +Simple transitive sentences +The following lexical items were used for the repli- +cation experiment: +• Nouns: pig, horse, soldier, farmer, android, +computer, grass, forest, comet, galaxy, cloud, +lightning +• Verbs: hears, pursues, imagines, recognizes, +touches, finds +mpnet +distilroberta +bert +SameAdv +0.54 +0.32 +0.95 +SamePred +0.49 +0.43 +0.75 +SubjObj_0A +1.46 +1.50 +0.70 +SubjObj_0B +1.49 +1.53 +0.66 +SubjObj_A0 +1.48 +1.54 +0.76 +SubjObj_AB +3.19 +3.23 +1.56 +SubjObj_B0 +1.40 +1.48 +0.50 +SubjObj_BA +3.07 +3.14 +1.34 +R-squared +0.81 +0.8 +0.45 +Table 9: A summary of the replication models predict- +ing z-scored pairwise cosine similarities between em- +beddings of sentences with intransitive verbs. All coef- +ficients are significant with p < 0.001. +• Adverbs: suddenly, predictably +A summary of the replication models is shown in +Table 9. +A.2.3 +Transitive sentences with long NP +modifiers +The following lexical and phrasal items were used +for the replication experiment: +• Nouns: horse, pig, donkey, elephant, bison, +moose +• NP modifiers: missing a hind leg, whose face +we all know, born under a bad sign, pictured +on page seventeen +• Verbs: hears, pursues, imagines, recognizes, +touches, finds +• Adverbs: suddenly, predictably +The overview of the model fits is shown in Table 10. +A.2.4 +Coordinated verbal phrases +The following lexical items were used for the repli- +cation experiment: +• Nouns: mouse, horse, fox, kangaroo, bison, +elephant +• Verbs: hears, pursues, imagines, recognizes, +touches, finds +A summary of the replication models is shown in +Tables 11 (individual-word-based models) and 12 +(overlap-based models). + +mpnet +distilroberta +bert +SameMod +1.18 +1.26 +1.83 +SameAdv +0.48 +0.26 +0.41 +SamePred +0.64 +0.64 +0.44 +SubjObj_0A +0.91 +1.00 +0.18 +SubjObj_0B +0.99 +1.09 +0.17 +SubjObj_A0 +1.10 +1.19 +0.24 +SubjObj_AB +2.13 +2.32 +0.42 +SubjObj_B0 +1.16 +1.25 +0.20 +SubjObj_BA +2.11 +2.28 +0.39 +R-squared +0.77 +0.84 +0.71 +Table 10: A summary of the replication models predict- +ing z-scored pairwise cosine similarities between em- +beddings of sentences with transitive verbs and lengthy +subject modifiers. All coefficients are significant with +p < 0.001. +mpnet +distilroberta +bert +V1Same +0.29 +0.18 +0.35 +V2Same +0.13 +0.08 +0.28 +V3Same +0.39 +0.40 +0.42 +N1Same +0.49 +0.48 +0.14 +N2Same +0.10 +0.25 +0.18 +N3Same +0.57 +0.52 +0.17 +R-squared +0.12 +0.11 +0.07 +Table 11: A summary of the replication models pre- +dicting z-scored pairwise cosine similarities between +embeddings of sentences with coordinated VPs from +binary predictors. All coefficients are significant with +p < 0.001. +A.2.5 +Predicative nominals with gerund +subjects +The following lexical items were used for the repli- +cation experiment: +• Gerund subjects: proposing, rejecting, prais- +ing, criticizing +• Pronomial and nominal objects: him, me, the +idea, the design +• Copula forms (same as in the original experi- +ment): is, was, will be, is going to be +• Nominal predicates: decision, defeat, loss, im- +provement +A summary of the replication models is shown in +Tables 13. +mpnet +distilroberta +bert +VerbOverlap +0.69 +0.52 +0.85 +NounOverlap +1.05 +1.20 +0.47 +R-squared +0.69 +0.76 +0.41 +Table 12: A summary of the replication models pre- +dicting z-scored pairwise cosine similarities between +embeddings of sentences with coordinated VPs from +overlap scores. +All coefficients are significant with +p < 0.001. +mpnet +distilroberta +bert +SameSubj +0.82 +0.70 +0.31 +SameCop +0.35 +0.30 +0.55 +SameAdj +0.58 +0.79 +0.50 +SamePred +0.99 +1.01 +0.52 +SameObjNoun +1.01 +1.04 +0.60 +SameObjPron +0.44 +0.50 +0.42 +R-squared +0.50 +0.54 +0.22 +Table 13: A summary of the replication models predict- +ing z-scored pairwise cosine similarities between em- +beddings of sentences with gerund subjects and nom- +inal predicates. +All coefficients are significant with +p < 0.001. +A.2.6 +Participant-set overlap vs. identical +participants +The following lexical items were used for the repli- +cation experiment: +• Basic nouns: horse, pig, donkey +• Extra nouns: elephant, bison, moose +• Verbs: gives, demonstrates, entrusts +• Adverbs: suddenly, predictably, openly +A summary of the replication models is shown in +Tables 14. + +mpnet +distilroberta +bert +SameAdv +1.05 +1.07 +0.64 +SamePred +0.93 +0.64 +0.83 +Overlap +0.90 +1.00 +0.91 +SPCRes +0.03 +0.02 +0.10 +R-squared +0.745 +0.738 +0.57 +R-squared +(w/o SPCRes) +0.744 +0.737 +0.56 +Table 14: A summary of the replication models predict- +ing z-scored pairwise cosine similarities between em- +beddings of sentences with ditransitive verbs. SPCRes +stands for SamePosCountRes, i.e. the residuals of the +number of identical words in identical positions re- +gressed on lexical overlap. All coefficients are signif- +icant with p < 0.001. + +5 +0 +5 +10 +15 +5 +0 +5 +10 +15 +20 +mpnet +10 +0 +10 +20 +10 +0 +10 +20 +30 +distilroberta +2 +0 +2 +4 +6 +2 +4 +6 +8 +10 +12 +bert +subj +robot +machine +tree +bush +planet +star +wind +rain +5 +0 +5 +10 +15 +5 +0 +5 +10 +15 +20 +mpnet +10 +0 +10 +20 +10 +0 +10 +20 +30 +distilroberta +2 +0 +2 +4 +6 +2 +4 +6 +8 +10 +12 +bert +obj +robot +machine +tree +bush +planet +star +wind +rain +5 +0 +5 +10 +15 +5 +0 +5 +10 +15 +20 +mpnet +10 +0 +10 +20 +10 +0 +10 +20 +30 +distilroberta +2 +0 +2 +4 +6 +2 +4 +6 +8 +10 +12 +bert +predicate +sees +chases +draws +meets +remembers +pokes +5 +0 +5 +10 +15 +5 +0 +5 +10 +15 +20 +mpnet +10 +0 +10 +20 +10 +0 +10 +20 +30 +distilroberta +2 +0 +2 +4 +6 +2 +4 +6 +8 +10 +12 +bert +adverb +quickly +slowly +Figure 2: UMAP projections of embeddings of sentences with transitive verbs colour coded according to subject, +object, predicate, and adverb. + +5 +0 +5 +10 +5 +0 +5 +10 +15 +mpnet +5 +0 +5 +10 +15 +10 +5 +0 +5 +10 +15 +20 +25 +distilroberta +5 +0 +5 +10 +15 +20 +15 +10 +5 +0 +5 +10 +15 +20 +25 +bert +subj +cat +dog +rat +giraffe +wombat +hippo +5 +0 +5 +10 +5 +0 +5 +10 +15 +mpnet +5 +0 +5 +10 +15 +10 +5 +0 +5 +10 +15 +20 +25 +distilroberta +5 +0 +5 +10 +15 +20 +15 +10 +5 +0 +5 +10 +15 +20 +25 +bert +obj +cat +dog +rat +giraffe +wombat +hippo +5 +0 +5 +10 +5 +0 +5 +10 +15 +mpnet +5 +0 +5 +10 +15 +10 +5 +0 +5 +10 +15 +20 +25 +distilroberta +5 +0 +5 +10 +15 +20 +15 +10 +5 +0 +5 +10 +15 +20 +25 +bert +modifier +with big shiny eyes +that my brother saw yesterday +whose photo was in the papers +worth a great deal of money +5 +0 +5 +10 +5 +0 +5 +10 +15 +mpnet +5 +0 +5 +10 +15 +10 +5 +0 +5 +10 +15 +20 +25 +distilroberta +5 +0 +5 +10 +15 +20 +15 +10 +5 +0 +5 +10 +15 +20 +25 +bert +adv +quickly +slowly +5 +0 +5 +10 +5 +0 +5 +10 +15 +mpnet +5 +0 +5 +10 +15 +10 +5 +0 +5 +10 +15 +20 +25 +distilroberta +5 +0 +5 +10 +15 +20 +15 +10 +5 +0 +5 +10 +15 +20 +25 +bert +pred +sees +chases +draws +meets +remembers +pokes +Figure 3: UMAP projections of embeddings of sentences with transitive verbs and long subject modifiers colour +coded according to subject, modifier, object, predicate, and adverb. + diff --git a/w9FPT4oBgHgl3EQfPzQl/content/tmp_files/load_file.txt b/w9FPT4oBgHgl3EQfPzQl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2b0b1077d61f16f882dc274b3b37da4e934f664 --- /dev/null +++ b/w9FPT4oBgHgl3EQfPzQl/content/tmp_files/load_file.txt @@ -0,0 +1,1257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf,len=1256 +page_content='Representation biases in sentence transformers Dmitry Nikolaev Sebastian Padó IMS, University of Stuttgart dnikolaev@fastmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='com pado@ims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='de Abstract Variants of the BERT architecture specialised for producing full-sentence representations of- ten achieve better performance on downstream tasks than sentence embeddings extracted from vanilla BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' However, there is still little understanding of what properties of inputs de- termine the properties of such representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' In this study, we construct several sets of sen- tences with pre-defined lexical and syntactic structures and show that SOTA sentence trans- formers have a strong nominal-participant-set bias: cosine similarities between pairs of sen- tences are more strongly determined by the overlap in the set of their noun participants than by having the same predicates, lengthy nominal modifiers, or adjuncts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' At the same time, the precise syntactic-thematic functions of the participants are largely irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 1 Introduction Transformer-based encoder-only models derived from the BERT architecture and pre-trained us- ing similar objective and training regimens (De- vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2019) have become the standard tool for downstream tasks at the level of individual tokens and token sequences (Tenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Whole-sentence representations can also be easily extracted from the outputs of these models by either using the embedding of the special [CLS] token, in cases where the model was trained on the next-sentence- prediction task, or averaging or max-pooling the embeddings of all tokens produced by the model (Zhelezniak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' While both approaches are widely used in practice, it has been argued that these representations are not well suited for sentence-level downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Several modifica- tions to the architecture and training regime were proposed, which are known collectively as sentence transformers (STs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Reimers and Gurevych, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' STs have achieved state-of-the-art performance on downstream tasks such as semantic search and question answering (Santander-Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Their analysis, however, has re- ceived considerably less attention than the analysis of the vanilla BERT model and its variants (Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Conia and Navigli, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' In fact, these models are often considered to be uninterpretable (Minaee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A common feature of STs is that they are fine- tuned to produce similar vector-space representa- tions for semantically similar sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This ob- jective induces a complex loss landscape shaped by the available training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The original Sentence- BERT model (Reimers and Gurevych, 2019) was trained on natural language inference data, and sen- tences were considered to be semantically similar if their NLI label was that of entailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' SOTA models were trained on a much larger web-crawled corpus including more than 1 billion sentence pairs mined from sources such as Reddit conversations, duplicate question pairs from WikiAnswers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1 The richness and variability of this dataset begs the question of what notion of semantic similarity is implicitly learned by the models trained on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' In this study, we begin addressing this question through analysis of natural-looking synthetic sen- tences with controlled syntactic and lexical content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We concentrate on three questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' First, we test if STs have part-of-speech biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We show that, all other things being equal, informa- tion provided by nouns plays more important role than the information provided by verbs, both in simple sentences and in sentences with coordinated verbal phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Second, we compare the relative importance of the overlap in the sets of participants in two sen- tences with that of how many participants have identical syntactic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We show that raw lexical overlap is relatively more important than having the same nouns in the same syntactic slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 1See the list at https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='co/ sentence-transformers/all-mpnet-base-v2 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='13039v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='CL] 30 Jan 2023 Third, we check how strongly sentence represen- tations are affected by other sentential elements, such as adverbials and nominal modifiers of differ- ent types and lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We show that, unlike BERT with token averaging, STs seem to largely disregard these components in favor of nominal participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The paper is structured as follows: § 2 presents the methodology that we follow in our analyses and the models we employ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' § 3 presents the case studies and their results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' § 4 provides an overall discussion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' § 5 surveys related work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' § 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 2 Methods and Experimental Setup We experiment with representations produced by three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Two are SOTA STs: all- mpnet-base-v2 (MPNET) is an instance of mpnet-base (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2020) fine-tuned on the 1B sentence-pair corpus using the training ar- chitecture from Reimers and Gurevych (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' all-distilroberta-v1 (DistilRoberta) is a distilled instance of roberta-base (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2019) fine-tuned in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The third model is the vanilla pre-trained bert-large-uncased (BERT), as a point of comparison for the first two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All models were downloaded from HuggingFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Standard APIs from the Sentence Transformers library2 were used to compute embeddings using MPNET and DistilRoberta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' for the vanilla BERT model, we averaged the embeddings of all sentence tokens, including [CLS] and [SEP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='3 We structure the presentation as a series of case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' For each case study, we construct a set of sentences controlled for lexical content and syntac- tic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Sentences are created in such a way as to be grammatically correct, look naturalistic, and as far as possible not bias the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='4 They are arguably less complex and variable than examples sampled from real-word corpora;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' however, we be- lieve that an analysis based on simple sentences is a reasonable first step towards a better understand- ing of model representations, as previous work has 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='sbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='net/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='html, Reimers and Gurevych (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3We experimented with omitting the special tokens, but this led to sentence representations dominated by punctuation signs and other undesired effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' In line with previous work (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2019), we also found that using [CLS] embeddings leads to bad results due to their high redundancy, and we do not discuss them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 4Sentence-generating and model-fitting scripts can be found in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' shown for sentiment analysis (Kiritchenko and Mo- hammad, 2018) and syntactic analysis (Marvin and Linzen, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' For each case study, we compute embeddings for all sentences, together with cosine similarities between embeddings of sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We analyze the similarities by means of regression modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' More precisely, we regress cosine similarities, z- scored to improve comparability between encoders, on the properties of sentence pairs, such as lexical overlap, presence of identical participants in identi- cal syntactic positions, or POS tags of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We inspect the coefficients of the resulting regres- sion fits to assess the relative importance of these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Since (almost) all properties are coded as binary variables, their magnitudes are directly comparable in terms of importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' For terminological clarity, we will use the term models to refer to the regression models we use to analyse the impact of sentence properties on rep- resentational similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We call the transformers computing these embeddings encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Where the features of sentence pairs can be straightforwardly related to simple properties of individual sentences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', in case when we are testing if they have the same subject or direct ob- ject), we also project sentence embeddings on a 2-D surface using UMAP (McInnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2018)5 and check if the spatial organisation of the points is in line with our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Lexical choice A potential confound of our ex- perimental setup is lexical choice, which is never completely neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' For example, by taking a se- mantically close pair of verbs, we can considerably reduce the effect of predicate mismatch between two sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Moreover, encoders can react id- iosyncratically to particular words and word com- binations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Including all combinations of words and their positions in sentence pairs as predictor vari- ables is not a solution, however, as it defeats the purpose of identifying structural patterns and, in the limit, amounts to replicating the encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We address this confound in three ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' First, we select nouns to be always at least as interchangeable as words of other parts of speech in terms of belonging to similar mid-to-high fre- quency bands and referring to conceptually simple, concrete objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This follows from our working hypothesis that encoders give preferential treatment 5We use the default settings and pairwise cosine dissimi- larities as distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' to nominal elements, whose (generally entity re- ferring) semantics is arguably easier to capture than, for example, that of (generally event refer- ring) verbs (Baroni and Lenci, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Second, we compare the analysis of the ST en- coders against the analysis of the vanilla BERT en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' As they are derived from averaging, vanilla BERT embeddings treat all words equally, so if our sentences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', undersell differences in adverbs because we chose two nearly synonymous ones, this should be visible in the small coefficient track- ing the impact of adverbs in the regression model based on BERT embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' As will be shown below, however, the hierarchy of coefficients for regression models of STs is very different from that for vanilla BERT, which arguably indicates that the role of lexical effects is minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Third, we re-run all reported models on sen- tences of the same structure with different lexical content;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' see the Appendix for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We observe high stability of coefficients across replications, higher for STs than for vanilla BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This further corroborates the validity of our generalisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3 Case Studies This section presents a series of case studies testing the sensitivity of embeddings produced by sentence transformers and BERT token averages to proper- ties of input sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We start with analysing simple intransitive sentences (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1) and simple transitive sentences (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We then make specific aspects of the structure more complex, analysing the effect of lengthy NPs (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='3) and coordinated VPs (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Finally, we look more closely at the syntax-semantics interface by inverting the proto- typical alignment of POS tags and syntactic func- tions (predicative nominals and gerund subjects, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5) and by testing the degree to which encoders track particular syntactic functions of verb argu- ments (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1 Simple Intransitive Sentences Data The main goal of the analysis of simple intransitive sentences is to check the relative con- tribution of their components to their embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We study a nearly-minimal sentence template with a nominal subject, an adverbial adjunct, and an in- transitive verb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We construct a set of 256 sentences of the form ‘[det] [subj] [adverb] [verb][punct]’, where det ranges over {a, the};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' subj ranges over mpnet distilroberta bert SameDet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='37 SameAdv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='45 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='58 SamePunct 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='84 SameSubj 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='27 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='48 Table 1: A summary of the models predicting z-scored pairwise cosine similarities between embeddings of sentences with intransitive verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' a set of nouns,6 adverb ranges over {quickly, slowly}, verb ranges over {appears, vanishes, stops, moves}, and punct, over {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Here and in subsequent experiments, the generation procedure assures that all sentence features are statistically independent, which is a crucial prerequisite for linear-regression modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Model The regression model matrix is based on 32,640 pairs of generated sentences, which differ in the value of at least one feature, with predictor vari- ables SameDeterminer, SameAdverb, SameVerb, SamePunct, and SameSubj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We regress z-score- transformed cosine similarities between sentence embeddings computed by three different encoders on these predictor variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The coefficients of the fitted models are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='7 Results Three observations from Table 1 hold for all subsequent analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (i) The coefficients are positive for all models and all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This means that sentence pairs which agree in some constituent are always more similar than sentence pairs that do not – as ex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (ii) The coefficient of determination (R2) is larger for ST-focused linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This means that the embeddings computed by the ST encoders are more dependent on the features of the sentences we track and less dependent on identities of lexical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (It can be noted that the fact that we achieve R2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='7 using only a few structural properties is remarkable in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=') (iii) The differences among coefficients of the ST-focused linear models are in general larger than 6{cat, dog, artist, teacher, planet, star, wind, rain} 7Replication models, fitted on sentences with the same structure but different lexical content, are shown in Table 8 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 10 0 10 0 10 mpnet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 bert subj cat dog artist teacher planet star wind rain 10 0 10 0 10 mpnet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 bert adv quickly slowly 10 0 10 0 10 mpnet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 bert pred appears vanishes stops moves 10 0 10 0 10 mpnet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 bert punct .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Figure 1: UMAP projections of embeddings of sen- tences with intransitive verbs (left: sentence trans- former, right: BERT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' those of the linear model analysing BERT: in the latter, the biggest coefficient (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='27 for SameSubj) is only ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 times higher than the smallest one (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='37 for SameDet), while for the ST models this ratio is above 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This is connected to the fact that BERT-derived sentence representations are more dependent on semantically impoverished elements, such as determiners and punctuation signs, which dampen the effect of other constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' For the sake of brevity, we do not analyse determiners and punctuation in subsequent experiments and keep them constant as the and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Turning to the comparison of coefficients inside models, we see that STs pay considerably more attention to subjects than to predicates: all things being equal, sentences with different predicates and adverbs but the same subject will be more similar than sentences with the same predicate and adverb and different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The influence of punctuation is surprisingly strong, being comparable to that of adverbs, while the effect of determiners is very weak, albeit statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A plot of UMAP projections of sentence em- beddings produced by MPNET and BERT, shown in Figure 1, underlines that while averaged BERT embeddings distinguish punctuation signs but do not distinguish subjects, the situation is reversed for the sentence transformer: it distinguishes sub- jects cleanly but largely abstracts away from other structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2 Transitive Sentences Data The transitive sentences used in the anal- ysis are generated using the following template: ‘The [subj] [adverb] [verb] the [obj].’ The range of nouns was slightly extended;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='8 the same adverbs as in the previous experiment were used, while verb ranged over {sees, chases, draws, meets, remem- bers, pokes}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This produces 672 different sentences and 225,456 sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Model The coding for SameAdv and SamePred remains as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The main focus in this study is on whether sentence similarities are dominated by the sentences having the same subject, the same direct object, or the same words in these two po- sitions even if their order were reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' To test for this, we added a categorical variable with the following values: 00 no overlap in subject and object (the baseline);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A0 same subject, different objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 0B same object, different subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 0A the subject of the first sentence is the object of the second;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' B0 the object of the first sentence is the subject of the second;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' BA subject and object are swapped;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' AB the same subject and object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Results A summary of the fitted models is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='9 It demonstrates that when it comes to simple transitive sentences, our understanding of their embeddings produced by sentence transform- ers remains high, despite the sentences being more complex (R2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='7), while BERT embeddings become more unpredictable (R2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Fur- thermore, while BERT again essentially treats all tokens more or less equally, with adverbs slightly discounted, STs prioritise participants (even B0 has higher coefficients than SamePred).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' On the other hand, neither BERT nor STs priori- tise the exact syntactic function of the participants: coefficients for A0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 0A, 0B vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' B0, and AB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 8To {cat, dog, teacher, artist, robot, machine, tree, bush, planet, star, wind, rain}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 9A summary of the replication model fits is provided in Table 9 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert SameAdv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='56 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='78 SubjObj_0A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='65 SubjObj_0B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='69 SubjObj_A0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='75 SubjObj_AB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='60 SubjObj_B0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='58 SubjObj_BA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='39 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='31 Table 2: A summary of the models predicting z-scored pairwise cosine similarities between embeddings of sentences with transitive verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are sig- nificant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' BA are largely comparable across all models with BA ≈ A0 + 0B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' That is, the effects of subjects and objects are largely independent of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A UMAP plot with the embeddings for the tran- sitive sentences is shown in Figure 2 in the Ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' It demonstrates that STs arrive at a much more fine-grained clustering of sentences, largely dominated by subjects and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' They largely discount predicates and adverbs which are quite prominent in averaged BERT embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='3 Transitive Sentences with Long NP Modifiers The previous analyses showed that representations computed by STs are highly attuned to verb par- ticipants but not to their particular syntactic roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This may mean that ST may be potentially misled by nouns in other positions in the sentence, which have less relevance to the described situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This study explores this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Data We repeat the analysis from § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2 using the template of the form ‘The [subj] [modifier] [adverb] [verb] the [obj]’, with a smaller set of sub- jects,10 and the modifier ranging over {with big shiny eyes, that my brother saw yesterday, whose photo was in the papers, worth a great deal of money}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Altogether this gives 1,440 sentences and 1,036,080 sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The modifiers have inter- nal syntactic structure and contain a non-negligible amount of lexical material that the models have to ‘skip over’ if their representations were focused on the participant structure of the matrix clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 10{cat, dog, rat, giraffe, wombat, hippo} mpnet distilroberta bert SameMod 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='62 SameAdv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='27 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='40 SubjObj_0A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='32 SubjObj_0B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='42 SubjObj_A0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='53 SubjObj_AB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='00 SubjObj_B0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='54 SubjObj_BA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='91 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='61 Table 3: A summary of the models predicting z-scored pairwise cosine similarities between embeddings of sentences with transitive verbs and lengthy subject modifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Model The same coding strategy as in the preced- ing section is used, augmented by a new binary vari- able, SameMod, tracking whether two sentences have the same modifier for the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Results Both the model coefficients, shown in Table 3, and the UMAP plot, shown in Figure 3 in the Appendix, indicate that BERT embeddings are highly sensitive to lengthy modifiers:11 the SameMod coefficient in the linear model is larger than the coefficients for the same predicate and the same subject-object combination added together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The situation is very different for STs: SameMod is more important than SamePred, especially for DistilRoberta, but, with one exception, not more important than even a partial overlap in participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Having the same participants, in either the same or swapped syntactic functions, is more than twice as important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We take this as evidence that STs have a specific bias towards matrix-clause participant sets, that is, the nouns that fill a thematic role of the main predicate, while their precise functions and nouns found in other positions in the sentence are less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='4 Coordinated Verbal Phrases The analyses presented above show that the main predicate of the sentence has only a limited influ- ence on the representations computed by STs, com- pared to its subjects and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Here, we show that this effect still holds if there is more than one 11The results of the replication fits are shown in Table 10 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert V1Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='21 V2Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='23 V3Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='41 N1Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='23 N2Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='30 N3Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='41 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='09 Table 4: A summary of the models predicting z-scored pairwise cosine similarities between embeddings of sentences with coordinated VPs from binary predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' main predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Data Using the same sets of nouns and transi- tive verbs as in the previous experiment, we con- struct sentences of the form ‘The man [verb1] the [noun1], [verb2] the [noun2], and [verb3] the [noun3]’, where triples of verbs and nouns are taken from the Cartesian product of the sets of all noun and verb combinations of size 3 without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' To alleviate a possible ordering bias, all verb and noun triples are shuffled for each sen- tence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This results in 400 sentences and 79,800 sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Models and results The analysis proceeds in three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' First, we check if positions 1, 2, and 3 have different importance by regressing the nor- malised cosine similarity on six binary variables N[oun]1Same, V[erb]1Same, N2Same, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The models, summarised in Table 4,12 show low coeffi- cients of determination (with R2 around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1), but they indicate that positions are of unequal impor- tance: BERT gives more weight to the last noun and the last verb, while STs focus on the first and the last N-V pair and largely ignore the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A significantly better fit can be achieved by re- placing binary predictors with overlap scores for nouns and verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' As Table 513 shows, this type of model, even though it contains only 2 variables instead of 6, obtains R2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='65 for STs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' It is also evident that all three models place more weight on noun overlap than on verb overlap, with Dis- tilRoberta showing the biggest difference between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 12A summary of the replication fits is given in Table 11 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 13See Table 12 in the Appendix for the replication fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert VerbOverlap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='64 NounOverlap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='88 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='52 Table 5: A summary of the models predicting z-scored pairwise cosine similarities between embeddings of sentences with coordinated VPs from overlap scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This raises the question of whether particular verb-noun collocations play a noticeable role, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', if a sentence containing chases the wombat will be considerably more similar to another sentence containing the exact phrase compared to a sentence containing chases and wombat but not as a trigram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Simply adding n-gram overlap scores to the model is not possible, however, because it is highly cor- related with both noun overlap and verb overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' In order to obviate this obstacle, we first construct an auxiliary linear model predicting trigram over- lap from noun and verb overlap and then use the residuals of this regression in the main model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The results are ambiguous: on one hand, the coefficient for residualised trigram overlap is sta- tistically significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' On the other hand, the effect is very weak (more than ten times weaker than that of either noun overlap or verb overlap), and the addition of trigram overlap to the model improves R2 by less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This seems to indicate that trigram overlap is not important for practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 Predicative Nominals with Gerund Subjects A potential weak point of our analysis is that parts of speech and syntactic functions are not decoupled: it is not yet clear whether the encoders pay attention to nouns or to subjects and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Data To address this issue, we construct another set of sentences where the subject is a gerund and the predicate is nominal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The template is ‘[gerund] [object] [copula] a [adjective] [predicate]’, where gerund ranges over {continuing, abandoning, starting, completing}, object ranges over {it, them, the project, the plan}, copula is one of {is, was, will be, is going to be}, adjectives are {big, real, negli- gible, insignificant}, and the predicative nominal ranges over {solution, mistake, failure, triumph}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This gives 1024 sentences and 523,776 sentence mpnet distilroberta bert SameSubj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='31 SameCop 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='55 SameAdj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='52 SameObjNoun 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='60 SameObjPron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='42 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='22 Table 6: A summary of the models predicting z-scored pairwise cosine similarities between embeddings of sentences with gerund subjects and nominal predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A variable copula provides an additional test as to whether the sentence encoders can recognise multi-word sequences with low semantic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Model The sentence pair encoding includes four binary variables (SameSubj, SameCop, SameAdj, SamePred) and a nominal variable for the direct ob- ject, indicating whether objects are different (base- line), are identical and pronominal (SamePron), or are identical and nominal (SameNoun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Results The results in Table 614 demonstrate that all models treat both nominal predicates and nom- inal direct objects as more important than gerund subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' STs, moreover, pay less attention to iden- tical pronominal objects and discount multi-word copula forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' R2 values for the ST model are lower than in the previous experiments (in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='55 range), which may potentially indicate a poor choice of lexical items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' however, replication ex- periments with a different set of words (except for copula forms) achieved comparable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This suggests that embeddings of sentences of this type are less easily explainable as additive combinations of individual words compared to the sentence types surveyed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='6 Revisiting Participant Sets: Ditransitive Sentences Our final experiment revisits the opposition be- tween lexical overlap in verbal phrases and exact argument-predicate matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' In this case, we fo- cus on ditransitive verbs with two arguments: a direct object and an oblique object which is an 14See an overview of replication fits in Table 13 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert SameAdv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='64 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='83 Overlap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='91 SPCRes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='10 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='57 R-squared (w/o SPCRes) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='56 Table 7: A summary of the models predicting z-scored pairwise cosine similarities between embeddings of sentences with ditransitive verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' SPCRes stands for SamePosCountRes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' the residuals of the number of identical words in identical positions regressed on lexical overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' integral part of the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='15 Data All permutations of the triple of basic nouns {cat, dog, rat} are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' For each per- mutation, all three nouns are, in turn, replaced with one of the members of the set of extra nouns {gi- raffe, wombat, hippo};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' the original permutations are also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This provides a set of unique triples of nouns where each pair of triples has from one to three nouns in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The Cartesian product of this set of triples with a set of ditransitive verbs ({describes, sells, shows}) and a set of adverbs ({happily, quickly, secretly) is used to fill the tem- plate ‘The [noun1] [adverb] [verb] the [noun2] to the [noun3].’ This procedures gives 540 sentences and 145,530 sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Model The sentence pairs are coded for same adverb, same predicate, the number of matching nouns in matching positions (SamePosCount), and lexical overlap minus 1 (the baseline value of 0 corresponds to overlap of 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' each successive value corresponds to increase in overlap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' As with over- lapping words and trigrams above, these predictors are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Therefore, we residualise Same- PosCount after regressing it on lexical overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Results Table 7 is inconclusive in a similar way to results from § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The coefficients for residu- alised SamePosCount are significant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' however, in the ST models, their size is very small, and Same- 15Many English ditransitive verbs can undergo the ‘dative alternation’, which swaps the oblique object with a preposi- tional phrase: Give the book to me/John vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Give me/John a book (Levin, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Of the verbs we use, show and sell partic- ipate in it, and the status of describe varies across speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' PosCount does not materially improve the predic- tive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We conclude, therefore, that syntactic positions do not matter a great deal, in line with our ‘participant set’ interpretation from § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 4 Discussion Our analysis arguably goes some way towards ex- plaining why sentence transformers beat vanilla BERT-based models with token averaging on sentence-modelling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Token averaging makes it impossible to distinguish between semantically rich and impoverished sentence elements, nor be- tween syntactically central vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' peripheral elements: punctuation signs and determiners contribute on the same level as the matrix-clause predicate and main participants, while lengthy modifiers, such as relative clauses, and multi-word copula forms dominate the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Sentence transformers, on the other hand, learn to discount elements that only serve a grammatical function or present background information and fo- cus instead on the semantic kernel of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The latter is in effect largely synonymous with the set of nominal elements in the main clause, first of all participants, but also predicative nominals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Importantly, despite their evident syntactic-analytic capabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', in our setting they can distinguish between participants of main and relatives clauses and between main and auxiliary verbs), STs seem to not pay much attention to the distinction between subjects and direct or indirect objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Instead they prioritise raw overlap in the set of nominal partic- ipants of the matrix clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This can be seen, by slightly abusing terminology of theoretical linguis- tics, as a focus on the aboutness/topic of sentences, what things they describe, and not on their predica- tion/comment, what they actually say about those things (Hu and Pan, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We believe that this focus is not inherent to the architecture of sentence transformers but reflects the nature of the datasets used for fine-tuning STs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The size of these datasets makes it impossible to convincingly reason about their contents, but their genres (QA pairs, Reddit threads, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=') makes it plausible to expect a high degree of topic-based overlap: questions and conversations tend to re- volve around entities (persons and things), with their actions and properties repeating less often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This naturally leads to a focus on nouns referring to prominent entities, which are known to appear preferentially as subjects or objects for reasons of coherence (Barzilay and Lapata, 2008), arguably a good match to the patterns we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 5 Related Work Analysis of transformer-based models for sentence- level tasks, such as NLI, question answering, or text classification, has largely followed the same approaches as found in the general BERTology (Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2020): probing, analysis of the ge- ometry of the embedding space, extraction of parts of input that are particularly important for model performance, and behavioural analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' In this vein, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (2021) and Peyrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (2021) analyse the attention patterns powering the performance of transformer models on different types of sentence classification, and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (2020) show that embed- dings of sentences computed by BERT-based mod- els, including siamese-fine-tuned sentence trans- formers, are anisotropic and can be improved via normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Chrysostomou and Aletras (2021) survey the existing methods for extracting ratio- nales from input sentences in the context of text classification and propose an improved approach, while Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (2021) demonstrate that sentence embeddings derived by averaging BERT token rep- resentations suffer from artefacts arising from po- sitional embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Zhelezniak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (2019) ar- gue that averaging should be replaced with max- pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Very similar to ours is the approach adopted by MacAvaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' (2022), who construct a series of probes to analyse the performance of several mod- els on the task of information retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' While their methodology relies on high-level document statis- tics and wholistic document manipulation (word and sentence shuffling, token-frequency similar- ity between the document and the query, textual fluency, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' ), our study analyses the role of lin- guistically motivated structural factors and thus complements their findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Opitz and Frank (2022) aim at directly decom- posing the representations produced by sentence transformers into several parts capturing different properties of sentences reflected in AMR annota- tions (presence of negation, concepts included in the sentence, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' While our study tries to as- certain what meaning components dominate the representations, Opitz and Frank assume that these components are known in advance and are equally important: sentence embeddings in their modified SBERT model are split into 15 segments, each of which corresponds to one AMR-based meaning component, plus a residual part to capture every- thing not covered by AMR annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 6 Conclusion This paper aims at making a contribution towards a better understanding of sentence transformers, which are often seen as black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' We have demonstrated that we can make surprisingly precise inferences about sentence-pair similarities using simple linguistic features such as lexical overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The crucial difference between bag-of-words dis- tributional models and current encoders is that STs have became quite adept at disregarding ‘irrelevant’ parts of the sentence and concentrating on its key elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Unlike vanilla BERT sentence embed- dings obtained by token averaging, STs yield more structured embeddings that focus on the matrix clause and are less tied to individual lexical items and strings of function words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This progress, however, comes with a particu- lar type of bias: the structures that lead to high sentence similarity in STs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' the overlap in nomi- nal ‘participant sets’, seem to mirror the dominant type of paraphrases found in the data the STs were tuned on, and STs are not compelled to look at finer structures of input sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' At least without further fine tuning, this would appear to make them unsuitable for downstream tasks that require knowl- edge about more fine-grained aspects of sentence structure, such as semantic roles (Conia and Nav- igli, 2022), or extra-propositional aspects, such as monotonicity, negation, or modality (Yanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Nakov, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' An interesting direction for future research would be to explore the ways of decomposing sen- tence representations into additive aspects such as participant structure, main predication, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The additional challenge here is that while theoretical semantics has a lot to say about aspects of sentence meaning (Pagin, 2016), there remains a lack of analysis linking the notion of one-dimensional se- mantic similarity (Agirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=', 2012) that underlies the optimisation of current sentence transformers with theoretically more substantial concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Limitations The limitations of the proposed analysis are the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The analysis is based on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This allows us to fully control the sentence struc- ture and use balanced lexical material, but it does not necessarily reflect the performance of models on real-world data, especially when sentences or text fragments are much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' However, synthetic data have generally shown to be a good first step toward understanding the behaviour of complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The analysis does not cover graded distinc- tions between words, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' we did not experi- ment with filling the slots with synonymous words, as opposed to completely unrelated words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This makes it impossible to decide if the models are sensitive to word identities or to their actual semantics, as long as these two notions are distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The outputs of the models are interpreted using linear regression analysis anchored to the properties of synthetic sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' This kind of analysis makes it possible to disentan- gle additive effects of different components of sentence structure and provides statistical- significance estimates, while high R2 values indicate that our findings have some valid- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' However, it cannot fully account for the lexical effects (which we tried to safeguard against by carefully selecting template fillers), non-linear effects, and hidden collinearity pat- terns (beyond those we addressed using resid- ualised analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' The range of models analysed in the paper is restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' It covers some amount of variabil- ity (sentence transformers vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' vanilla BERT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' two different variants of a base model for STs, one of them distilled), but other combinations of model architecture and training/fine-tuning regime 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Linguistics: ACL- IJCNLP 2021, pages 103–119, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Vitalii Zhelezniak, Aleksandar Savkov, April Shen, Francesco Moramarco, Jack Flann, and Nils Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Ham- merla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Don’t settle for average, go for the max: Fuzzy sets and max-pooled word vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' CoRR, abs/1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='13264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert SameDet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='26 SameAdv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='96 SamePred 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='49 SamePunct 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='64 SameSubj 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='65 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='43 Table 8: A summary of the replication models predict- ing z-scored pairwise cosine similarities between em- beddings of sentences with intransitive verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coef- ficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1 Dimensionality-reduction plots A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1 Simple transitive sentences A UMAP plot of embeddings of simple transitive sentences encoded accordings to their properties is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2 Transitive sentences with long NP modifiers A UMAP plot of embeddings of transitive sen- tences with lengthy subject modifiers encoded ac- cordings to their properties is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2 Replication-model fits A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='1 Simple intransitive sentences The following lexical items were used for the repli- cation experiment: Nouns: wolf, bear, fruit, vegetable, building, car, lightning, wave Verbs: stabilizes, bursts, grows, shrinks Adverbs: suddenly, predictably A summary of the replication models is shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2 Simple transitive sentences The following lexical items were used for the repli- cation experiment: Nouns: pig, horse, soldier, farmer, android, computer, grass, forest, comet, galaxy, cloud, lightning Verbs: hears, pursues, imagines, recognizes, touches, finds mpnet distilroberta bert SameAdv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='95 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='75 SubjObj_0A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='70 SubjObj_0B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='66 SubjObj_A0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='76 SubjObj_AB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='56 SubjObj_B0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50 SubjObj_BA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='34 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='45 Table 9: A summary of the replication models predict- ing z-scored pairwise cosine similarities between em- beddings of sentences with intransitive verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coef- ficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' Adverbs: suddenly, predictably A summary of the replication models is shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='3 Transitive sentences with long NP modifiers The following lexical and phrasal items were used for the replication experiment: Nouns: horse, pig, donkey, elephant, bison, moose NP modifiers: missing a hind leg, whose face we all know, born under a bad sign, pictured on page seventeen Verbs: hears, pursues, imagines, recognizes, touches, finds Adverbs: suddenly, predictably The overview of the model fits is shown in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='4 Coordinated verbal phrases The following lexical items were used for the repli- cation experiment: Nouns: mouse, horse, fox, kangaroo, bison, elephant Verbs: hears, pursues, imagines, recognizes, touches, finds A summary of the replication models is shown in Tables 11 (individual-word-based models) and 12 (overlap-based models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert SameMod 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='83 SameAdv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='41 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='64 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='17 SubjObj_A0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='24 SubjObj_AB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='42 SubjObj_B0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='20 SubjObj_BA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='39 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='71 Table 10: A summary of the replication models predict- ing z-scored pairwise cosine similarities between em- beddings of sentences with transitive verbs and lengthy subject modifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert V1Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='35 V2Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='28 V3Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='42 N1Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='14 N2Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='18 N3Same 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='17 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='07 Table 11: A summary of the replication models pre- dicting z-scored pairwise cosine similarities between embeddings of sentences with coordinated VPs from binary predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='5 Predicative nominals with gerund subjects The following lexical items were used for the repli- cation experiment: Gerund subjects: proposing, rejecting, prais- ing, criticizing Pronomial and nominal objects: him, me, the idea, the design Copula forms (same as in the original experi- ment): is, was, will be, is going to be Nominal predicates: decision, defeat, loss, im- provement A summary of the replication models is shown in Tables 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert VerbOverlap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='85 NounOverlap 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='47 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='41 Table 12: A summary of the replication models pre- dicting z-scored pairwise cosine similarities between embeddings of sentences with coordinated VPs from overlap scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert SameSubj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='31 SameCop 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='55 SameAdj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='52 SameObjNoun 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='60 SameObjPron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='42 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='22 Table 13: A summary of the replication models predict- ing z-scored pairwise cosine similarities between em- beddings of sentences with gerund subjects and nom- inal predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='6 Participant-set overlap vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' identical participants The following lexical items were used for the repli- cation experiment: Basic nouns: horse, pig, donkey Extra nouns: elephant, bison, moose Verbs: gives, demonstrates, entrusts Adverbs: suddenly, predictably, openly A summary of the replication models is shown in Tables 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' mpnet distilroberta bert SameAdv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='64 SamePred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='83 Overlap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='91 SPCRes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='10 R-squared 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='57 R-squared (w/o SPCRes) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='56 Table 14: A summary of the replication models predict- ing z-scored pairwise cosine similarities between em- beddings of sentences with ditransitive verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' SPCRes stands for SamePosCountRes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' the residuals of the number of identical words in identical positions re- gressed on lexical overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' All coefficients are signif- icant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQfPzQl/content/2301.13039v1.pdf'} +page_content=' ' metadata={'source': 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